1 Lecture L6: Pause and Refresh

By reading the following summaries of the first five lectures, the rest of the book may become intelligible without studying the details of those lectures. In the first lecture we have introduced the basic structures of algebra such as groups, rings, fields, vector spaces, ideals, modules, polynomials, rational functions, euclidean domains, principal ideal domains, and unique factorization domains. In the second lecture, after introducing power series, meromorphic series, and valuations, we show the equivalence of well-ordering and Zorn’s Lemma and use them to establish the existence of vector space basis, transcendence basis, algebraic closure, and maximal ideals. The third lecture deals with the power series theorems of Newton, Hensel, and Weierstrass. The fourth and fifth lectures deal with ideals, modules, varieties, and models which are the avatars of varieties full of local rings.

§1: SUMMARY OF LECTURE L1 ON QUADRATIC EQUATIONS

For sets (= collections of objects) S and T , a map φ : S → T is an assignment which to every x ∈ S, i.e., to every element (= object) x of S, assigns φ(x) ∈ T ; this may be written x 7→ φ(x); the element φ(x) is called the image of x under φ; we put dom(φ) = S and ran(φ) = T and call these the domain and range of φ respectively. The composition of maps φ : S → T and ψ : T → U is the map ψφ : S → U given by (ψφ)(x) = ψ(φ(x)). The map φ is injective, or is an injection, if φ(x) = φ(y) ⇒ x = y.A subset of S is a set R whose objects are amongst the objects of S; we write R ⊂ S; we may also write S ⊃ R and call S an overset of R. We put φ(R) = {φ(x): x ∈ R} = the set of all φ(x) with x varying over R, and call this the image of R under φ. We put im(φ) = φ(S) and call this the image of φ; the map φ is surjective, or is a surjection, if φ(S) = T . The map φ is bijective, or is a bijection, if it is injective as well as surjective. For a bijection φ : S → T we have the inverse map φ−1 : T → S given by φ−1(y) = x ⇔ φ(x) = y. Without assuming the map φ : S → T to be bijective, for any y ∈ T we put φ−1(y) = {x ∈ S : φ(x) = y}, and for any U ⊂ T we put φ−1(U) = {x ∈ S : φ(x) ∈ U}; moreover, for any A ⊂ S and B ⊂ T with φ(A) ⊂ B, by φ|(A,B) we denote the map A → B which sends every x ∈ A to φ(x) in B, and we call this the restriction of φ to (A, B); if B = φ(A) then we may denote this by φ|A and call it the restriction of φ to A. The empty set is denoted by ∅. For subsets R1 and R2 of a set S, the com- plement of R2 in R1 is denoted by R1 \ R2, i.e., R1 \ R2 = {x ∈ R1 : x 6∈ R2}; needless to say that x 6∈ R2 means x is not an element of R2, just as x 6= y means x is not equal to y, and so on. For subsets R1 and R2 of a set S, their intersection is denoted by R1 ∩ R2 and their union is denoted by R1 ∪ R2, i.e., R1 ∩R2 = {x ∈ S : x ∈ R1 and x ∈ R2} and R1 ∪R2 = {x ∈ S : x ∈ R1 or x ∈ R2}. Similarly for more than two subsets R1,...,Rm of a set S we have ∩1≤i≤mRi = {x ∈ R : x ∈ Ri for all i} and ∪1≤i≤mRi = {x ∈ S : x ∈ Ri for some i}. More generally we could have a family (Ri)i∈I of subsets Ri of a set S indexed by an indexing set I, i.e., i 7→ Ri gives a map of I into the set of all subsets of S, and then we put ∩i∈I Ri = {x ∈ S : x ∈ Ri for all i ∈ I} and ∪i∈I Ri = {x ∈ S : x ∈ Ri for some i ∈ I}.A partition of a set S is a collection of nonempty subsets of S such that S is their union and any two of them have an empty intersection.

2 §1: SUMMARY OF LECTURE L1 ON QUADRATIC EQUATIONS 3

The set whose elements are x1, . . . , xe (which may or may not be distinct) is denoted by {x1, . . . , xe}. By N+ ⊂ N ⊂ Z ⊂ Q ⊂ R ⊂ C we denote the sets of all positive integers, nonnegative integers, integers, rational numbers, real numbers, and complex numbers respectively. The size of any set S, i.e., the number of elements in it, is denoted by |S|; note that then |S| ∈ N or |S| = ∞ according as S is finite or infinite; later on we shall give a more precise meaning to different types of infinities, and then |S| will denote the “cardinal number” of S. Clearly Q |∅| = 0. For Sn defined below we have |Sn| = n! = 1≤i≤n i, and hence |S0| = 1; convention: an empty product is 1 and an empty sum is 0. A group G is a set with a binary operation which to every pair of elements x, y in G associates a product xy ∈ G such that: (i) (xy)z = x(yz) for all x, y, z in G (associativity); (ii) there is 1 ∈ G with 1x = x1 = x for all x ∈ G (existence of identity); (iii) for every x ∈ G there is x−1 ∈ G with xx−1 = x−1x = 1 (existence of inverse). A subgroup of a group G is a subset H which is a group under the same operation as G; we then write H ≤ G. If H ≤ G with H 6= G then we write H < G and call H a proper subgroup of G. A normal subgroup of a group G is a subgroup H such that for all x ∈ G we have xHx−1 = H where xHx−1 = {xyx−1 : y ∈ H}; we then write H/G. If H/G with H 6= G, then H is called a proper normal subgroup of G. A group G is simple if G 6= 1 and G has no proper normal subgroup 6= 1, where 1 denotes the identity group having only one element. The size |G| of a group is its order; the order of x ∈ G is the smallest r ∈ N+ with r x = 1; if there is no such r ∈ N+ then the order of x is ∞. A group G is cyclic if it is generated by a single element x, i.e., if every element of G is a power of x; we then denote G by Zr where r is the order of x which may be ∞; clearly Zr is simple ⇔ r is a prime number. A homomorphism of a group G into a group J is a map φ : G → J such that φ(1) = 1 and φ(xy) = φ(x)φ(y) for all x, y in G; the kernel of φ is defined by ker(φ) = φ−1(1), and we have im(φ) ≤ J and ker(φ) /G; note that φ is injective iff (= if and only if) ker(φ) = 1 and when that is so we call φ a monomorphism; if φ is surjective, i.e., if im(φ) = J, then we call φ an epimorphism; if φ is bijective then we call it an isomorphism; if J = G and φ is an isomorphism then we call φ an automorphism of G. For H ≤ G we put xH = {xy : y ∈ H} and call this a left coset of H in G (similar definition of a right coset), and by G/H we denote the set of all left cosets of H in G, and note that this is a partition of G; also we put [G : H] = |G/H| and call this the index of H in G. If H/G then G/H becomes a group by defining (xH)(yH) = (xy)H, and we call G/H the factor group of G by H; now x 7→ xH gives an epimorphism G → G/H with kernel H; we call this the canonical epimorphism of G onto G/H. A finite group G is solvable if there is a chain 1 = G0 /G1 / ··· /Gr = G such that Gi/Gi−1 is cyclic of prime order for 1 ≤ i ≤ r. The set of all bijections of a set S onto itself forms a group under composition which we call the symmetric group on S and denote it by Sym(S). If |S| = n ∈ N then Sym(S) = Sn. Any permutation σ on n letters, i.e., σ ∈ Sn, can be written as a product of a certain number ν of transpositions, and the parity of ν, i.e., its evenness or oddness, depends only on σ; we call the permutation σ even or odd 4 LECTURE L6: PAUSE AND REFRESH according as ν is even or odd, and we define the signature sgn(σ) of σ to be 1 or −1 according as σ is even or odd; note that a transposition is an element τ in Sn = Sym(S) such that for some i 6= j in S we have τ(i) = j, τ(j) = i, and τ(l) = l for all l ∈ S \{i, j}. The set all even σ in Sn is a normal subgroup of Sn which we call the alternating group and denote it by An. We have [Sn : An] = 1 or 2 according as n ≤ 1 or n > 1. For n ≥ 5, An is simple and Sn is unsolvable. See (X1), (X2), and (E1) to (E6) of L1§11. A group G is commutative or abelian if xy = yx for all x, y in G. An additive abelian group is a commutative group in which the operation is written as a sum x + y instead of a product, 0 is written for the identity, and −x is written for the inverse of x. To contrast with this, a usual group, with the operation written as a product or multiplication, may be called multiplicative. Deleting (iii) from the the definition of a group we get the definition of a monoid, and by deleting (ii) and (iii) we get the definition of a semigroup. The terms com- mutative or abelian semigroup, additive abelian semigroup, multiplicative semi- group, subsemigroup, and oversemigroup are obvious generalizations of the corre- sponding terms for groups; similarly for a monoid; note that the identity element of a submonoid is required to coincide with the identity group of the monoid. A semigroup H is cancellative means for any x, y, z in it we have the implication: xy = xz and yx = zx ⇒ y = z. If H is a subsemigroup of a group G then clearly H is cancellative. As examples, Z is an additive abelian group, N is a submonoid of Z, and N+ is a subsemigroup of Z. A ring R is an additive abelian group which is also a commutative multiplicative monoid such that the two operations are connected by the distributive laws saying that for all x, y, z in R we have x(y + z) = xy + xz and (y + z)x = yx + zx; we call R a null ring if in it we have 1 = 0, i.e., equivalently if |R| = 1. A domain is a nonnull ring whose nonzero elements form a cancellative multiplicative monoid. A field is a nonnull ring whose nonzero elements form a multiplicative group. If R ⊂ S are rings under the same operations (and have the same zero and identity elements) then R is a subring of S or S is an overring of R. Similarly for subfields and overfields, and subdomains and overdomains. As an additive abelian group, every subgroup I of a ring R is a normal subgroup and so we can form the factor group R/I; a typical member of R/I is a residue class (= additive coset) a + I with a ∈ R. By an ideal in a ring R we mean an additive subgroup I of R such that for all a ∈ R and x ∈ I we have ax ∈ I, and when that is so we make R/I into a ring by defining (a + I)(b + I) = (ab) + I for all a, b in R. We call R/I the residue class ring of R modulo I, and we call I a maximal ideal or a prime ideal in R according as R/I is a field or a domain; for more standard definitions of prime ideal and maximal ideal see (D1). Likewise, we call I a nonzero ideal or a nonunit ideal according as I 6= {0} or I 6= R; note that every ideal contains the zero ideal I = {0} and is contained in the unit ideal I = R; moreover, the zero ideal is prime ⇔ R is a domain; likewise, I is the unit ideal ⇔ R/I is the null ring. For any a ∈ R we put aR = the set of all multiples az of a with z ∈ R, and for any W ⊂ R we put WR = the set of all finite linear combinations a1z1 + ··· + aeze with a1, . . . , ae in W and z1, . . . , ze in R, and we note that these are ideals in R (by convention an empty sum is zero and hence 0 ∈ WR); they are called ideals generated by a and W respectively; an ideal of the form aR is called a principal ideal in R and a called a generator of it; in case §1: SUMMARY OF LECTURE L1 ON QUADRATIC EQUATIONS 5

W = {w1, w2,... }, we may denote WR by (w1, w2,... )R and call w1, w2,... its generators. The zero ideal, and more generally the (additive abelian) zero group may be denoted by 0 instead of {0}. It is easy to see that every ideal in Z is of the form pZ with p ∈ N; moreover pZ is a maximal ideal in Z iff p is a prime number, and then Z/pZ is the Galois field GF(p). The set of all automorphisms of a group G is clearly a subgroup of Sym(G) and we denote it by Aut(G). For rings S and T , a (ring) homomorphism φ : S → T is a homomorphism of additive abelian groups such that φ(1) = 1 and φ(xy) = φ(x)φ(y) for all x, y in S; now ker(φ) is an ideal in S, and the definitions of monomorphism, epimorphism, isomorphism, and automorphism carry over from the group case. Conversely, if φ : S → S/I is the canonical ring epimorphism where I is an ideal in a ring S then ker(φ) = I. The set of all ring automorphisms of a ring S is clearly a subgroup of Sym(S) and we denote it by Aut(S); momentarily letting this Aut be written as Ring-Aut(S) and writing Group-Aut(S) for the automorphism group of the additive abelian group S we clearly have Ring-Aut(S) ≤ Group-Aut(S) ≤ Sym(S); it is usually clear from the context which automorphism group is being considered. Finally for a subring R of a ring S, by an R-automorphism of S we mean an automorphism φ of S which leaves R elementwise fixed, i.e., φ(x) = x for all x ∈ R; these form a subgroup of Aut(S) and we denote it by AutR(S). Dropping the commutativity of multiplication in a field (resp: ring, domain) we get the notion of a skew-field (resp: skew-ring, skew-domain). In the definitions of fields and rings we have spelled out two distributive laws, although they follow from each other, exactly because in case of skew-fields and skew-rings they do not. Moreover, in the definition of a field, by requiring the left-distributive law x(y+z) = xy+xz, but not requiring the right-distributive law (y+z)x = yx+zx, we get the notion of a near-field. The concepts of subskew-field, overskew-field, and so on, are defined in an obvious manner. Also the above definitions of homomorphism, monomorphism, and so on, have obvious generalizations to semigroups, monoids, and so on. See section L1§5 on Modules and Vector Spaces for the definitions of: a module V over a ring R, scalar multiplication in V , linearly independent and lin- early dependent elements of V, submodule of V , submodule generated by a set of elements or a subset, generators of a submodule, finitely generated submodule, basis of a submodule, R-homomorphism (or R-linear map) V → V ∗ where V ∗ is another R-module (= module over R), R-monomorphism, R-isomorphism, left and right modules over a skew-ring, vector space over a field K (= module over K), subspace (= submodule of a vector space), vectors (= elements of a vector space), the dimension of a vector space V over a field K denoted by dimK V , and so on. An overring of a ring R is an R-module in an obvious sense; thus R is an R- module, and an ideal in R is exactly an R-submodule of R. In particular, an overring L of a field K is a K-vector space, and we put [L : K] = dimK L. This applies to an overfield L of K, and then we call [L : K] the field degree of L/K, i.e., of L over K. For a ring or skew-ring R, by R+ we denote the underlying additive abelian group of R, and by R× we denote the set of all nonzero elements of R; note that K× is a multiplicative group in case of a field or skew-field K. Extending this notation to a vector space V we denote the set of all nonzero elements in it by V ×. For an overfield L of a field K, the symbol [L : K] usually denotes the field degree rather than the index of K+ in L+. 6 LECTURE L6: PAUSE AND REFRESH

See section L1§6 on Polynomials and Rational Functions for the defini- tions of: indeterminate (= variable) over a ring, univariate polynomial ring R[X] over a ring R, degree of a polynomial, monic polynomial, nonconstant polynomial, Cauchy multiplication rule, univariate rational function field K(X) over a field K, derivative, quotient rule, multivariate polynomial ring R[X1,...,Xm], degree and partial derivatives of a multivariate polynomial, multivariate rational function field K(X1,...,Xm), substitution epimorphism R[X1,...,Xm] → R[x1, . . . , xm] = the ring generated over R by elements x1, . . . , xm in an overring of R, algebraic or transcendental element over R, algebraic set of elements over R, alge- braically dependent or algebraically independent sets of elements over R, field K(x1, . . . , xm) generated over a field K by elements x1, . . . , xm in an overfield of K, transcendence basis of an overfield L of K, and the number of elements in it called the transcendence degree of L/K (= L over K) denoted by trdegK L. See section L1§7 on Euclidean Domains and Principal Ideal Domains for the definition of: the group U(R) of all units in a ring R (or two-sided units in a skew-ring R), associates and irreducible elements in a ring, UFD = unique factorization domain, PID = principal ideal domain, euclidean function, ED = euclidean domain, special ED, quasispecial subset, and quasispecial ED. See section L1§8 on Root Fields and Splitting Fields for the definitions of: R-homomorphism (resp: R-monomorphism, R-epimorphism, R-isomorphism) between overrings of a ring R, root field of a univariate irreducible polynomial over a field K, splitting field of a nonconstant univariate polynomial f over K denoted by SF(f, K), separable polynomial f, the Galois group Gal(L∗,K) of the splitting field L∗ = SF(f, K) of a nonconstant univariate separable polynomial f over K, the natural isomorphism Gal(L∗,K) → Gal(f, K) = the Galois group of f over K which is the group of all relation preserving permutations of the roots of f, and ∗ for any H ≤ Gal(L ,K) the fixed field fixL∗ (H) of H which is a field L between K and L∗. In (T1) we state the theorem which includes the assertion that H 7→ L = fixL∗ (H) gives an inclusion reversing bijection of the set of all subgroups H of Gal(L∗,K) onto the set of all fields L between K and L and the inverse bijection is given by L → Gal(L∗,L). After stating other relevant theorems in (T2) to (T5), in (T6) we deduce Galois’ Unsolvability Theorem which says that if a field k contains n! distinct n!-th roots of 1 then, for n ≥ 5, the generic n-th degree polynomial over K = k(X1,...,Xn) cannot be solved by radicals. In (D1) we define divisibility x|y and prime elements in a ring. In (R2) to (R4) we define: gcd and GCD (= greatest common divisor), and coprime and relatively prime elements. In (D2) to (D4) we define: characteristic of a ring R denoted by ch(R), minimal polynomial of an algebraic element over a field, lcm and LCM (= least common multiple), and reduced form. In (R7) we define the (relative) algebraic closure of a field in an overfield. In (X2) we define the decomposition of a permutation into disjoint cycles. In (E1) we define the modified discriminant. In (N1) we define derivations and in (E11) and (E12) we describe their properties. Proper Containment. By a proper subset A of a set B we mean A ⊂ B with A 6= B. We indicate this by writing A $ B or B % A, and by saying that A is properly contained in B or B properly contains A. This agrees with the above definition of proper subgroup, and with the definitions in L4§8 and L5§5(Q16). §2: SUMMARY OF LECTURE L2 ON CURVES AND SURFACES 7

§2: SUMMARY OF LECTURE L2 ON CURVES AND SURFACES

In L2§1 we introduce the concepts of cartesian product, partial order, linear or- der, well order, cardinals, and ordinals. The cartesian product S1×· · ·×Sm of sets S1,...,Sm is the set of all m-tuples (γ1, . . . , γm) with γi ∈ Si. If S1 = ··· = Sm = S m then we write S for S1×· · ·×Sm. Also we define concepts of algebraically closed field, an (absolute) algebraic closure of a field, and the (relative) algebraic closure of a field in an overfield. To prove the existence of algebraic closures, we introduce the Axiom of Choice, Zorn’s Lemma, and Well Ordering. These are based on the concepts of posets (= partially ordered sets), losets (= ordered sets = linearly ordered sets), and wosets (= well ordered sets). Cardinals are equivalence classes of sets under bijections, and ordinals are equivalence classes of well ordered sets under order preserving bijections. The power set Pb(T ) is the set of all subsets of a set T , and the restricted power set Pb×(T ) = Pb(T ) \ {∅}. In L2§2 on Power Series and Meromorphic Series we introduce the notation QF(E) for the quotient field of a domain E. The quotient field of the power series ring K[[Z1,...,Zd]] over a field K is the meromorphic series field K((Z1,...,Zd)). We define the orders and derivatives of meromorphic series. We use the geometric series identity (1 − X)(1 + X + X2 + ... ) = 1 to show that the group U(S) of all units in the ring of power series S = R[[Z1,...,Zd]] over a ring R are exactly those power series Q(Z1,...,Zd) whose constant term Q(0,..., 0) belongs to U(R). In L2§3 on Valuations we generalize the concept of the order of a meromorphic series by defining a valuation of a field K to be a map v : K → G ∪ {∞} where G is an ordered abelian group G such that for all x, y in K we have: (1) v(xy) = v(x) + v(y); (2) v(x + y) ≥ min(v(v(x), v(y)); (3) v(x) = ∞ ⇔ x = 0. We call G the assigned value group of v, and by the value group of v we mean × the subgroup of G given by Gv = {v(x): x ∈ K }. By the valuation ring of v we mean the ring Rv = {x ∈ K : v(x) ≥ 0} and we note that this is clearly a subdomain (= subring which is a domain) of K which is its quotient field. We say that v is trivial over a subfield k of K, or that v is a valuation of K/k, to mean that x ∈ k× ⇒ v(x) = 0. For instance ord gives a valuation of the meromorphic series field K((Z1,...,Zd)) over any field K which is trivial over K and whose value group is Z. Here by an ordered abelian group we mean an additive abelian group G which is also an ordered set such that for all x, y, x0, y0 in G we have: x ≤ y and x0 ≤ y0 ⇒ x+x0 ≤ y+y0. For instance G could be the set R[d] of lexicographically ordered d-tuples of real numbers r = (r1, . . . , rd), s = (s1, . . . , sd),... , where lexicographic order means: r ≤ s ⇔ either ri = si for 1 ≤ i ≤ d, or for some j with 1 ≤ j ≤ d we [d] have ri = si for 1 ≤ i < j and rj < sj. Or G could be a subgroup of R such as G = Z or Q or R. The field K((X)) is generalized by taking any field K and any ordered abelian G group G and considering the field K((X))G = {A ∈ K : Supp(A) is well ordered} where KG is the set of all maps G → K and where the support of A ∈ KG, denoted by Supp(A), is defined by putting Supp(A) = {g ∈ G : A(g) 6= 0}. For any A ∈ K((X))G we define ord(A) = min Supp(A) if A 6= 0 and ord(A) = ∞ if A = 0, and we note that now ord is a valuation of K((X))G/K with value group G. 8 LECTURE L6: PAUSE AND REFRESH

In (T1) we state Newton’s 1660 Theorem on the algebraic closure of k((X)) when k is an algebraically closed field with ch(k) = 0, and in (T2) we formulate its generalization for ch(k) = p > 0. In L2§5 on Zorn’s Lemma and Well Ordering we prove that these two are equivalent and they are equivalent to the Axiom of Choice. Then by using Zorn’s Lemma we prove the: (R5) existence of maximal ideals, (R6) existence of algebraic closure, (R7) uniqueness of algebraic closure, existence of vector space bases as well as transcendence bases, (R9) linear order on cardinals, (R10) well order on ordinals. Let us state the contents of Remarks (R5) to (R8) in greater detail thus. (R5) Any nonunit ideal in a ring R is contained in some maximal ideal in R. (R6) Any field K has an algebraic closure, i.e., an algebraically closed field K which is algebraic over K. (R7) The K of (R6) is unique in the sense that given any field isomorphism φ : K → L and any algebraic closure L of L there exists an isomorphism φ : K → L such that φ(x) = φ(x) for all x ∈ K. (R8) Let K be a field and let L be either a vector space over K or an overfield of K. Let W be a subset of L. In the vector space case (resp: overfield case): let us call W independent if every finite subset of W is linearly (resp: algebraically) independent over K, and let us call W generating if L coincides with KW (resp: if L is algebraic over K(W )); in both cases call W a basis if it is independent and generating; given any other subset U of L, let us say that U is dependent on W if every u ∈ U belongs to KW (resp: is algebraic over K(W )). Then we prove the facts (W7) and (W8) stated below. ( W is a basis ⇔ it is a minimal generating set (W7) ⇔ it is a maximal independent set where the minimality means that W is a generating set but there is no generating set W 0 with W 0 ⊂ W and W 0 6= W , and the maximality means that W is an independent set but there is no independent set W 0 with W ⊂ W 0 and W 6= W 0. (W8) W is generating ⇒ there exists a basis W 0 with W 0 ⊂ W . In L2§7 on Definitions and Exercises we define: real and complex numbers, ordered fields, torsion subgroups and divisible groups, rational and real completions, Cauchy sequences and Dedekind cuts, rational and real ranks. In (D7) we define approximate roots of polynomials and in (E15) to (E17) list their basic properties. More About Fields. In L3§12(E6) we show that if G(Z1,...,ZN ) is any given polynomial over an infinite field k such that G(a1, . . . , aN ) = 0 for all a1, . . . , aN in k then G(Z1,...,ZN ) = 0. In L4§10(E1) we elucidate the above Remark (R6). In L5§5(Q27)(C41) we give a criterion for the minimal polynomial of an algebraic element to be separable. In L5§5(Q29)(T109) we discuss the linear disjointness of fields defined in the preamble of (Q29); we discuss this further in L5§6(E22). In L5§5(Q29)(C48) we give a dimension formula for subspaces of a vector spaces. In L5§5(Q32) we discuss separable and inseparable polynomials, separable and in- separable elements, purely inseparable elements, separable and inseparable field extensions, purely inseparable field extensions, finite fields, perfect fields, and the existence of primitive elements for finite separable algebraic field extensions. For any power q of any prime number p, in L5§5(Q32) we show that up to isomorphism there is a unique field of q elements; this is called the Galois field GF(q). §3: SUMMARY OF LECTURE L3 ON TANGENTS AND POLARS 9

§3: SUMMARY OF LECTURE L3 ON TANGENTS AND POLARS

In L3§1 we introduce the concepts of rectangular matrices with entries in a ring, and groups of square matrices whose determinants are units in the ring of entries. In L3§2 we study the geometry of quadrics and their pole-polar properties. In L3§3 to L3§5 this is expanded into the geometry of hypersurfaces, homogeneous coordinates, projective spaces, tangents, and singularities. In L3§6 we prove Basic Hensel’s Lemma which says that if a monic polynomial in Y whose coefficients are power series in X factors into coprime factors after putting X = 0 then it factors before putting X = 0. Using this and using the completing the square method of solving quadratic equations conceived by the fifth century Indian mathematician Shreedharacharya, in L3§7 we establish Newton’s Theorem on fractional power series expansion which was stated in L2§3(T1) as a theorem on the algebraic closure of k((X)). Newton’s Theorem motivates the idea of integral dependence which we take up in L3§7 toL3 §9. An element t in an overring S of a ring R is said to be integral over R if t satisfies a monic polynomial equation over R; a univariate polynomial over R is monic means its leading coefficient, i.e., the coefficient of the highest degree term in it, is 1. The set of all elements in S which are integral over R is a subring R0 of S and it is called the integral closure of R in S; if R0 = R then we say that R is integrally closed in S; if R is integrally closed in its total quotient ring QR(R), as defined in L4§7, then R is said to be normal. In particular a domain R is normal means it is integrally closed in its quotient field K. A domain R is overnormal means for every pair of monic polynomials in an indeterminate Y over K we have: A(Y )B(Y ) ∈ R[Y ] ⇒ A(Y ) ∈ R[Y ] and B(Y ) ∈ R[Y ]. A domain R is intval means it is the intersection of a nonempty family of valuation rings of K, i.e., valuation rings of valuations of K. It turns out that for any domain we have: UFD ⇒ intval ⇔ overnormal ⇔ normal. The GCD of all the coefficients of a nonzero polynomial over a UFD is its content. Gauss Lemma says that the content of a product is the product of their contents. This yields the fact that a finite variable polynomial ring over a UFD is a UFD. The proofs of some of the above assertions may be found in L4§10(E2), L4§10(E3), and L4§12(R7). In L3§11 we deduce the multivariable version of Hensel’s Lemma as a consequence of the Abstract WPT = Weierstrass Preparation Theorem and its companion the Abstract WDT = Weierstrass Division Theorem. These deal with the univariate power series ring S = R[[Y ]] over a complete Hausdorff quasilocal ring R. By a quasilocal ring we mean a ring R with a unique maximal ideal M(R); note that a ring R is quasilocal iff nonunits in it form an ideal (which then equals M(R)); also note that the valuation ring Rv of any valuation v of a field K is a quasilocal ring with M(Rv) = {x ∈ K : v(x) > 0}. The Weierstrass degree of any f ∈ S is defined by wideg(f) = ord(F ) where F ∈ (R/M(R))[[Y ]] is obtained by applying the residue class map R → R/M(R) to the coefficients of f. Now WPT says that every f ∈ S with wideg(f) = d ∈ N can uniquely be written as f = δf ∗ where δ is a unit in S and f ∗ is a distinguished polynomial of degree d. i.e., ∗ d ∗ d−1 ∗ ∗ ∗ f = Y + f1 Y + ··· + fd with f1 , . . . , fd in M(R), and WDT says that any g ∈ S can be uniquely written as g = qf + r where q ∈ S and r ∈ R[Y ] with deg(r) < d. 10 LECTURE L6: PAUSE AND REFRESH

The above abstract versions of WPT and WDT are converted into their concrete versions by taking R to be the m-variable power series ring k[[X1,...,Xm]] over a field k. To apply these to the n-variable power series ring S = k[[X1,...,Xn]] over a field k we need to ensure that wideg of a nonzero element f of S relative to Xn is finite so that we can take Y = Xn and m = n − 1. In case the field k is infinite, we do this by rotating coordinates, i.e., by making a k-linear automorphism as explained in L3§12(D4). Alternatively, without assuming k to be infinite, again as explained in L3§12(D4), we achieve the same thing by making a more general polynomial automorphism. About the polynomial and power series rings R[X1,...,Xm] and R[[X1,...,Xm]] in indeterminates X1,...,Xm over a ring R with positive integer m let us note that, according the famous Hilbert Basis Theorem as proved in (T3) and (T4) of L4§3, if the ring R is noetherian then the rings R[X1,...,Xm] and R[[X1,...,Xm]] are also noetherian, where we note that in honor of Emmy Noether a ring R in which every ideal is finitely generated is said to be noetherian. It follows that the power series ring k[[X1,...,Xm]] over a field k is a local ring, by which we mean a noetherian quasilocal ring. In L5§5(Q4) we prove the famous Krull Intersection Theorem which n says that every local ring R is Hausdorff, i.e., in it we have ∩n∈NM(R) = 0. Thus in a local ring R, for every x ∈ R× = the set of all nonzero elements of R we have ordRx ∈ N, where for any x in any quasilocal ring R we have put ordRx = n n max{n ∈ N : x ∈ M(R) } with the understanding that if x ∈ ∩n∈NM(R) then ordRx = ∞. Note that if R is the power series ring k[[X1,...,Xm]] over a field k then for any

X i1 im f = f(X1,...,Xm) = fi1...im X1 ...Xm with fi1...im ∈ k i1,...,im we have

ordRf = ord(f) = max{i1 + ··· + im :(i1, . . . , im) ∈ Supp(f)} m where Supp(f) = {(i1, . . . , im) ∈ N : fi1...im 6= 0}. In (D1) to (D3) and (E1) to (E4) of L3§12 we discuss the theory of matrices including: ranks, minors, cofactors, transposes, adjoints, and linear maps. In (D5) and (E12) to (E14) of L3§12 we generalize Hensel’s Lemma by extending the notions of ord, completeness, and Hausdorffness from quasilocal rings to more general situations in the following manner Let I be an ideal in a ring R and let V be an R-module. V is I-Hausdorff, or i Hausdorff relative to I, means ∩i∈ I V = 0, where for any ideal J in R, by JV we N P denote the submodule of V given by JV = { jivi : ji ∈ J and vi ∈ V }. For any i h ∈ V we define the (R,I)-order of h by putting ord(R,I)h = max{i ∈ N : h ∈ I V } and we note that then V is I-Hausdorff means for any h ∈ V we have: ord(R,I)h = ∞ ⇔ h = 0. A sequence x = (xi)1≤i<∞ in V is I-Cauchy, or Cauchy relative to I, means for every E ∈ N there exists NE ∈ N such that for all i > NE and j > NE E we have xi − xj ∈ I V . The sequence x is I-convergent, or convergent relative to I, means it converges to an I-limit ξ ∈ V , i.e., for every E ∈ N there exists NE ∈ N E such that for all i > NE we have ξ − xi ∈ I V ; we indicate this by some standard notation such as xi → ξ as i → ∞ or limi→∞xi = ξ. If V is I-Hausdorff then a limit when it exists is unique. V is I-complete means every I-Cauchy sequence in it has a limit in it. If R is quasilocal with I = M(R) and V = R then the adjective “relative to I” may be dropped from the above definitions. §4: SUMMARY OF LECTURE L4 ON VARIETIES AND MODELS 11

§4: SUMMARY OF LECTURE L4 ON VARIETIES AND MODELS

In L4§1 we open up the lecture with Sylvester’s 1840 theory of the resultant of two univariate polynomials f, g of degrees n, m which by eliminating the variable Y produces an n+m by n+m matrix ResmatY (f, g) in their coefficients, the vanishing of whose determinant ResY (f, g) gives a condition for them to have a common solution. In turn the vanishing of the discriminant DiscY (f) = ResY (f, fY ) gives a condition for f to have a multiple root. Having already cited the Hilbert Basis Theorem proved in L4§3 which follows the general chit-chat about curves, surfaces, and varieties in L4§2, let us now describe L4§5 on ideals and modules consisting of Observations (O7) to (O27). Before coming to (O7), in the preamble of L4§5 we talk about intersections and sums of submodules of a module V over a ring, and make the definitions of the colon (or quotient) modules (C : B)V and (C : a)V , where C is a submodule of V with B ⊂ R and a ∈ R, by putting (C : a)V = {v ∈ V : av ∈ C} and (C : B)V = ∩b∈B(C : a)V . In case of ideals we also consider their products, and we define the colon (or quotient) ideals (C : B)R and (C : a)R, where C is a submodule of V with B ⊂ V and a ∈ V , by putting (C : a)R = {r ∈ R : rv ∈ C} and (C : B)R = ∩b∈B(C : a)R. Moreover, for a submodule U of V and an ideal I in R, we define their radicals by e e putting radV U = {r ∈ R : r V ⊂ U for some e ∈ N+} and radRI = {r ∈ R : r ∈ U for some e ∈ N+}. Also we define zerodivisors and nilpotents. Finally we define primary and irreducible ideals as well as submodules. In (O7), (O8), (O9), (O12), (O13), and (O18) we discuss primary ideals and submodules as well as annihilators and colons and relate them to radicals. In (O10) and (O11) we discuss basic isomorphism theorems. In (O14) we define noetherian modules in terms of the three equivalent con- ditions of NNC (= noetherian condition saying that every submodule is finitely generated) and ACC (= ascending chain condition which says that every ascending chain of submodules stops) and MXC (= maximal condition which says that every nonempty family of submodules has a maximal element). In (O15) we show that every submodule U of a noetherian module V over a ring R has an irredundant primary decomposition U = U1 ∩ · · · ∩ Uh where U1,...,Uh are a finite number of primary submodules of V ; here irredundant means none of the Ui can be deleted and the radicals P1,...,Ph of U1,...,Uh are distinct prime ideals in R. In (O16) we introduce the spectrum spec(R) of a ring R as the set of all prime ideals in R. The set of all maximal ideals in R is denoted by mspec(R). The set of all members of spec(R) which contain an ideal J of R is called the spectral variety of J and is denoted by vspecRJ. The sets of all maximal and minimal members of vspecRJ are denoted by mvspecRJ and nvspecRJ respectively. In L5§5(Q7)(C11) we also put nspec(R) = nvspecRJ. The set of all spectral varieties in R is denoted by svt(R). The spectral ideal of any subset W of spec(R) is defined by putting ispecRW = ∩P ∈W P . The set of all radical ideals in R, i.e., the set of all ideals in R which are their own radicals, is denoted by rd(R). In L5§5(Q21) we also put id(R) (resp: nid(R)) = the set of all (resp: all nonunit) ideals in R. If R is a finite variable polynomial ring over a field then ispecR gives a bijection svt(R) → rd(R) whose inverse is given by vspecR; this is part (T22.8) of the Hilbert Nullstellensatz L4§8(T22) proved in L5§5(Q11). 12 LECTURE L6: PAUSE AND REFRESH

In (O16) and (O17) we define the associator assRV and the tight associator tassRV of an R-module V as certain subsets of spec(R), and show that for the above irredundant primary decomposition U = U1 ∩ · · · ∩ Uh of the submodule U of the noetherian R-module V we have assR(V/U) = tassR(V/U) = {P1,...,Ph}. Members of assR(V/U) are called associated primes of U in V . In (O19) we use bracketed colon to prove partial uniqueness of the primary components Ui in the above decomposition. A multiplicative set in a ring R is a subset S of R with 1 ∈ S such that S contains the product of every pair of elements in it. For any submodule U of an R-module V we put [U : S]V = ∪s∈S(U : s)V and call this the isolated S-component of U in V ; the uniqueness follows by showing that [U : S]V = ∩Pi∩S=∅Ui. For any prime ideals Q1,...,Qn in R with n ∈ N+ we put [U :(Q1,...,Qn)]V = [U : S]V where S = ∩1≤i≤n(R \ Qi)]V and call this the isolated (Q1,...,Qn)-component of U in V , and we note that now {1 ≤ i ≤ h : Pi ∩ S = ∅} = {1 ≤ i ≤ h : Pi ⊂ Qj for 1 ≤ j ≤ n}. In (O20) we generalize the idea of primary decomposition to quasiprimary decomposition without assuming the noetherian condition. We also introduce the notation ZR(V ) = the set of all zerodivisors of a module V over a ring R, and SR(V ) = the multiplicative set R \ ZR(V ) in R. In (O21) we study the properties of the length `R(V ) of a module V over a ring R which we define to be the maximum length n ∈ N of a finite sequence 0 = V0 $ V1 ⊂ · · · $ Vn = V of submodules of V which we call a normal series of V of length n. When there does not exist a bound for the lengths of such sequences then we put `R(V ) = ∞. By a simple module we mean a module of length 1. In (O22) we define heights and depths of ideals and dimensions of rings. By the dimension dim(R) of a ring R we mean the maximum length n ∈ N of a finite sequence (δ) P0 $ P1 $ ··· $ Pn of prime ideals in R. We call such a sequence a prime sequence in R of length n. When there does not exist a bound for such sequences then we put dim(R) = ∞ provided R is not the null ring; if R is the null ring then we put dim(R) = −∞. By the height htRP (resp: depth dptRP ) of a prime ideal P in R we mean the maximum length of a prime chain (δ) in R with P = Pn (resp: P = P0); again the maximum can be ∞. We define the height htRJ and the depth dptRJ of a nonunit ideal J in R by putting htRJ = min{htRP : P ∈ vspecRJ} ∈ N ∪ {∞} and dptRJ = max{dptRP : P ∈ vspecRJ} ∈ N ∪ {∞}. Let us complete the definitions of height and depth by putting htRR = −∞ = dptRR. In (O25), analogous to noetherian modules, we define artinian modules in terms of the two equivalent conditions of DCC (= descending chain condition which says that every descending chain of submodules stops) and MNC (= minimal condition which says that every nonempty family of submodules has a minimal element). A ring is artinian if it is artinian as a module over itself. We show that a module has finite length iff it is artinian as well as noetherian. We show that a ring is artinian iff it is either the null ring or a zero-dimensional noetherian ring. We show that a finitely generated module over a noetherian (resp: artinian) ring is noetherian (resp: artinian). In (O26) we define ideals A, B in a ring R to be comaximal to mean that A + B = R. We show that if A1,...,An are a finite number of pairwise comaximal Q ideals in R then 1≤i≤n Ai = ∩1≤i≤nAi, and given any xi ∈ R for 1 ≤ i ≤ n there exists x ∈ R with x ≡ xi mod Ai for 1 ≤ i ≤ n, i.e., x − xi ∈ Ai for 1 ≤ i ≤ n. §4: SUMMARY OF LECTURE L4 ON VARIETIES AND MODELS 13

In (O27) we define an affine ring over a ring R to be a finitely generated ring extension of R. In L4§6 we recapitulate primary and quasiprimary decompositions. In L4§7 we introduce the localization RS of a ring R at a multiplicative set S in R, together with a ring homomorphism (called the canonical homomorphism or the localization map) φ : R → RS thus. We define RS to be the set of all equivalence classes of pairs (u, v) ∈ R × S under the equivalence relation given by: (u, v) ∼ (u0, v0) ⇔ v00(uv0 − u0v) = 0 for some v00 ∈ S. The equivalence class u containing (u, v) is denoted by u/v or v , and we add and multiply the equivalence classes by the rules u1 + u2 = u1v2+u2v1 and u1 × u2 = u1u2 . Also we “send” any v1 v2 v1v2 v1 v2 v1v2 u ∈ R to u/1 ∈ RS. This makes RS into a ring and u 7→ u/1 gives the canonical ring homomorphism φ : R → RS. Clearly φ(S) ⊂ U(RS) and ker(φ) = [0 : S]R. The localization of R at SR(R) is the total quotient ring QR(R) of R; clearly [0 : SR(R)] = 0 and hence we may and we do regard QR(R) to be an overring of R. Now: ker(φ) = 0 ⇔ S ⊂ SR(R). If S ⊂ SR(R) then there is a unique R-injection of RS into QR(R); we call its image the localization of R at S in QR(R) and continue to denote it by RS; let us call this the GOOD case; thus in the good case RS is an overring of R. For any prime ideal P in R we let RP stand for RR\P and we call it the localization of R at P . Some basic properties of localization are proved in Theorems (T9) to (T17) of L4§7 including L4§7.1. In particular, in (T12) we prove the salient fact which says that: I 7→ J = φ(I)RS gives a bijection of the set of all ideals I in R for which −1 [I : S]S = I onto the set of of all ideals in RS and its inverse is given by J 7→ φ (J); this bijection commutes with the ideal theoretic operations of radicals, intersections and quotients. The geometry started in §§2-5 of L3 is picked up again in §§8-9 of L4. Common solutions of a finite number of polynomial equations in a finite number of variables with coefficients in a field form an affine algebraic variety. Some properties of these varieties are stated in Theorems (T19) to (T22) of L4§8 and proved in (Q10), (Q11), and (Q32) of L5§5. In L4§8.1 we relate these varieties to the spectral varieties discussed in the above item (O16) and then in L4§8.2, via localization, we relate them to modelic specs which are collections of local rings. Now a simple point is reincarnated as a regular local ring which means a local R whose embedding dimension emdim(R) equals its dimension, where emdim(R) is defined to be the smallest number of generators of M(R). In L4§8.3, L4§9, L4§9.1, and L4§9.2 we formulate some properties of modelic specs as Theorems (T23) to (T30) which are proved in (R7) to (R9) of L4§12 together with (Q7), (Q14), and (Q17) of L5§5. In L4§9.2 we show how modelic specs give rise to modelic blowups and in L4§9.3 we use them for simplifying singularities. In L4§11 we start a new column called Problems by doing which the student may get “mild satisfaction or Ph.D. thesis or fame.” In (R1) to (R5) of L4§12 we deal with Laplace development of determinants, block matrices, and Cramer’s rule for solving liner equations. In (R6) of L4§12 we concoct a resultant theory which makes it valid over rings with zerodivisors. This is done by converting the product rule for polynomials into matrix multiplication and expanding this into a product formula for the resultant matrix. 14 LECTURE L6: PAUSE AND REFRESH

§5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES

In L5§1 we define the direct sums of modules and related objects. A module V over a ring R is an internal direct sum of a family (Vi)l∈I of submodules if it is P 0 the sum V = i∈I Vi and the sum is direct, i.e., for every finite subset I of the P 0 indexing set I we have: i∈I0 vi = 0 with vi ∈ Vi for all i ∈ I ⇒ vi = 0 for all 0 P i ∈ I . The fact that the sum is direct is expressed by writing V = Vi (internal L i∈I direct sum) or V = i∈I Vi (internal). In case I is a finite set, say I = {1, 2, . . . , n}, we may write V = V1 ⊕ V2 ⊕ · · · ⊕ Vn (internal). The cartesian product W = U1 × · · · × Un of a finite number of R-module U1,...,Un is converted into an R-module by componentwise addition and scalar multiplication, and then we call W the direct sum of U1,...,Un and indicate this by writing W = U1 ⊕ U2 ⊕ · · · ⊕ Un; to distinguish this from internal direct sum we may call it external direct sum. If U1 = ··· = Un = U then, as in the case of cartesian products, we put W = U n and call it the (module theoretic) direct sum of n copies of U. More generally, the cartesian product of a family (Ui)i∈I of sets Ui, where I is any indexing set (which need not be finite), is defined by Q S putting i∈I Ui = the set of all maps φ : I → i∈I Ui such that φ(i) ∈ Ui for all i ∈ I. In case the Ui are R-modules, this becomes an R-module, called the direct product, by defining componentwise addition and scalar multiplication. The set Q of all φ ∈ Ui whose support supp(φ) = {i ∈ I : φ(i) 6= 0} is finite is called the i∈I L direct sum of the family and is denoted by Ui; note that this is a submodule Q i∈I of the direct product i∈I Ui and is hence an R-module; to distinguish this from the internal direct sum we may again call it the external direct sum. Q For any j ∈ I we get an obvious injective map µj : Uj → Ui called the Q i∈I natural injection, and a surjective map νj : i∈I Ui → Uj called the natural projection. Clearly νiµi is the identity map of Ui. All this works for sets Ui. In the module case these maps are R-homomorphisms, and they give rise to an obvious R-monomorphism (again called the natural injection) µj : Uj → ⊕i∈I Ui and an obvious R-epimorphism (again called the natural projection) νj : ⊕i∈I Ui → Uj. L L Letting U i to be the image of µi we have i∈I Ui (external) = i∈I U i (internal) . Q I If Ui = a set U for all i ∈ I then the cartesian product i∈I Ui is denoted by U and is called the set-theoretic I-th power of U, and if U is an R-module then the module U I is called the module theoretic I-th power of U. Finally, if the I module Ui = a module U for all i ∈ I then the module ⊕i∈I Ui is denoted by U⊕ and is called the module theoretic restricted I-th power of U. Other related definitions such as diagonal direct sum may be found in L5§1. In L5§2 we define a graded ring S over a ring R to be an overring of R such that P S = i∈I Si as an internal direct sum of a family (Si)i∈I of R-modules with S0 = R where the type I is any additive abelian monoid, and for all fi ∈ Si and fj ∈ Sj with i, j in I we have fifj ∈ Si+j. An ideal in S is homogeneous if it is generated by 0 homogeneous elements, i.e., elements of ∪i∈I Si. A ring homomorphism φ : S → S 0 of graded rings of type I is graded if φ(Si) ⊂ Si for all i. If I = N then we call S a naturally graded ring. In L5§3 we define a semihomogeneous ring to be a naturally graded ring S such that S = S0[S1]; if S1 is a finitely generated module over R = S0 then we call S a homogeneous ring. Also we study the ideal theory of graded rings. §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 15

The very long L5§5 on More About Ideals And Modules is divided into Quests (Q1) ro (Q35) which we proceed to summarize. Referring to (O14) to (O17) of the above Summary of L4, let U = U1 ∩ · · · ∩ Uh be an irredundant primary decomposition of a submodule U of a finitely generated module V over a noetherian ring R, and let P1,...,Ph be the prime ideals in R which are the respective radicals of U1,...,Uh. In the Zerodivisor Theorem (Q1)(T1) we show that then ZR(V/U) = P1 ∪ · · · ∪ Ph. In (Q2) we define a faithful module over a ring R to be an R-module D such that annRD = 0. Note that annRD = {r ∈ R : rd = 0 for all d ∈ D}. In (Q3) we define the Jacobson radical jrad(R) of a ring R to be the intersection of all its maximal ideals, and we define a Zariski ring to be pair (R,J) where J is an ideal in a ring R which is contained in the Jacobson radical of R. In the very versatile Nakayama Lemma (Q3)(T3) we show that: if U is a submodule of a finitely generated module V over a ring R such that V = U + JV for an ideal J in R with J ⊂ jrad(R), then V = U. The particularly noteworthy case of this when R is a local ring with J = M(R) was proved by the master algebraist Krull. In (Q4) we prove: the Krull Intersection Theorem (T7) saying that in any n local ring R we have ∩n∈NM(R) = 0, its Domainized Version (T10) saying that n for any nonunit ideal J in any domain R we have ∩n∈NJ = 0, and its variation which is called the Artin Rees Lemma (T4) and which says that, given any ideals V, W, J in a noetherian ring R, there exists a ∈ N such that for all c ≥ b ≥ a in N we have J cW ∩ V = J c−b(J bW ∩ V ). In (Q5) we explain Nagata’s idealization principle which converts module situations to ideal situations, and use it prove the module incarnations of the above results of (Q4). We also prove a Characterization Theorem (T20) for Zariski Rings. In (Q6) we prove Cohen’s Theorem saying that a ring is noetherian if all the prime ideals in it are finitely generated, and Eakin’s Theorem saying that a ring is noetherian if some noetherian overring is a finitely generated module over it. In (Q7) we prove Krull’s Generalized Principal Ideal Theorem (T27) saying that: if an ideal I in a noetherian ring R is generated by n elements x1, . . . , xn with n ∈ N then htRI ≤ n, and if xi is a nonzerodivisor mod x1, . . . , xi−1 for 1 ≤ i ≤ n then htRP = n for every P in nvspecRI. The Principal Ideal Theorem (T24) is the n = 1 case. We deduce the Extended Dim-Emdim Theorem (T29) saying that for any local ring R we have that emdim(R) ≥ dim(R) = the smallest number of elements in R which generate an M(R)-primary ideal. We also deduce the Dimension Lemma (T30) about the polynomial ring S = R[X1,...,Xm] as well as the power series ring S = R[[X1,...,Xm]] in a finite number of variables X1,...,Xm over a noetherian ring R, saying that: (1) S is noetherian, and for any nonunit ideal I in R and j ∈ {0, . . . , m}, upon letting Ij = IS + (X1,...,Xj)S we have Ij ∩ R = I with htSIj = j + htRI, and if I ∈ spec(R) then we also have Ij ∈ spec(S); (2) dim(S) = m + dim(R); (3) if R local and we are in the power series case then S is local with M(S) = M(R)S + (X1,...,Xm)S and emdim(S) = m + emdim(R), and hence: R is regular ⇔ S is regular; (4) If the ring R is a field then R[X1,...,Xm] is an m-dimensional noetherian ring and R[[X1,...,Xm]] is an m-dimensional regular local ring; (5) in the polynomial case with m = 1, for any I = J ∩ R with J ∈ spec(S) we have I ∈ spec(R) and: (i) if J = IS then htSJ = htRI, whereas (ii) if J 6= IS then htSJ = 1 + htRI and there exists E F (X1) ∈ S of degree E > 0 such that the coefficient of X1 in F belongs to R \ I. 16 LECTURE L6: PAUSE AND REFRESH

In the Relative Independence Theorem (Q8)(T34) we show that n elements in a noetherian ring generate an ideal of height n iff they are independent over that ring in the following sense. Considering the n variable polynomial ring R[Y1,...,Yn] in n variables Y1,...,Yn over a ring R with n ∈ N, for any A ⊂ R and i ∈ N we let A[Y1,...,Yn]i = the set of all homogeneous members of R[Y1,...,Yn] of degree i all whose coefficients belong to A (including the zero polynomial in case 0 ∈ A) S and we put A[Y1,...,Yn]∞ = A[Y1,...,Yn]i. Let J be an ideal in R. For i∈N elements x1, . . . , xn in R let I = (x1, . . . , xn)R. The elements x1, . . . , xn are said to be independent over (R,J), or independent relative to (R,J), if I ⊂ J and: (†) F (Y1,...,Yn) ∈ R[Y1,...,Yn]∞ with F (x1, . . . , xn) = 0 ⇒ F (Y1,...,Yn) ∈ J[Y1,...,Yn]∞. The elements x1, . . . , xn are said to be independent over R, or independent relative to R, if I 6= R and the elements are independent over (R, radRI). Considering the R-epimorphism α : R[Y1,...,Yn] → R which sends Yi to xi for 1 ≤ i ≤ n, we show that (]) ker(α) = (Y1 −x1,...,Yn −xn)R[Y1,...,Yn] and condition (†) is equivalent 0 to the condition: († ) ker(α) ∩ R[Y1,...,Yn]∞ ⊂ J[Y1,...,Yn]∞. Assuming I ⊂ J, we show that condition (†) is equivalent to the condition: i+1 (‡) F (Y1,...,Yn) ∈ R[Y1,...,Yn]i with i ∈ N and F (x1, . . . , xn) ∈ I ⇒ F (Y1,...,Yn) ∈ J[Y1,...,Yn]i. In case R is the power series ring K[[X1,...,Xm]] in a finite number of variables over a field K, elements x1, . . . , xn in M(R) are said to be analytically independent over K if there is no nonzero f(Y1,...,Yn) ∈ K[[Y1,...,Yn]] with f(x1, . . . , xn) = 0. In the Analytic Independence Theorem (Q8)T35) we show that: (1) if the elements x1, . . . , xn are independent over (R,M(R)) then they are analytically independent over K, and (2) if m = n and the ideal I is M(R)-primary then the elements x1, . . . , xn are analytically independent over K. In (Q9) we show that the localization of any normal ring at a multiplicative set in it not containing any zerodivisor is again normal. A prime ideal Q in an overring S of a ring R lies above a prime ideal P in R means P = Q ∩ R; we may also say that P lies below Q in R. In (Q9) we mostly deal with an integral extension of a ring, i.e., an overring S of a ring R such that S is integral over R. We show that for any prime ideal P in R lying below a prime ideal Q in S we have: P is maximal ⇔ Q maximal. In the Lying Above Theorem (T40) we show that for any prime ideal P in R, there exists a prime ideal Q in S lying above P . In the Going Up Theorem (T41) we show that given any prime ideals P ∗ ⊂ P in R, and given any prime ideal Q∗ in S lying above P ∗, there exists a prime ideal Q in S lying above P for which we have Q∗ ⊂ Q. In the Going Down Theorem (T44) we show that if R is a normal domain and S is a domain, then given any prime ideals P ∗ ⊂ P in R, and any prime ideal Q in S lying above P , there exists a prime ideal Q∗ in S lying above P ∗ with Q∗ ⊂ Q. In the Dimension Corollary (T45) we show that: (1) above any ascending finite sequence of prime ideals in R there lies an ascending finite sequence of prime ideals in S; (2) any strictly ascending finite sequence of prime ideals in S lies above a strictly ascending finite sequence of prime ideals in R; (3) dim(S) = dim(R); (4) for any prime ideal Q in S lying above any prime ideal P in R we have dptSQ = dptRP and htSQ ≤ htRP ; (5) if S is a domain and R is a normal domain, then for any prime ideal Q in S lying above any prime ideal P in R we have htSQ = htRP . §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 17

In (Q10) we prove the Noether Normalization Theorem (T46) which says that any affine ring A over a field k has a normalization basis over k, by which we mean a finite number of elements Y1,...,Yr in A which are algebraically in- dependent over k and which are such that the ring A is integral over the subring k[Y1,...,Yr]. By a saturated chain of prime ideals in a ring R we mean a finite sequence ∗ P0 $ P1 $ ··· $ Pn of prime ideals in R such that there is no prime ideal P in ∗ R with Pi $ P $ Pi+1 for some i ∈ {0, . . . , n − 1}; we call n the length of the chain; if P ⊂ Q are prime ideals in R with P = P0 and Q = Pn then we say that P0 $ P1 $ ··· $ Pn is a saturated prime ideal chain between P and Q; if there is no prime P in R with P $ P0 and there is no prime Q in R with Pn $ Q then we say that P0 $ P1 $ ··· $ Pn is an absolutely saturated prime ideal chain in R. By an infinite chain of prime ideals in a ring R we mean a sequence of prime ideals (Pi)i∈N in R such that for all i ∈ N we have Pi $ Pi+1; if P ⊂ Q are prime ideals in R such that P ⊂ Pi ⊂ Q for all i ∈ N then we say that this is an infinite prime ideal chain between P and Q. 0 0 For a subdomain R of a domain R , by trdegRR we denote the transcendence degree of QF(R0) over QF(R). More generally, for a prime ideal P 0 in a ring R0 lying above a prime ideal P in a subring R, after identifying R/P with a subdomain 0 0 0 0 0 0 of R /P , by trdegR/P R /P we denote the transcendence degree of QF(R /P ) 0 0 0 over QF(R/P ); in case R is a subfield k of R , we write trdegkR /P in place of 0 0 0 trdegk/0R /P ; in case P and P are the respective maximal ideals in quasilocal 0 0 0 0 rings R and R, we may write restrdegRR in place of trdegR/P R /P and call this the residual transcendence degree of R0 over R. Using (T46) we establish the Extended Dimension Theorem (T47) which subsumes L4§8(T19) and which says that for any affine ring A over a field k we have the following: (T47.1) For any nonunit ideal J in A, letting J1,...,Jm be the distinct minimal primes of J in R we have dim(A/J) = dptAJ = max1≤i≤mtrdegk(A/Ji) ∈ N, and for any prime ideal P in A we have dim(A/P ) = dptAP = trdegk(A/P ) ∈ N. Hence in particular, if A is a domain then dim(A) = trdegkA ∈ N. (T47.2) Every pair of prime ideals P ⊂ Q in A has the property which says that: (∗) there is no infinite chain of prime ideals between P and Q, and any two saturated chains of prime ideals between P and Q have the same length. Moreover: (∗∗) this common length equals dim(A/P ) − dim(A/Q). (T47.3) If A is a domain then the length of every absolutely saturated chain of prime ideals in R equals the dimension of R. (T47.4) If A and H is a nonunit ideal H in A then htAH + dptAH = dim(A). (T47.5) If A is a domain then, for any prime ideal P in A, the length of every saturated prime ideal chain in A between 0 and P equals htAP , and the length of every saturated prime ideal chain in A between P and any member of mvspecAP equals dptAP . (T47.6) If A is a domain and A0 is an affine domain over A then for any prime ideal P 0 in A0 lying above any prime ideal P in A, upon considering the localizations 0 0 0 0 0 R = AP 0 and R = AP , we have dim(R ) + restrdegRR = dim(R) + trdegRR . (T47.7) in the situation of (T46) we have dim(A) = r. In (Q10) and (Q11) we prove the Inclusion Relations Theorem L4§8(T21) and the Hilbert Nullstellensatz L4§8(T22). 18 LECTURE L6: PAUSE AND REFRESH

In (Q11), the Jacobson Spectrum And Nilradical of a ring R are defined by the equation jspec(R) = {P ∈ spec(R): P = ∩Q∈mvspecRP Q} and the equation nrad(R) = ∩P ∈spec(R)P . The ring R is said to be a Jacobson ring if jspec(R) = spec(R). Using jspec we supplement L4§8(T21) in (Q11)(T50). Using nrad we prove the Minimal Primes Theorem(Q11)(T51) which says that, given any ideal J in any ring R, every member of vspecRJ contains some member of nvspecRJ, and we have radRJ = ∩P ∈nvspecRJ P . A ring A is catenarian means every pair of prime ideals P ⊂ Q in it satisfies the above condition (∗) of (T47.2). A ring A is universally catenarian means it is noetherian and the polynomial ring over it in any finite number of variables is catenarian. The dimension formula (resp: the dimension inequality) holds for a domain A relative to an overdomain A0 means for every prime ideal P 0 in 0 0 0 0 A , considering the localizations R = AP 0 and R = AP with P = P ∩ A, we 0 0 0 0 0 have dim(R ) + restrdegRR = dim(R) + trdegRR (resp: dim(R ) + restrdegRR ≤ 0 dim(R)+trdegRR ). The dimension formula (resp: the dimension inequality) holds in a domain A means it holds for A relative to every affine domain over A. Thus (T47.6) says that the dimension formula holds in any affine domain over a field. In (Q12) we sharpen Comment (Q7)(C13) on Univariate Ideal Extensions, which in the present summary corresponds to part (5) of the Dimension Lemma (Q7)(T30), by proving the Multiple Ring Extension Lemma (T55) which says that: the dimension inequality holds in any noetherian domains A relative to any 0 0 affine domain A over A, and if A = A[x1, . . . , xm] for some m ∈ N and x1, . . . , xm in A0 then for any prime ideal P 0 in A0 lying above any prime ideal P in A, considering 0 0 0 0 the localizations R = AP 0 and R = AP we have restrdegRR ≤ m with trdegRR ≤ 0 m, and if either trdegRR = m or the polynomial ring B = A[X1,...,Xm] in 0 0 0 m variables is catenarian, then dim(R ) + restrdegRR = dim(R) + trdegRR . The m = 1 case of (T55) is designated as the Simple Ring Extension Lemma (T54). In the Catenarian Ring Theorem (T58) we show that a ring A is universally catenarian iff A is a noetherian catenarian ring and the dimension formula holds in A/H for every prime ideal H in A. In (Q13) on Associated Graded Rings And Leading Ideals, given any ideal I in a ring R, we define the associated graded ring of (R,I) to be the naturally graded ring grad(R,I) = L (In/In+1) with external direct sum of -modules. n∈N Z m n m+n The multiplication is defined by grad(R,I)(fm) · grad(R,I)(fn) = grad(R,I)(fmfn) for m n n n all fm ∈ I and fn ∈ I with m, n in N. Here grad(R,I) : I → grad(R,I) is the n n n+1 composition of the natural surjection λn : I → (I /I ) and the natural injection n n+1 n n µn :(I /I ) → grad(R,I). We put grad(R,I)n = grad(R,I)(I ). For any h ∈ R n we put lefo(R,I)(h) = grad(R,I)(h) with the understanding that if ord(R,I)h = ∞ then lefo(R,I)(h) = 0; we call this the leading form of h relative to (R,I). For any ideal J in R we put ledi(R,I)(J) = the homogeneous ideal in grad(R,I) generated by (lefo(R,I)(h))h∈J and we call this the leading ideal of J relative to (R,I). Any family of generators x = (xl)l∈L of I induces a unique graded ring epimorphism x x Θ(R,I) : k[(Xl)l∈L] → grad(R,I) with k = R/I such that Θ(R,I)(z) = µ0(z) for all x z ∈ k and Θ(R,I)(Xl) = yl for all l ∈ L where (Xl)l∈L are indeterminates over k 1 x and yl = grad(R,I)(xl); we say that Θ(R,I) is induced by (R,I) and x. Note that grad(R,I) is a semihomogeneous (homogeneous if L is finite) ring over the ring grad(R,I)0 with grad(R,I) = grad(R,I)0[grad(R,I)1] = grad(R,I)0[(yl)l∈L]. §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 19

In case R is a quasilocal ring with I = M(R), in the above paragraph we write R in place of (R,I), except that we write grad(R) and grad(R)n in place of grad(R,I) and grad(R,I)n respectively. A valuation function on a ring R is a map v : R → Z ∪ {∞} such that for all x, y, z in R we have: (i) v(xy) = v(x) + v(y), (ii) v(x + y) ≥ min(v(x), v(y)), and (iii) v(z) = ∞ ⇔ z = 0, with the usual understanding about ∞. Thus in the notation of L3§11, a valuation function is like a valuation with R and Z playing the roles of the field K and the ordered abelian group G respectively. If v is a valuation function on a nonnull ring R then clearly R is a domain and we get a unique valuation v : QF(R) → Z ∪ {∞} such that for all x, y in R× we have v(x/y) = v(x) − v(y); we say that the valuation v is induced by v; usually we denote v also by v. Theorem (T60) says that for any given ideal I in a ring R we have the following: (1) I 6= R and ord(R,I) is a valuation function on R iff R is I-Hausdorff and grad(R,I) is a domain. (2) Upon letting R∗ = R/J ∗ and I∗ = I/J ∗ with J ∗ = n ∗ ∗ ∗ ∩n∈NI , we have that R is I -Hausdorff, and if grad(R,I) is a domain then R ∗ 0 is a domain and ord(R∗,I∗) is a valuation function on R . (3) Let φ : R → R be a ring epimorphism, with J = ker(φ), such that the leading ideal grad(R,I)(J) is a n (homogeneous) prime ideal in grad(R,I); then ∩n∈N(J + I ) is a prime ideal in R. Given any local ring R with dim(R) = d and emdim(R) = e, let x = (x1, . . . , xm) be a finite number of generators of M(R), let X = (X1,...,Xm) be indeterminates over the field k = R/M(R), and recall that Θx : k[X] → grad(R) is a graded ring P R epimorphism where k[X] = k[X]n is the naturally graded polynomial ring n∈N with k[X]n = the set of all homogeneous polynomials of degree n (including 0). x Theorem (T61) says that then we have: (1) ker(ΘR) ∩ k[X]1 = 0 ⇔ m = e. x x (2) ker(ΘR) = 0 ⇔ m = d. (3) If m = e then: R is regular ⇔ ker(ΘR) = 0. (4) R is regular ⇔ there is a permutation σ of {1, . . . , m} together with elements x aij in k such that the ideal ker(Θ ) in k[X] is generated by the m − e elements P R Xσ(i) − 1≤j≤e aijXσ(j) for e + 1 ≤ i ≤ m. In (Q14) we define a Completely Normal Domain to be a domain R such that: x ∈ QF(R) with dxn ∈ R for some d ∈ R× and all n ∈ N ⇒ x ∈ R. Complete Normality Lemma (T63) says that for any given ring R we have the following: (1) If R is a completely normal domain then R is a normal domain. If R is a noetherian normal domain then R is a completely normal domain. (2) If I n is an ideal in R such that for all z ∈ R we have ∩n∈N(zR+I ) = zR, and grad(R,I) is a completely normal domain, then R is a completely normal domain. (3) For any ideal I in R we have that: I 6= R iff grad(R,I) is nonnull. (4) If R is noetherian and I is any ideal in R then grad(R,I) is noetherian. (5) If R is noetherian and has an ideal I with I ⊂ jrad(R) such that grad(R,I) is a normal domain, then R is a normal domain. We proved L4§8(T24.1) as the Krull Intersection Theorem (Q4)(T7), we shall prove L4§8(T24.3) as Another Ord Valuation Lemma (Q15)(T71), and we prove L4§8(T24.2) as the Ord Valuation Lemma (Q14)(T64) which says that given any regular local ring R, for all nonzero elements x and y in R we have ordR(xy) = ordRx+ordRy, and hence R is a normal domain and we get a valuation ordR : QF(R) → Z of QF(R) by putting ordR(x/y) = ordRx − ordRy. 20 LECTURE L6: PAUSE AND REFRESH

To summarize (Q15) on Regular Sequences And CM Rings, let R be a ring, let x = (x1, . . . , xn) be a sequence of elements in R of length n ∈ N, for 0 ≤ m ≤ n let Im be the ideal (x1, . . . , xm)R, and let V be a module over R. We say that x is a V -regular sequence, if InV 6= V and for 1 ≤ m ≤ n we have xm ∈ SR(V/(Im−1V )). Moreover if J is an ideal in R with In ⊂ J then we say that x is a V -regular sequence in J, and if for every z ∈ J ∩ SR(V/(InV )) we have InV + zV = V then we say that x is a maximal V -regular sequence in J. We define the (R,J)-regularity of V by putting reg(R,J)V = the maximum length of a V -regular sequence in J. If R is local with J = M(R) then we call this the R-regularity of V and denote it by regRV ; in this case we write reg(R) in place of regRR and we call it the regularity of R. We define the generating number of V in R by putting gnbRV = the smallest number of generators of V . An ideal I in R is said to be an ideal-theoretic complete intersection in R if I 6= R and d htRI = gnbRI. A finitely generated R-module is free means it is isomorphic to R for some d ∈ N. In case R is local and V is finitely generated, V is said to be a CM (= Cohen-Macaulay) module if regRV = dim(R/(0 : V )R). A local ring R is said to be a CM ring if it is a CM module over itself, i.e., if reg(R) = dim(R). A noetherian ring R is said to be a CM ring if, for every maximal ideal P in R, the localization RP is a CM ring. An ideal I in a noetherian ring R is said to be unmixed if for every prime P in assR(R/I) we have htRP = htRI. The unmixedness theorem is said to hold in a noetherian ring R if every ideal in R, which can be generated by as many elements as its height, is unmixed. In Cohen-Macaulay Theorem (T70) we show that for any noetherian ring R we have the following: (1) If R is CM and x = (x1, . . . , xn) is any R-regular sequence in R with n ∈ N, then upon letting In = xR we have htRIn = n and for every P ∈ assR(R/In) we have htRP = n, and hence in particular In is an ideal-theoretic complete intersec- tion in R, and In is unmixed. (2) If R is CM then for any nonunit ideal I in R we have reg(R,I)R = htRI. (3) If R is CM and I is a nonunit ideal I in R which is an ideal-theoretic complete intersection in R then I is generated by an R-regular sequence. (4) If R is CM then its localization RS at any multiplicative set S in it is CM, and hence in particular its localization RP at any prime ideal P in it is CM. (5) If R is CM then R is catenarian. (6) If R is CM and local then the length of any absolutely saturated prime ideal chain in R equals dim(R) and for any nonunit ideal I in R we have htRI = dim(R) − dim(R/I). (7) R is CM ⇔ the unmixedness theorem holds in R. (8) R is a finite variable polynomial ring over a field ⇒ R is CM. (9) R is a finite variable power series ring over a field ⇒ R is CM. (10) R is CM ⇒ every finite variable polynomial ring over R is CM. (11) R is CM ⇒ every finite variable power series ring over R is CM. (12) R is CM ⇒ every homomorphic image of R is universally catenarian. In Another Ord Valuation Lemma (T71) we show that given a local ring R, for any x ∈ M(R)\ZR(R) we have that: (i) R/(xR) is a local ring whose dimension is one less than the dimension of R, (ii) if R/(xR) is regular then ordRx = 1, and (iii) if ordRx = 1 and R is regular then R/(xR) is regular. §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 21

In (Q16) on Complete Intersections And GST Rings we start by making the following definitions. A local ring R is said to be a complete intersection if there exists a regular local ring T and an epimorphism φ : T → R such that ker(φ) is an ideal-theoretic complete intersection in T . By parameters for R we mean a sequence of elements in M(R) whose length equals dim(R) and which generates an M(R)-primary ideal; by a parameter ideal in R we mean an ideal generated by parameters. By regular parameters for R we mean parameters which generate M(R). The socle soc(R) of R is defined by putting soc(R) = (0 : M(R))R; this is an ideal in R with M(R) ⊂ (0 : soc(R))R; hence it is a finite dimensional vector space over R/M(R); we define the socle-size socz(R) of R to be the vector space dimension of soc(R) over R/M(R). We define the CM type cmt(R) of R to be the minimum of socz(R/Q) taken over all parameter ideals Q in R. The local ring R is said to be GST (= Gorenstein) if every parameter ideal in it is irreducible. A noetherian ring R is said to be a local complete intersection if its localiza- tion RP is a complete intersection for every maximal ideal P in it. A noetherian ring R is said to be GST if its localization RP is GST for every maximal ideal P in it. A noetherian ring R is said to be regular if its localization RP is regular for every maximal ideal P in it. Given any ring R with total quotient ring K, for any −1 −1 R-submodule J of K we define its inverse module J by putting J = (R : J)K and we note that J −1 is again an R-submodule of K; by a fractional R-ideal J we mean an R-submodule J of K such that tJ ⊂ R for some t ∈ SR(R). By the conductor cond(R, Re) of a ring R in an overring Re of R we mean the ideal in R given by cond(R, Re) = (R : Re)R. In Socle Size Lemma (T75) we show that for any local ring R we have the following: (1) If dim(R) = d > 0 and there are parameters (x1, . . . , xd) for R such e e that upon letting Qe = (x1, . . . , xd)R we have socz(R/Qe) = 1 for all e ∈ N+ then reg(R) > 0. (2) If socz(R/Q) = 1 for every parameter ideal Q in R then R is CM. (3) If R is CM then for every parameter ideal Q in R we have cmt(R) = socz(R/Q). In GST Ring Characterization Theorem (T76) we show that a local ring is GST iff it is CM and some parameter ideal in it is irreducible. In Complete Intersection Theorem (T77) we show that if a noetherian ring is a local complete intersection then it is GST; in particular, if a noetherian ring is regular then it is a complete intersection and hence it is GST. In One Dimensional GST Ring Characterization Theorem (T78) we show that for any one-dimensional CM local ring R the following 7 conditions are equiv- alent, where P = M(R). (i) R is GST. −1 (ii) `R(P /R) = 1. (iii) For every t ∈ SR(R) ∩ P , the ideal tR is irreducible. (iv) For every t ∈ SR(R) ∩ P we have `R(tR : P )R/(tR)) = 1. (v) For every t ∈ SR(R) ∩ P we have socz(R/(tR)) = 1. −1 −1 (vi) For every fractional R-ideal J with J ∩ SR(R) 6= ∅ we have J = (J ) . −1 (vii) For every ideal I in R with I ∩ SR(R) 6= ∅ we have `R(I /R) = `R(R/I). In Something Is Twice Something Theorem (T79) we show if R is a GST local ring of dimension one such that the integral closure R∗ of R in the total quotient ring K of R is a finitely generated R-module, then upon letting C be ∗ ∗ ∗ the conductor of R in R we have `R(R /R) = `R(R/C) and hence `R(R /C) = 2`R(R/C) where all the lengths are nonnegative integers. 22 LECTURE L6: PAUSE AND REFRESH

In (Q17) on Projective Resolutions of Finite Modules we prove (T24.4) of L4§8 which says that the localization of a regular local ring at any prime ideal in it is again regular. This is part (T81.7) of the Dim-Pdim Theorem (Q17)(T81) where the projective dimension pdimRV of a module V over a ring R is defined to be the smallest length of a projective R-resolution of V if such exists; otherwise we put pdimRV = ∞. An exact sequence of R-modules is a sequence(f, W, n) of R-homomorphisms fi : Wi+1 → Wi for 0 ≤ i ≤ n ∈ N such that im(fi) = ker(fi−1) for 1 ≤ i ≤ n with ker(fn) = 0 and im(f0) = 0; the length of the sequence is n; by a projective R-resolution of V we mean such a sequence for which W0 = V and the module Wi is projective for 1 ≤ i ≤ n. By a projective R-module we mean an R-module D having the property which says that if α : E → F is any R-epimorphism of R-modules and β : D → F is any R-homomorphism then β = αγ for some R-homomorphism γ : D → E. By a finite module we mean a finitely generated module. By the global dimension gdim(R) we mean the smallest nonnegative integer n, if it exists, such that pdimRV ≤ n for every finite R-nodule V ; otherwise we put gdim(R) = ∞. In part (T81.7) of the Dim-Pdim Theorem (Q17)(T81) we show that if R is a noetherian ring with dim(R) = d ∈ N then: R is regular ⇔ gdim(R) = d ⇔ gdim(R) < ∞ ⇔ pdimRV < ∞ for every finite R-module V . What we proved previously in L4§5(O24)(22•) is now designated as the Prime Avoidance Lemma (C31.1) which says that if J ⊂ P1 ∪· · ·∪Pn where J is an ideal in a ring R and P1,...,Pn are prime ideals in R with n ∈ N+ then for some i ∈ {1, . . . , n} we have J ⊂ Pi. In (Q18) on Direct Sums of Algebras, Reduced Rings, And PIRs we define an R-algebra or algebra over a ring R to be a ring S together with a ring homomorphism φ : R → S which may be called the underlying homomorphism; S becomes an R-module by taking rs = φ(r)s for all r ∈ R and s ∈ S. Let φ1 : R → R1, . . . , φn : R → Rn be R-algebras with n ∈ N. Defining componentwise multiplication, the external direct sum module R1 ⊕· · ·⊕Rn becomes a ring so that the diagonal direct sum φ1⊕· · ·⊕φn : R → R1⊕· · ·⊕Rn is a ring homomorphism; the R-algebra with this underlying homomorphism is called the R-algebra-theoretic or multiplicative direct sum of R1,...,Rn. Every ring is a Z-algebra, and hence for any rings R1,...,Rn we get the notion of the ring-theoretic or Z-algebra theoretic direct sum R1 ⊕ · · · ⊕ Rn. We speak of a finite direct sum of rings instead of ring-isomorphic to a direct sum of rings. A PIR (= Principal Ideal Ring) is a ring in which every ideal is principal, and a SPIR (= Special PIR) is a zero-dimensional local ring which is a PIR. A reduced ring is a ring having no nonzero nilpotent element. In (Q18.1)(4) we prove the following: Let T = R1 ⊕ · · · ⊕ Rn with n ∈ N+ be a direct sum of rings. For 1 ≤ i ≤ n let fi : T → Ri be the natural projection. Let R be a nonnull reduced noetherian ring with height-zero primes Q1,...,Qn such that for 1 ≤ i ≤ n there is an epimorphism φi : R → Ri with ker(φi) = Qi. Then Φ = φ1 ⊕ · · · ⊕ φn : R → T is a ring monomorphism with fi(Φ(R)) = Ri for 1 ≤ i ≤ n and in a natural manner QR(T ) may be identified with QR(Φ(R)). In (Q18.4)(12) we prove the following: A finite direct sum of PIRs is itself a PIR. Every PIR is isomorphic to a finite direct sum of PIDs and SPIRs. A PIR whose zero ideal is primary is an SPIR or a PID according as its dimension is zero or positive. A nonnull PIR is an SPIR or a PID ⇔ its zero ideal is primary ⇔ it has exactly one height-zero prime ideal. §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 23

In (Q19) on Invertible Ideals, Conditions for Normality, And DVRs, we introduce Serre Condition (Rn): P ∈ spec(R) with htRP ≤ n ⇒ RP is regular, and Serre Condition (Sn): P ∈ spec(R) with reg(RP ) < n ⇒ RP is CM. These are conditions on a ring R. An R-submodule I of the QR(R) is invertible if IJ = R for some R-submodule J of QR(R); this is so iff II−1 = R.A DVR is the valuation ring of a real discrete valuation, i.e, a valuation whose value group is order isomorphic to Z. A valuation is trivial if its value group is 0. In Krull Normality Lemma (T82.2) we show that if P is a nonzero maximal ideal in a normal noetherian domain with P −1 6= R then P is invertible. In Krull Normality Theorem (T83) we show that a noetherian domain R is normal iff: the local ring RP is normal for every height-one prime P in R and we have R = T R . {P ∈spec(R):htRP =1} P In Normal Local Ring Lemma (T84) we show that for any local ring R we have the following. (1) If M(R) is an associated prime of the zero ideal in R then QR(R) = R and hence R is normal. (2) If R is reduced and normal then R is a domain. (3) If R is normal and M(R) is not an associated prime of the zero ideal in R then R is a domain. In Conditions for DVR (T85) we show that for any domain R we have the following. (1) R is a DVR ⇔ R is a one dimensional normal local domain ⇔ R is a one dimensional regular local domain ⇔ R is a positive dimensional local PID (where a local PID means a local ring which is also a principal ideal domain) ⇔ R is a local PID in which M(R)e with e = 0, 1, 2,... are exactly all the distinct nonzero ideals. (2) If R is the valuation ring of a nontrivial valuation v then: R is noetherian ⇔ v is real discrete. (3) If R is local then: R is the valuation ring of a nontrivial valuation ⇔ R is regular of dimension one. In Serre Criterion (T88) we show that a noetherian ring R is a reduced ring iff it satisfies conditions (R0) and (S1), whereas a noetherian ring R is a reduced normal ring iff it satisfies conditions (R1) and (S2). In (Q20) on Dedekind Domains And Chinese Remainder Theorem we introduce several related terms as follows. A DD (= Dedekind Domain) is a normal noetherian domain in which every nonzero prime ideal is maximal. A DFI (= Domain with Factorization of Ideals) is a domain in which every nonzero ideal is a product of nonzero prime ideals. A UFI (= domain with Unique Factorization of Ideals) is a DFI in which the said factorization is unique (up to order). A GFI (= domain with Group Factorization of Ideals) is a domain R such that the set of all nonzero fractional R-ideals form a (multiplicative commutative) group. In Characterization of DD (T89) we show that: DD ⇔ DFI ⇔ UFI ⇔ GFI. In Properties of DD (T91) we show that in a Dedekind Domain we have the following: (1) If R is semilocal then R is a PID. (2) If I is any nonzero ideal in R then R/I is a PIR. (3) Every ideal in R is generated by two elements. (4) The localization of R at any multiplicative set in R is again a DD. In Chinese Remainder Theorem (T93) we show that given any elements (xi)1≤i≤n and ideals (Ai)1≤i≤n in a Dedekind Domain R with xi − xj ∈ Ai + Aj for all i 6= j where n ∈ N+, there exists x ∈ R with x − xi ∈ Ai for all i. 24 LECTURE L6: PAUSE AND REFRESH

In (Q21) on Real Ranks of Valuations And Segment Completions we introduce the real rank ρ(v) of a valuation v as the order type (taken under reverse inclusion) of the set of all nonmaximal prime ideals in its valuation ring. By (Q21)(3) this equals the real rank ρ(Gv) of the value group Gv which by L2§7(D5) is the order type (under inclusion) of the set of all nonzero isolated subgroups of Gv, where isolated subgroup is a subgroup which is also a segment, i.e., a subset H of Gv such that: g ∈ Gv and h ∈ H with |g| ≤ |h| ⇒ g ∈ H. It follows that a valuation is real (i.e., its value group is order isomorphic to a subgroup of R) iff its valuation ring has no nonzero nonmaximal prime ideal. By a segment cut of a loset E we mean a nonempty subset L of E such that for all (x, y) ∈ L × (E \ L) we have x < y. By the segment completion of E we mean the set Esc of all segment cuts of E which we (linearly) order by inclusion. 0 For every x ∈ E we put DE(x) = {z ∈ E : z ≤ x} and DE(x) = {z ∈ E : z < x} sc 0 sc and this gives us order preserving injections DE : E → E and DE : E → E . We say that E is segment-complete if the map DE is surjective. Given any F ⊂ E, sc by putting DF,E(H) = {z ∈ E : z ≤ x for some x ∈ H} for all H ∈ F , we get sc sc sc . an order preserving injection DF,E : F → E ; we write F = E to mean that sc DF,E(F ) = DE(E). By the core of E we mean the set C(E) of all x ∈ E such 0 that either DE(x) = ∅ or x has an immediate predecessor in E; note that, for any x ∈ E, an immediate predecessor of x in E means an element y in E such that y < x and there is no z ∈ E with y < z < x. We call E segment-full if E is order . isomorphic to T sc for some loset T . We call E core-full if C(E)sc = E. We call E brim-full if every nonempty subset of E has a max in E, and every element of E is the max in E of some subset of C(E). By L2§3 any ordered abelian group is the value group of some valuation, and hence to characterize the real ranks of valuations it suffices to characterize the real ranks of ordered abelian groups. In Characterization Theorem for Real Ranks (T97.2) we prove that: an order type is the real rank of some ordered abelian group ⇔ it is the order type of the segment-completion of some loset ⇔ it is the order type of a segment-full loset ⇔ it is the order type of a core-full loset ⇔ it is the order type of a brim-full loset. In (Q22) on Specializations And Compositions of Valuations we define these concepts. Given any valuations v and w of a field K with Rv ⊂ Rw, we say that v is a specialization of w or w is a generalization of v, and we indicate this by writing w & v or v % w. By L4§12(R7), associated with any valuation ring R of a field K (i.e., which means the valuation ring of some valuation of K) we have the divisibility valuation γR of K with valuation ring R and value group × Γ(R) = K /U(R); if R = Rv then we may write γv (resp: Γv) instead of γRv (resp: Γ(Rv)). By L4§12(R7) we get the following assertions (6) to (8) of (Q22): (6) If v is any valuation of a field K then P 7→ (Rv)P gives an inclusion reversing bijection of spec(R) onto the set of all valuation rings of K containing Rv and this set coincides with the set of all subrings of K containing Rv, and hence these sets are losets under inclusion. (7) If w is any valuation of a field K then, letting πv : Rv → ∆v = Rv/M(Rv) be −1 the residue class epimorphism, we have that S 7→ R = πw (S) gives a bijection of the set of all valuation rings of ∆w onto the set of all valuation rings of K contained in Rw; in case S = Ru for a valuation u of ∆w, we denote the valuation γR of K by w u and call it the composition of w by u. (8) In the situation of (6) we have γv = w (v/w). §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 25

In (Q23) on UFD Property of Regular Local Domains, we prove the UFD Theorem (T102) which says that every regular local domain is a UFD. This we deduce from the Parenthetical Colon Lemma (T98) which says that for any ideals I,J in a noetherian ring R, upon letting L = (I : J)R, we have the following: (1) L = I ⇔ for every P ∈ assR(R/I) we have J 6⊂ P . (2) If R is a local ring with J = M(R) 6= 0 and L = I, then L = (I : xR)R for some x ∈ J \ J 2. (3) If R is a local domain and I,J are nonzero principal ideals in R then: there 0 0 0 0 exist nonzero principal ideals I ,J in R for which we have L = (I : J )R and I0 6⊂ M(R)L. In (Q24) on Graded Modules And Hilbert Polynomials, we introduce these polynomials in terms of what are called Hilbert functions. P Considering a graded ring S = i∈I Si with R = S0 and additive abelian monoid I, by a graded module over S (or a graded S-module) we mean an S-module P V = i∈I Vi with internal direct sum of R-modules such that for all si ∈ Si and vj ∈ vj with i, j in I we have sivj ∈ Vi+j. The family (Vi)i∈I of R-submodules of V is called the gradation of V , and I is called the type of V . Elements of Vi are called homogeneous elements of degree i. Collectively, elements of ∪i∈I Vi are called homogeneous elements. The direct sum assumption tells us that every P v ∈ V has a unique expression v = i∈I vi with vi ∈ Vi; the element vi is called the homogeneous component of v of degree i, and collectively the elements vi are called the homogeneous components of v. Note that vi = 0 for almost all i. An S-submodule U of V is said to be a homogeneous submodule if it satisfies one and hence all of the following three mutually equivalent conditions: P (i) U = i∈I (U ∩ Vi) (as additive groups). (ii) The homogeneous components of every element of U belong to U. (iii) U is generated (as an S-module) by homogeneous elements. As in the case of an element, we call Vi the homogeneous component of V of degree i, and we note that for the homogeneous component Ui of U of degree i we have Ui = U ∩ Vi. We use i-th homogeneous component as a synonym for homoge- neous component of degree i. Also we may say component instead of homogeneous component. By a finite graded S-module we mean a graded S-module V which is finitely generated as an S-module. In view of (1), this is equivalent to saying that V is generated by a finite number of homogeneous elements. Note that a homogeneous ideal in S is nothing but a homogeneous S-submodule of S where we regard S as a graded S-module in an obvious manner. Now let S be a naturally graded ring, i.e., suppose that I = N. Assume that R = S0 is an artinian ring. Given any graded module V over S, we put µ bh (S, V, n) = `R(Vn) µ and we call the function bh (S, V ): N → N ∪ {∞}, defined by n 7→ `R(Vn), the modulized Hilbert function of S at V . In particular, given any homogeneous ideal Q in S, we note that S/Q is a graded module over S, and we put µ bh(S, Q, n) = bh (S, S/Q, n) = `R(Sn/Qn) and we call the function bh(S, Q): N → N ∪ {∞}, defined by n 7→ `R(Sn/Qn), the Hilbert function of S at Q. Note that if R is a field then `R(Vn) = [Vn : R] and `R(Sn/Qn) = [Sn/Qn : R]. In (Q26) to (Q28) we prove the following: 26 LECTURE L6: PAUSE AND REFRESH

HILBERT FUNCTION THEOREM (T103). Let X1,...,XN with N ∈ N be indeterminates over an artinian ring R. Let S = R[X1,...,XN ] be the naturally graded polynomial ring with R = S0. Let V be a finite graded module over S. Then for every nonnegative integer n we have

`R(Vn) < ∞ for all n ∈ N and there exists a unique univariate polynomial hµ(S, V, Z) in an indeterminate Z with coefficients in Q such that hµ(S, V, n) = bhµ(S, V, n) for all n >> 0. Moreover, upon letting tµ(S, V ) = the Z-degree of hµ(S, V, Z) we have tµ(S, V ) < N, and if d = tµ(S, V ) ∈ N then we have µ X g (S, V )Z(Z − 1) ... (Z − i + 1) hµ(S, V, Z) = i with gµ(S, V ) ∈ i! i Z 0≤i≤d and upon letting µ µ g (S, V ) = gd (S, V ) (and assuming d = tµ(S, V ) ∈ N) we have gµ(S, V ) > 0, and if R is a field then (without assuming tµ(S, V ) ∈ N) we have ( dim(S/(0 : V ) ) − 1 if dim(S/(0 : V )S) > 0 tµ(S, V ) = S −∞ otherwise.

µ µ µ µ We respectively call the quantities h (S, V, Z), t (S, V ), gi (S, V ), g (S, V ), the modulized Hilbert polynomial, the modulized Hilbert transcendence, the modulized i-th Hilbert subdegree, and the modulized Hilbert degree of S at V . [Note that, “for all n >> 0” means for all large (enough) n, i.e., for all integers n such that n ≥ n0 for some n0 ∈ N. Also note that if V = S/Q where Q is a homogeneous ideal in S then (0 : V )S = Q]. Given any homogeneous ideal Q in S, with S and R as in (T103), we put h(S, Q, Z) = hµ(S, S/Q, Z)  t(S, Q) = tµ(S, S/Q) µ gi(S, Q) = gi (S, S/Q)  g(S, Q) = gµ(S, S/Q) and we again respectively call these quantities the Hilbert polynomial, the Hilbert transcendence, the i-th Hilbert subdegree, and the Hilbert degree of S at Q. In (Q25) we illustrate the case when Q is the ideal of a hypersurface in some projective space. In (Q26) to (Q28) we include some material to be used in the proof of (T103), which is completed in parts (Q28.3) and (Q28.5) of (Q28). §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 27

In (Q27) on Homogeneous Normalization, we homogenize Normalization Theorem (Q10)(T46) by proving assertions (T104) to (T106) stated below. We P define the positive portion Ω(V ) of a graded module V = Vi over a graded P P i∈I ring S = i∈I Si by putting Ω(V ) = {i∈I:i>0} Vi, and we call a submodule U of V irrelevant or relevant according as Ω(S) ⊂ radV U or Ω(S) 6⊂ radV U. In particular these definition apply to ideals in homogeneous rings. In (C40) we make further comments about them. IRRELEVANT IDEAL LEMMA (T104). Let S be a homogeneous ring over a ring R, and let z1, . . . , zm be a finite number of homogeneous elements in S \ R. Then S is integral over R[z1, . . . , zm] iff the ideal (z1, . . . , zm)S is irrelevant. HOMOGENEOUS NORMALIZATION THEOREM (T105). In the naturally graded polynomial ring S = k[X1,...,XN ] in indeterminates X1,...,XN over a field k with N ∈ N we have the following. (T105.1) Given any homogeneous element ZN in S \ k, there exist homogeneous elements Z1,...,ZN−1 in S \ k such that S is integral over k[Z1,...,ZN ]. (T105.2) Given any nonunit homogeneous ideal J in S, there exists a positive integer c together with nonzero homogeneous polynomials Y1,...,YN in S of degree c such that S is integral over D = k[Y1,...,YN ] and J ∩D = (Yr+1,...,YN )D where r = dim(S/J). (T105.3) Given any finite sequence J0 ⊂ · · · ⊂ Jm of nonunit homogeneous ideals in S with m ∈ N, there exists a positive integer c together with nonzero homogeneous polynomials Y1,...,YN in S of degree c such that S is integral over

D = k[Y1,...,YN ] and Ji ∩ D = (Yri+1,...,YN )D for 0 ≤ i ≤ m where we have ri = dim(S/Ji). Moreover, for 0 ≤ i ≤ m and any such polynomials Y1,...,YN , upon letting φi : S → S/Ji be the residue class epimorphism, the ring φi(S) is integral over the subring φi(D), and the latter ring is naturally isomorphic to the polynomial ring Ti = k[X1,...,Xri ] by the isomorphism ψi : Ti → φi(D) such that ψi(κ) = φi(κ) for all κ ∈ k and ψi(Xj) = φi(Yj) for 1 ≤ j ≤ ri. HOMOGENEOUS RING NORMALIZATION THEOREM (T106). Given any homogeneous ring over a field k, and given any finite sequence I1 ⊂ · · · ⊂ Im of nonunit homogeneous ideals in A with m ∈ N, there exists a positive integer c together with nonzero homogeneous elements Y1,...,Yr in A of degree c with dim(A) = r such that the elements Y1,...,Yr are algebraically independent over k and such that the ring A is integral over the subring D = k[Y1,...,Yr] and

Ij ∩ D = (Yrj +1,...,Yr)D for 1 ≤ j ≤ m where rj = dim(A/Ij). Moreover, for 1 ≤ j ≤ m and any such elements Y1,...,Yr, upon letting φj : A → A/Ij be the residue class epimorphism, the ring φj(A) is integral over the subring φj(D), and the latter ring is naturally isomorphic to the polynomial ring Tj = k[X1,...,Xrj ] by the isomorphism ψj : Tj → φj(D) such that ψj(κ) = φj(κ) for all κ ∈ k and ψj(Xi) = φj(Yi) for 1 ≤ i ≤ rj. COMMENT (C40). [Irrelevant Ideals and Integral Dependence]. To apply (T104) we use the following facts. Let S be a homogeneous ring over a ring R, let z1, . . . , zm be a finite number of homogeneous elements in S \ R, and for 1 ≤ i ≤ m di let ti = zi with di ∈ N+. Then S/R[z1, . . . , zm] is integral iff S/R[t1, . . . , tm] is integral, and the ideal (z1, . . . , zm)S is irrelevant iff the ideal (t1, . . . , tm)S is irrelevant. If zi ∈ Sdi with di ∈ N+ for 1 ≤ i ≤ m then upon letting d = d1 . . . dm d/di and ti = zi we get ti ∈ Sd. So in checking irrelevancy or integralness we may assume z1, . . . , zm are homogeneous of the same positive degree. 28 LECTURE L6: PAUSE AND REFRESH

In (Q29) on Linear Disjointness And Intersection of Varieties we prove Dimension of Intersection Theorem (T112) which says that, if the intersection U of any two varieties V and W in the N-dimensional affine space over a field k is nonempty, then dim(U) ≥ dim(V ) + dim(W ) − N. As preparation for (T112), we prove Codimension of Product Theorem (T111) which says that, given any two irreducible affine varieties V and V 0 (in different affine spaces), the codimension of any irreducible component of their product V × V 0 equals the sum of their codimensions; the codimension of a variety V in affine N-space means N−dim(V ). As preparation for (T111), we prove some properties of linear disjointness; given any subrings R and R0 of a ring S, such that R ∩ R0 contains a subfield k of S, 0 we say that R and R are linearly disjoint over k to mean that: if (xi)1≤i≤m are any finite number of elements of R which are linearly independent over k and 0 0 (xj)1≤j≤m0 are any finite number of elements of R which are linearly independent 0 0 over k, then the mm elements xixj of S are linearly independent over k. This linear disjointness is particularly significant for the ring-theoretic composi- tum of R and R0. The ring-theoretic compositum of subrings R,R0 of a ring S is defined to be the smallest subring of S which contains both R and R0. Equiv- alently it may be defined to be the intersection of all subrings of S which contain both R and R0. For any subring k of the ring R ∩ R0, the said compositum clearly coincides with k[R,R0], where we recall that for any subring k of a ring S, and any subset E of S, by k[E] we denote the subring of S consisting of all finite sums P i1 in ai1...in z1 . . . zn with n, i1, . . . , in in N, ai1...in in k, and z1, . . . , zn in E. In all this, instead of two subrings R,R0 of S, we may take any set (or family) of subrings of S. Here we may everywhere change “ring” to “field” with the understanding that k[E] is replaced by k(E) and the finite sums are replaced by quotients of finite sums with nonzero denominators. Usually the adjective “field-theoretic” is dropped from the phrase “field-theoretic compositum.” In (Q30) on Syzygies And Homogeneous Resolutions we give Hilbert’s original version of (Q17). In (Q17) we discussed projective resolutions of modules. In their original incarnation, as was conceived by David Hilbert, they started out as free resolutions of homogeneous ideals in polynomial rings; in that set-up, what we have called the pdim (= projective dimension) of a module corresponds to the hdim (= homogeneous dimension) of a graded module; for details see the text of (Q30). All this gives rise to syzygies, originally an astronomical term to indicate the waning and waxing of the phases of the moon, adapted by Sylvester to indicate a mathematical object. Actually in (Q17) we have already worked with syzygies except that we did not give them a name. So what are they? Now in the definition of a projective R-resolution (f, W, n) given in (Q17), by not requiring the last module Wn+1 to be projective, we get hold of the definition of a preprojective R-resolution. Given any noetherian ring R, any finite R-module V , and any n ∈ N, we can find a preprojective R-resolution (f, W, n) of V of length n, and we call Wn+1 an n-th 0 R-syzygy of V . We can see that if Wn+1 is any other n-th R-syzygy of V then the 0 0 modules Wn+1 and Wn+1 are R-equivalent, in symbols Wn+1 ∼R Wn+1, by which we mean that for some finite projective R-modules U, U 0 we have an isomorphism 0 0 n Wn+1 ⊕ U ≈ Wn+1 ⊕ U of R-modules. By syzRV we denote the collection of all n-th R-syzygies of V . §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 29

In (Q31) on Projective Modules Over Polynomial Rings, we give two proofs (due to Suslin) of the freeness of any finite projective module over a finite variable polynomial ring over a ground ring K. The first applies when K is a PID, and the second works when K is a field. The main technique in the second proof is the idea of completing unimodular rows. A unimodular row over a ring R n is an n-tuple r = (r1, . . . , rn) ∈ R which generates the unit ideal in R. This is completable means it is the first column of some matrix A = (Aij) in GL(n, R), i.e., Ai1 = ri for 1 ≤ i ≤ n. The general linear group GL(n, R) consists of all n×n matrices over R whose determinant is a unit in R. The special linear group SL(n, R) consists of all n × n matrices over R whose determinant is 1. The special elementary group SE(n, R) is the subgroup of SL(n, R) generated by all elementary matrices, i.e., matrices from the identity matrix by changing at most one zero entry to a nonzero entry. The general elementary group GE(n, R) is the subgroup of GL(n, R) generated by SL(n, R) and all restricted dilatation matrices. As a byproduct of the second proof we establish Suslin’s Theorem (T130) which says that for n > 2 and R = K[X1,...,XN ] is a finite variable polynomial ring over a field K we have SL(n, R) = SE(n, R). In (T131) we give a presentation of Cohn’s example showing that (T130) is not true for n = N = 2. The proof of (T130) uses Theorem (T129) on properties of Mennike Symbols which are certain equivalence classes of SL(3, n) mod SE(3, n). In (T132) to (T134) we discuss permutation matrices, i.e, n × n matrices of the form T (n, σ) = (T (n, σ)ij) where σ(i) = 1 or 0 according as i = σ(j) or i 6= σ(j) for some permutation σ ∈ Sn = the symmetric group on n letters. In (Q32) on Separable Extensions And Primitive Elements, we establish Theorem L4§8(T20) by proving that, given any affine domain T = k[x1, . . . , xN ] over an algebraically closed field k, upon letting L = k(x1, . . . , xN ), there exist P k-linear combinations yi = 1≤j≤N Aijxj with Aij ∈ k such that y1, . . . , yr is a transcendence basis of L/k and L = k(y1, . . . , yr+1). In (T144) we give a stronger version of this result. Firstly, in (T143) we show that suitable k-linear combinations y1, . . . , yr of x1, . . . , xN constitute a separating transcendence basis of L/k, i.e., a transcendence basis so that L is separable algebraic over K = k(y1, . . . , yr) as de- fined below. Secondly, in (T140) we show that a suitable k-linear combination yr+1 of x1, . . . , xN provides a primitive element of L over K, i.e, an element for which L = K(yr+1). Actually, in (T143) we show that suitable k-linear combinations y1, . . . , yr of x1, . . . , xN provide a separating normalization basis of T/k, i.e., a separating transcendence basis of L/K such that the ring T is integral over the ring R = k[y1, . . . , yr]. Using (T140), in (T141) we show that the integral closure of a normal noetherian domain in a finite separable algebraic extension of its quotient field is a finite module over that domain, and hence the integral closure S of R in L is a finite R-module; it follows that S, which is also the integral closure of T in L, is a finite T -module. As an aid to (T141), in (T136) we embed the integral ∗ closure in a ring obtained by inverting the modified discriminant DiscY (f) of a univariate polynomial

n n−1 f(Y ) = a0Y + a1Y + ··· + an which is defined by putting

∗ n(n−1)/2 DiscY (f) = (−1) DiscY (f). 30 LECTURE L6: PAUSE AND REFRESH

According to L1§1 and L1§8, a univariate polynomial f = f(Y ) over a field K is separable if it is devoid of multiple roots; if f is not separable then it is inseparable. An element y in an overfield of K is separable over K if f(y) = 0 for some separable f over K, i.e., equivalently, if y/K is algebraic and its minimal polynomial over K is separable; if y/K is algebraic but not separable then y/K is inseparable; if ch(K) = p 6= 0 (i.e., if the characteristic of K is a prime number p) pu and y ∈ K for some u ∈ N+ then y/K is purely inseparable; if ch(K) = 0 then y/K is purely inseparable means y ∈ K. A field extension L/K is separable (resp: purely inseparable) means every element of L is separable (resp: purely inseparable) over K. A field extension L/K is inseparable means it is algebraic but not separable. A field is perfect means every algebraic extension of it is separable; otherwise it is imperfect. In (T139) we show that every characteristic zero field is perfect and so is every finite field. In (T142) we show that every algebraic function field over a perfect ground field is separably generated according to the following definitions. By an algebraic function field L (of r variables) over a ground field k we mean a finitely generated field extension of k (with trdegkL = r). The field extension L/k is separably generated means there exists a transcendence basis y1, . . . , yr of L/k such that L is a separable algebraic field extension of K = k(y1, . . . , yr); we then call y1, . . . , yr a separating transcendence basis of L/k; in case of r = 1 we call y1 a separating transcendental of L/k. In Quests (Q33) to (Q35) we continue the geometrizing project of L3§§2-5 and L4§§8-9. Out of these Quests, (Q33) is on Restricted Domains And Projective Normalization, (Q34) is on Homogeneous Localization, and (Q35) is on Simplifying Singularities by Blowups. In (Q33) we define a restricted domain to be a noetherian domain T such that the integral closure of T in any finite algebraic extension of the quotient field L of T is a finite module over T , and we define a strongly restricted domain to be a restricted domain T such that every affine domain over T is again a re- stricted domain. In Theorem (T145) we show that any affine domain over a field is a restricted domain, i.e., a field is a strongly restricted domain. In Theorems (T146) and (T147) we respectively use this to discuss “normalizations” of affine and projective varieties in finite algebraic field extensions of their function fields. In (Q34.1) we define the Projective Spectrum proj(A) of any subintegrally P graded ring A = i∈I Ai, where I is an additive submonoid of Z, by putting proj(A) = the set of all relevant homogeneous prime ideals in A. In (Q34.2) we define the homogeneous quotient field K(D) of a semihomo- P geneous domain D = Dn by putting n∈N [ × K(D) = {yn/zn : yn ∈ Dn and zn ∈ Dn } n∈N and we note that this is a subfield of QF(D). For any P ∈ proj(D) we define the homogeneous localization D[P ] of D at P to be the subring of K(D) given by [ D[P ] = {yn/zn : yn ∈ Dn and zn ∈ Dn \ P } n∈N and we note that this is a quasilocal domain with quotient field K(D) and [ M(D[P ]) = {yn/zn : yn ∈ P ∩ Dn and zn ∈ Dn \ P }. n∈N §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 31

Assuming D 6= 0, we define the modelic proj W(D) of D by putting ( the set of all homogeneous localizations D W(D) = [P ] with P varying over proj(A). Note that then

W(D) = W(D0,D1) and if (xl)l∈Λ is any family of generators of D1 as a module over D0 then

W(D) = W(D0;(xl)l∈Λ) and hence in particular if D1, as a module over D0, is generated by a finite number × of generators x1, . . . , xN+1 in D1 then N+1 W(D) = ∪i=1 V(D0[x1/xi, . . . , xN+1/xi]) where according to the notation of L4§§8-9: we define the modelic spec V(A) of any domain A by putting

V(A) = the set of all localizations AP with P varying over spec(A), we define the modelic blowup W(A, P ) of any subdomain A of a field K and any nonzero A-submodule P of K by putting [ W(A, P ) = V(A[P x−1]), 06=x∈P and we define the modelic proj W(A;(xl)l∈Λ) of any family (xl)l∈Λ of elements in K, with xj 6= 0 for some j ∈ Λ, by putting [ W(A;(xl)l∈Λ) = V(A[(xl/xj)l∈Λ]);

j∈Λ with xj 6=0 if Λ is a finite set, say Λ = {1, . . . , n}, then we may write W(A; xl, . . . , xn) instead of W(A;(xl)l∈Λ) and call it the modelic proj of (x1, . . . , xn) over A. In (Q35) we define the dominating modelic blowup W(R,P )∆ of a nonzero ideal P in a quasilocal domain R by putting W(R,P )∆ = {R0 ∈ W(R,P ): R0 > R} where a quasilocal ring R0 dominates a quasilocal ring R, in notation R0 > R or R < R0, means R is a subring of R0 with M(R) = M(R0) ∩ R. We are particularly interested in the case when R is a regular local domain and P = R∩M(S) for some S ∈ V(R) such that S has a simple point at R by which we mean that R/P is a regular local domain. Any R0 ∈ W(R,P )∆ is now called a monoidal transform of (R,S), and by the (R, S, R0)-transform of any nonzero principal ideal J in 0 0 0 0 d 0 R we mean the ideal J in R given by J (M(R)R ) = JR where d = ordRJ. Also we call (R0,J 0) a monoidal transform of (R, J, S). We define E(R,J) by putting E(R,J) = the set of all T ∈ V(R) with ordT J = d, and we call this the equimultiple locus of (R,J); for any i ∈ N we let Ei(R,J) denote the set of all 0 0 i-dimensional members of E(R,J). Upon letting d = ordR0 J , in (T158)(2) we show that if S ∈ E(R,J) then d0 ≤ d; geometrically this says that the multiplicity does not increase when we blowup an equimultiple simple center. For any field K we put R(K) = the set of all valuation rings of K 32 LECTURE L6: PAUSE AND REFRESH i.e., valuation rings with quotient field K, and for any subdomain A of K we put R(K/A) = the set of all valuation rings of K/A i.e., valuation rings of K which contains A; we respectively call these sets the Riemann-Zariski space of K and the Riemann-Zariski space of K/A. Now W(R,P ) is a complete model of K/R with QF(R) = K, i.e., every V ∈ R(K/P ) dominates a unique R0 ∈ W(R,P ) (which is called the center of V on W(R,P )); it follows that if V dominates R then R0 ∈ W(R,P )∆; in this case we call R0 the monoidal transform of (R,S) along V ; similarly we also call (R0,J 0) the monoidal transform of (R, J, S) along V . In case of S = R we may say quadratic transform of R or (R,J) instead of monoidal transform of (R,S) or (R, J, S) respectively. To explain the above concept of “complete model” in greater detail we reproduce the definitions made in L4§9. For any field K we put R0(K) = the set of all quasilocal domains with quotient field K and for any subdomain A of K we put R0(K/A) = the set of all members of R0(K) which contain A and we call these the quasitotal Riemann-Zariski space of K and the quasitotal Riemann-Zariski space of K/A respectively. We also put R00(K) = the set of all quasilocal domains which are subrings of K and R00(K/A) = the set of all members of R00(K) which contain A and we call these the total Riemann-Zariski space of K and the total Riemann- Zariski space of K/A respectively. Domination converts R(K),..., R00(K/A) into posets (= partially ordered sets). A valuation v dominates a quasilocal ring R means Rv dominates R. By a premodel of K (resp: K/A) we mean a nonempty subset E of R0(K) (resp: R0(K/A)). The premodel E is irredundant means any member of R(K) (resp: R(K/A)) dominates at most one member of E. A quasilocal domain S can dominate at most one member R of an irredundant premodel E, and if R exists then we call it the center of S on E. By a semimodel (resp: model) of K/A we mean an irredundant premodel E of K/A which can be expressed as a union E = ∪l∈ΛV(Bl) for some family (resp: finite family) (Bl)l∈Λ of subrings Bl of K with quotient field K such that Bl is an overring of (resp: affine ring over) A. Note that if B is any subring of K with quotient field K such that B is an overring of (resp: affine ring over) A then V(B) is a semimodel (resp: model) of K/A; we call it an affine semimodel (resp: affine model) of K/A. By a projective model of K/A we mean premodel E of K/A such that E = W(A; x1, . . . , xn) for some finite number of elements x1, . . . , xn in an overfield of K at least one of which is nonzero. A quasilocal ring S dominates a set E of quasilocal rings (or E is dominated by S) means S dominates some member of E; we may indicate this by writing E < S (or S > E). A set E0 of quasilocal rings dominates a set E of quasilocal rings (or E is dominated by E0) means every member of E0 dominates E; we may indicate this by writing E < E0 (or E0 > E). A set E0 of quasilocal rings properly dominates a set E of quasilocal rings (or E is properly dominated by E0) means E0 dominates E and every member of E is dominated by some member of E0. A semimodel or model of K/A is complete means it is dominated by R(K/A). §5: SUMMARY OF LECTURE L5 ON PROJECTIVE VARIETIES 33

Every projective model of K/A is easily seen to be a complete model of K/A. A model E (of K/A) is said to be normal (resp: noetherian) if every R ∈ E is normal (resp: noetherian). A model E is said to be nonsingular if every R ∈ E is a regular local ring. By the dimension dim(E) of a model E we mean max{dim(R): R ∈ E}; note that then dim(E) ∈ N ∪ {∞}. What we said about normalization in the above summary of (Q33) can now be said more precisely thus. Given any quasilocal subring R of any domain K, by N(R, K) we denote the set of all S ∈ V(R) such that S dominates R where R is the integral closure of R in K, and we call this the normalization of R in K. Likewise, given any set E of quasilocal subrings of a domain K, we introduce the normalization of E in K by putting N(E, K) = ∪R∈EN(R, K). In (T145) and (T147) we show that if E is a projective model of K/k where K is an algebraic function field over a field k, and K is a finite algebraic field extension of K, then N(E, K) is a projective model of K/k. Given any irreducible curve C in the projective plane over k, as an affine equation of C we can take a bivariate irreducible polynomial f = f(X,Y ) in k[X,Y ]\k[X]. Now we can take E to be the “modelic” projective line V(k[X]) ∪ V(k[1/X]). We may then take K to be k(X) and identify K with the function field of C given by QF(k[X,Y ]/fk[X,Y ]). Now upon letting C = N(E, K), by Conditions (Q19)(T85) we see that C is a “modelic” nonsingular projective curve. This is Dedekind’s 1882 method of desingularizing C. Riemann’s 1851 method amounts to saying that, again by (Q19)(T85), N(E, K) coincides with R(K/k). Max Noether’s 1877 method, which is the only one so far capable of higher-dimensional generalization, uses monoidal transformations. To give the definitions of some concepts which are relevant in Noether’s method, let R be an n-dimensional regular local domain, let E ⊂ V(R), and let I,J be nonzero principal ideals in R. We say that E has a normal crossing at R if there exists a regular system of parameters x1, . . . , xn for R such that for every S ∈ E we have ySR = R ∩ M(S) for some subset yS of {x1, . . . , xn}, and we say that E has a strict normal crossing at R if E has a normal crossing at R and E contains at most two elements. We say that I has a normal crossing at R if {S0 ∈ V(R) : dim(S0) = 1 and IS0 6= S0} has a normal crossing at R. We say that (E,I) has a normal crossing at R if E ∪ {S0 ∈ V(R) : dim(S0) = 1 and IS0 6= S0} has a normal crossing at R, and we say that (E,I) has a strict normal crossing at R if (E,I) has a normal crossing at R and E contains at most two elements. Given S ∈ V(R), we say that (S, I) has a normal crossing at R if ({S},I) has a normal crossing at R. We say that (J, I) has a quasinormal crossing at R if I has a normal crossing at R and for every nonzero principal prime ideal P in R with J ⊂ P we have that PI has a normal crossing at R. We say that I has a quasinormal crossing at R if (I,R) has a quasinormal crossing at R. Given S ∈ V(R), we say that (S, I) has a pseudonormal crossing at R if S has a simple point at R and for every nonzero principal prime ideal P in R with I ⊂ P we have that {S, RP } has a normal crossing at R. We say that (E,I) has a pseudonormal crossing at R if I has a quasinormal crossing at R and for every S ∈ E we have that (S, I) has a pseudonormal crossing at R.(R,J) is resolved means there exists a nonnegative 0 0 0d integer d and a nonzero principal ideal J in R with ordRJ ≤ 1 such that J = J . Domination and Subgroups. In domination, R < S indicates subset. In subgroups, H ≤ G indicates subset while H < G indicates proper subset. 34 LECTURE L6: PAUSE AND REFRESH

§6: DEFINITIONS AND EXERCISES

In Exercises (E1) to (E6) we shall give hints for completing the proofs of the six Galois Theory Theorems (T1) to (T6) of L1§8. In Exercises (E7) to (E50) we shall deal further with questions of Galois Theory, Group Theory, and Field Theory. In the remaining Exercises we shall deal with questions arising out of Lecture L5. For the basic definitions of separable and inseparable extensions etc., used in the following Exercises, see Lecture L1 and §5(Q32) of Lecture L5.

EXERCISE (E1). [Fundamental Theorem of Galois Theory]. Prove (T1) of L1§8. In other words, let L∗/K be a Galois extension, i.e., L∗ is the splitting field of a nonconstant separable univariate polynomial f(Y ) over the field K. Let Λ be the set of all subfields L of L∗ with K ⊂ L and let us put G(L) = Gal(L∗,L), i.e, G(L) is the Galois group of L∗/L which means G(L) is the group of all automorphisms g of L∗ such that g(y) = y for all y ∈ L; note that L∗/L is a Galois extension because L∗ is clearly the splitting field of f(Y ) over L. Let Γ be the set of all subgroups H ∗ of G(L ) and let us put F (H) = fixL∗ (H), i.e., F (H) is the fixed field of H which means F (H) = {z ∈ L∗ : g(z) = z for all g ∈ H}. Prove that then (1) L 7→ G(L) gives a bijection Λ → Γ whose inverse is given by H 7→ F (H). Moreover show that (2) for every L ∈ Λ we have |G(L)| = [L∗ : L]. HINT. Given any L and L0 in Λ with L ⊂ L0, by L5§5(Q32)(T140) we can find x ∈ L0 with L0 = L(x). Let e(Y ) be the minimal polynomial of x over L. Then Q ∗ e(Y ) = z∈W (Y − z) where W is a subset of some overfield of L with x ∈ W . By L5§5(Q32)FACT(15) we know that e(Y ) is separable and hence upon letting m to be the Y -degree of e(Y ) we have |W | = m ∈ N+. First for a moment suppose that L0 = L∗. Then by (E2) below we have W ⊂ L∗ and g 7→ g(x) gives a bijection G(L) → W , and hence |G(L)| = [L∗ : L]. This proves (2). The said bijection also shows that if H < G(L) then upon letting W 0 = {g(x): 0 0 0 0 0 Q g ∈ H} and m = |W | we get m ∈ N+ with m < m. Let e (Y ) = g∈H (Y −g(x)). 0 0 0 0 Then e (Y ) ∈ F (H)[Y ] with e (x) = 0; since degY e (Y ) = m < m and e(Y ) is the minimal polynomial of x/L, we must have e0(Y ) 6∈ L[Y ]; therefore L $ F (H). This shows that (3) for every L ∈ Λ and H < G(L) we have L $ F (H). Next for a moment suppose that L 6= L0. Then we can find x0 ∈ W with x0 6= x. By (E7) below there exists g ∈ G(L) with g(x) = x0, and hence G(L0) < G(L). This shows that (4) for any L, L0 in Λ with L $ L0 we have G(L0) < G(L). In view of L5§6(D2), to prove (1) it suffices to show that (1*) for every L ∈ Λ we have F (G(L)) = L, and (2*) for every H ∈ Γ we have G(F (H)) = H. To prove (2*), given any H ∈ Γ, let L = F (H); then L ∈ Λ with H ⊂ G(L), and hence by (3) we must have H = G(L). To prove (1*), given any L ∈ Λ, let L0 = F (G(L)); then L ⊂ L0 with G(L0) = G(L), and hence by (4) we must have L = L0.

DEFINITION (D1). [Normal Extensions]. By a normal (field) extension of a field K we mean an algebraic extension L∗ of K such that whenever a nonconstant irreducible univariate polynomial e(Y ) ∈ K[Y ] has a root in L∗ then it has all its roots in L∗, i.e., if e(x) = 0 for some x ∈ L∗ and e(x0) = 0 for some x0 in an overfield of L∗ then x0 ∈ L∗. It is clear that if L∗ is a normal extension of K then L∗ is a splitting field over K of a family J of nonzero univariate polynomials over K. It §6: DEFINITIONS AND EXERCISES 35 is also clear that if L∗ is a finite normal extension of K then L∗ is a splitting field of one single polynomial f(Y ); namely L∗ is generated over K by a finite number of algebraic elements x1, . . . , xr and we can take f(Y ) to be the product of their minimal polynomials. The converse is proved in the following Exercise.

EXERCISE (E2). [Splitting Fields]. Let f(Y ) be a nonconstant univariate polynomial with coefficients in a field K, let L∗ be a splitting field of f over K, let L be a field between K and L∗, let e(Y ) be a nonconstant irreducible polynomial in Y with coefficients in L, and let x, x0 be roots of e(Y ) in an overfield of L∗ with x ∈ L∗. Show that then x0 ∈ L∗, and the L-isomorphism of root fields φ : L(x) → L(x0) given by x 7→ x0 can be extended to some g ∈ Gal(L∗,L). HINT. You are only being asked to give a more TRANSPARENT version of a part of (R6) to (R8) of L2§5 which was repeated in several sentences of L2§6 ending with the phrase “φ can be extended to an isomorphism of the splitting field LJ of 0 0 0 J = {f} over L(x) onto the splitting field LJ 0 of J = {f} over L(x ).” Note that since g is an L-isomorphism and L∗ is a splitting field of f over L, we must have g(L∗) = L∗. Therefore L∗(x0) = L∗ and g ∈ Gal(L∗,L). Turning to the more TRANSPARENT version, clearly L∗ and L∗(x0) are split- ting fields of f over L(x) and L(x0) respectively, and hence φ can be extended to an isomorphism g : L∗ → L∗(x0). Since L∗ is a splitting field of f over K, its image g(L∗) must be a splitting field of f over K. Therefore x0 ∈ g(L∗), and hence L∗(x0) = L∗. It follows that g ∈ Gal(L∗,K) with g(x) = x0.

DEFINITION (D2). [Lagrange Resolvent]. Let n ∈ N+ and let K be a field which contains n distinct n-th roots of 1, and note that then by L5§5(Q27)(C41) and L5§5(Q32)(T137) n is not divisible by the characteristic of K, and K contains a primitive n-th root ζ of 1, i.e., there exists ζ ∈ K such that ζn = 1 but ζm 6= 1 for all m < n in N+. Let L be any n-cyclic extension of K, i.e., a Galois extension of K such that Gal(L, K) is a cyclic group of order n, and let σ be a generator of the said Galois group. For any i ∈ N and θ ∈ L, we define the Lagrange resolvent (ζi, θ) of ζi and θ relative to σ by putting X (ζi, θ) = σj(θ)ζij 0≤j≤n−1 and we note that clearly σ((ζi, θ)) = (ζi, θ)ζ−i and hence σ((ζi, θ)n) = (ζi, θ)n. Now for every i ∈ {1, . . . , n − 1} we have ζi − 1 6= 0 with X ζin − 1 ζij = = 0 ζi − 1 0

By L5§5(Q32)(T140) we can find u ∈ L with L = K(u), and then by the above i equation we get L = K((ζ , u)i∈A(u)). It follows that if p is any prime divisor of n and v(p) is the largest integer such that pv(p) divides n then for some i(p) ∈ A(u) v(p) i(p) n/pv(p) we must have GCD(p , i(p)) = 1. Let up = (ζ , u) . Then upon letting Q α = up, where the product is taken over all prime divisors p of n, we get L = K(α) with 0 6= αn = a ∈ K.

EXERCISE (E3). [Cyclic Extensions of degree prime to characteristic]. Prove (T2) of L1§8. In other words, let n ∈ N+ and let K be a field which contains n distinct n-th roots of 1. In (D2) above we have shown that any n-cyclic extension of K is obtained by adjoining an n-th root of some nonzero element a of K, i.e., it is of the form K(α) with 0 6= αn = a ∈ K. Show that conversely, for any 0 6= a ∈ K, the splitting field L of Y n − a over K is an m-cyclic extension of K where m is the largest divisor of n for which the polynomial Y m − a is irreducible in K[Y ]. [Note that L1§8(T3) follows from this Exercise and the above Exercise (E1)]. n Q HINT. Now Y − a = η∈U (Y − αη) with α ∈ L and U = the group of all n-th roots of 1 in K. Therefore L = K(α) and g 7→ g(α)/α gives a monomorphism Gal(L, K) → U. By (D2) above, U is cyclic of order n. So we are done by (E7) below.

DEFINITION (D3). [Elementary Symmetric Functions]. To relate the roots and the coefficients of a polynomial, consider the monic polynomial F (Y ) of degree n ∈ N+ in an indeterminate Y with coefficients in L = k(Z1,...,Zn), where Z1,...,Zn are indeterminates over a field k, given by Y n X i n−i F (Y ) = (Y − Zi) = Y + (−1) Ei(Z1,...,Zn)Y . 1≤i≤n 1≤i≤n We call Ei = Ei(Z1,...,Zn) ∈ k[Z1,...,Zn] the i-th elementary symmetric function of Z1,...,Zn and we note that  E = Z + ··· + Z  1 1 n  E2 = Z1Z2 + ··· + Z1Zn + Z2Z3 + ··· + Zn−1Zn  E3 = Z1Z2Z3 + ··· + Zn−2Zn−1Zn . .   En−1 = Z1 ...Zn−1 + ··· + Z2 ...Zn  En = Z1 ...Zn and more generally X Y Ei = Zb B∈A(i) b∈B where A(i) = the set of all subsets of {1, . . . , n} of size i with n n(n − 1) ... (n − i + 1) |A(i)| = = . i i! In particular 0 and 1 are the only elements of k which occur as coefficients in Ei. Thus Ei makes sense when k is replaced by any nonnull ring. In the expressions of §6: DEFINITIONS AND EXERCISES 37

F (Y ) and Ei we may substitute for Z1,...,Zn any values z1, . . . , zn in any overfield of k to get f(Y ) and ei, and then ei will be the i-th elementary symmetric functions of the roots z1, . . . , zn of f(Y ) whose coefficients will be ±ei depending on the parity of i. The functions Ei are obviously symmetric in the following sense.

DEFINITION (D4). [Symmetric Functions]. Continuing with (D3) above, let Sn be the group of all permutations of {1, . . . , n}. Call a function u(Z1,...,Zn) ∈ L symmetric if for every σ ∈ Sn we have u(Zσ(1),...,Zσ(n)) = u(Z1,...,Zn). Let K = k(E1,...,En). Then clearly L/K is a splitting field of the separable polynomial F (Y ) over K. Consequently E1,...,En is a transcendence basis of L/K, and L/K is Galois with

Gal(L, K) → Gal(F,K) ⊂ Sym({Z1,...,Zn}) → Sn where the first arrow is the isomorphism which sends every g ∈ Gal(L, K) to its restriction g|{Z1,...,Zn}, and the second arrow is the isomorphism which sends every τ ∈ Sym({Z1,...,Zn}) to the unique µ(τ) ∈ Sn such that τ(Zi) = Zµ(τ)(i) for 1 ≤ i ≤ n. Given any σ ∈ Sn, we get a k-automorphism ν(σ) of L which sends every u(Z1,...,Zn) ∈ L to u(Zσ(1),...,Zσ(n)) ∈ L, and clearly ν(σ) is a K-automorphism with µ(ν(σ))|{Z1,...,Zn}) = σ. Therefore

Gal(F,K) = Sym({Z1,...,Zn}). Since K is the fixed field of Gal(L, K), we see that K coincides with the set of all symmetric functions. Consequently, every symmetric function u(Z1,...,Zn) can be expressed uniquely as a rational function of the elementary syymetric function, i.e, there exists a unique n-variable rational function

r(X1,...,Xn) ∈ k(X1,...,Xn) such that u(Z1,...,Zn) = r(E1,...,En).

As polynomial rings, k[Z1,...,Zn] and k[E1,...,En] are normal domains with quotients field L and K respectively. Since the elements Z1,...,Zn are obviously integral over k[E1,...,En], the entire ring k[Z1,...,Zn] is integral over the subring k[E1,...,En], and hence we get

K ∩ k[Z1,...,Zn] = k[E1,...,En].

Thus, assuming the symmetric rational function u(Z1,...,Zn) to be a polynomial, we must have r(X1,...,Xn) ∈ k[X1,...,Xn].

Giving weight j to Xj for 1 ≤ j ≤ n, we can write X r(X1,...,Xn) = ri(X1,...,Xn) i∈N where ri is isobaric [cf. L4§1(O3)] of weight i (or r0 = 0). Also let us write X u(X1,...,Xn) = ui(X1,...,Xn) i∈N where ui is homogeneous of degree i (or u0 = 0). Comparing terms of equal degree we must have

ui(Z1,...,Zn) = ri(E1,...,En) for all i ∈ N. 38 LECTURE L6: PAUSE AND REFRESH

EXERCISE (E4). [Newton’s Symmetric Function Theorem]. Prove (T4) of L1§8. In other words, show that for the generic n-th degree polynomial n n−1 F (Y ) = Y + X1Y + ··· + Xn over K = k(X1,...,Xn), where X1,...,Xn are indeterminates over a field k, we have Gal(F,K) = Sn. HINT. See (D4) above.

EXERCISE (E5). [More on Newton’s Symmetric Function Theorem]. Give a direct proof (without Galois theory) of the General Symmetric Function Theorem which was established in (D4) above when k is a field, but you are to do this for any nonnull ring k. In other words, let Z1,...,Zn be indeterminates over a nonnull ring k with n ∈ N+, and show that every symmetric polynomial u(Z1,...,Zn) ∈ k[Z1,...,Zn] can be uniquely expressed as

u(Z1,...,Zn) = r(E1,...,En) with r(X1,...,Xn) ∈ k[X1,...,Xn] where X1,...,Xn are indeterminates and E1,...,En are the elementary symmetric functions of Z1,...,Zn. Also show that in the notation of (D4) we have

ui(Z1,...,Zn) = ri(E1,...,En) for all i ∈ N and hence in particular the weight of r(X1,...,Xn) equals the (total) degree of u(Z1,...,Zn); the weight of r is defined to be the largest weight of a monomial occurring in r if r 6= 0, and −∞ if r = 0. [Note that L1§8(T5) is proved in L1§11(E5) and, in view of this, L1§8(T6) follows from the above Exercises (E1), (E3), and (E4)]. HINT. The assertion about the ui follows from the assertion about u, which we shall prove by double induction on n and the degree d of u. For n = 1 the assertion is obvious because then E1 = Z1. So let n > 1 and assume true for n − 1. Again the assertion is obvious for d ≤ 0. So let d > 0 and assume for all smaller values of the degree. Let F1,...,Fn−1 be obtained by putting Zn = 0 in E1,...,En−1 respectively. Clearly u(Z1,...,Zn−1, 0) is a symmetric polynomial in Z1,...,Zn−1 of degree ≤ d and hence by the n − 1 case we get

u(Z1,...,Zn−1, 0) = r(F1,...,Fn−1) where r(X1,...,Xn−1) ∈ k[X1,...,Xn−1] has weight ≤ d. Let

u(Z1,...,Zn) = u(Z1,...,Zn) − r(E1,...,En−1) ∈ k[Z1,...,Zn] and let d be the degree of u. Then d ≤ d. Clearly u(Z1,...,Zn−1, 0) = 0 and hence u(Z1,...,Zn) is divisible by Zn. Also clearly u(Z1,...,Zn) is symmetric and hence it is divisible by Z1 ...Zn. Thus

u(Z1,...,Zn) = v(Z1,...,Zn)En where v(Z1,...,Zn) ∈ k[Z1,...,Zn] is symmetric of degree e ≤ d − n. Therefore by the induction hypothesis on d, there is a unique s(X1,...,Xn) ∈ k[X1,...,Xn] of weight e such that v(Z1,...,Zn) = s(E1,...,En). Now by letting

r(X1,...,Xn) = s(X1,...,Xn)Xn + r(X1,...,Xn−1) we see that r(X1,...,Xn) ∈ k[X1,...,Xn] has weight d and

u(Z1,...,Zn) = r(E1,...,En). §6: DEFINITIONS AND EXERCISES 39

EXERCISE (E6). [Relations Preserving Permutations]. Let there be given Q any nonconstant monic separable polynomial f = f(Y ) = 1≤i≤n(Y − αi) = n P n−1 Y + 1≤i≤n aiY with coefficients ai in a field K. Show that then Gal(f, K) is the set of all relations preserving permutations of the roots, i.e., the set of all τ ∈ Sym({α1, . . . , αn}) such that for every n-variable polynomial P (X1,...,Xn) ∈ K[X1,...Xn] with P (α1, . . . , αn) = 0 we have P (τ(α1), . . . , τ(αn)) = 0.

EXERCISE (E7). [Cyclic Groups]. Verify FACTS (5) to (9) of L5§5(Q32). In particular show that for any cyclic group W of finite order n, the mapping V 7→ |V | = m gives a bijection of the set of all subgroups of W onto the set of all positive integers which divide n, and the groups V and W/V are cyclic of order m and n/m respectively.

DEFINITION (D5). [Artin-Schreier Extensions]. Replacing the multiplica- tive group of roots of unity by the underlying additive group of the field GF(p), the cyclic extension of (E3) is converted to an Artin-Schreier extension, i.e., to the splitting field L of the Artin-Schreier polynomial f(Y ) = Y p − Y − a over a field K of characteristic p 6= 0 for some a ∈ K. We claim that Gal(L, K) is a cyclic group of order p or 1 according as f(Y ) is irreducible in K[Y ] or not. In other words, if αp − α = a for some α ∈ K then f(Y ) factors into linear factors in K[Y ], and otherwise it is irreducible in K[Y ] with Gal(L, K) cyclic of order p. Note that if αp − α = a for some α ∈ L then for every i ∈ GF(p) ⊂ K we have (α + i)i − (α + i) = a; it follows that if α 6∈ K then g 7→ g(α) − α gives an isomor- phism of the multiplicative group Gal(L, K) onto the underlying additive group of GF(p). This proves the claim.

EXERCISE (E8). [Cyclic Extensions of degree equal to characteristic]. In (D5) above we have shown that the splitting field of the separable polynomial f(Y ) = Y p − Y − a over a field of characteristic p 6= 0 with a ∈ K is p-cyclic or 1-cyclic according as f(Y ) is irreducible in K[Y ] or not. Conversely show that any p-cyclic extension L of a field K of characteristic p 6= 0 is the splitting field of a separable irreducible polynomial of the form f(Y ) = Y p − Y − a with a ∈ K. HINT. Let σ be a generator of Gal(L, K). Since [L : K] is a prime number, i P l for any u ∈ L \ K we have L = K(u). Let ui = σ (u) and sl = 1≤i≤p ui. Also j−1 let M = (Mij) be the p × p matrix with Mij = ui−1 , and let N = (Nij) be the p × p matrix with Nij = Mij or sj−1 according as i 6= 1 or i = 1. Then N is obtained by adding the last p − 1 columns of M to its first column, and hence their determinants are equal. By L5§5(Q32)(T135) we see that the determinant of M is the Vandermonde Determinant V (u1, . . . , up) which is nonzero. Therefore the determinant of N is nonzero. But N11 = p = 0, and hence for some j ∈ {2, . . . , p} we must have N1j 6= 0. Let l = j − 1. Then l ∈ {1, . . . , p − 1} with sl = N1j, and hence sl 6= 0. But sj is a symmetric function of u1, . . . , up, and hence sl ∈ K. Let

−1 X l α = −sl iui. 0≤i≤p−1 40 LECTURE L6: PAUSE AND REFRESH

Then −1 X l −1 X l l σ(α) = −sl iui+1 = −sl [(i + 1)ui+1 − ui+1] = α + 1 0≤i≤p−1 0≤i≤p−1 and hence upon letting f(Y ) = Y p − Y − a with a = αp − α we see that f(Y ) is a separable irreducible polynomial over K and its splitting field is L.

DEFINITION (D6). [Field Polynomials and Norms and Traces]. Out of the n elementary symmetric functions, the first and the last (i.e., the n-th), which are called the trace and the norm, are the easiest to use because they are respectively additive and multiplicative. We shall now introduce them formally. So let L/K be a finite algebraic field extension, and let n be the field degree, i.e., [L : K] = n ∈ N+. Given any z ∈ L, let m = [K(z): K] and let e(Y ) be the minimal polynomial of z over K. Then Y m X m−i e(Y ) = (Y − yi) = Y + biY 1≤i≤m 1≤i≤m with bi in K and yi in an overfield of L such that yj = z for some j. Now n/m ∈ N+ and we define the field polynomial of z relative to the field extension L/K to be the monic polynomial f(Y ) of degree n obtained by putting n/m Y n X n−i f(Y ) = e(Y ) = (Y − zi) = Y + aiY 1≤i≤n 1≤i≤n with ai in K and zi in an overfield of L such that zj = z for some j. Note that i (−1) ai is the i-th elementary symmetric function of z1, . . . , zn. We define the norm NL/K (z) and the trace TL/K (z) of z relative to L/K by putting n NL/K (z) = (−1) an and TL/K (z) = −a1 i.e., NL/K (z) = z1 . . . zn and TL/K (z) = z1 + ··· + zn.

DEFINITION (D7). [Degrees of Separability and Inseparability]. In the above situation, the roots y1, . . . , ym are distinct iff z is separable over K. If z is inseparable over K then we must have ch(K) = p 6= 0, and we may ask how many times does each root repeat. To answer this we define the exponent of inseparability of e(Y ), i.e., of any monic irreducible polynomial e(Y ) of degree m ∈ + over a N  field K of characteristic p 6= 0, to be the largest  ∈ N such that e(Y ) ∈ K[Y p ], and we note that then  m = µp with µ ∈ N+. We call µ and p the degrees of separability and inseparability of e(Y ) respectively.

EXERCISE (E9). [Exponent of Inseparability]. Concerning the exponent of inseparability  defined above, upon letting  x = zp ∈ L show that  e(Y ) = d(Y )p §6: DEFINITIONS AND EXERCISES 41 where Y µ X µ−i d(Y ) = (Y − xi) = Y + ciY 1≤i≤µ 1≤i≤µ with ci = bip in K and xi in an overfield of L such that xj = x for some j. Also show that d(Y ) is the minimal polynomial of the separable algebraic element x  over K, and Y p − x is the minimal polynomial of the purely inseparable element z  over K(x). Thus the roots x1, . . . , xµ are distinct, and each of them is repeated p  times amongst the roots y1, . . . , ym of e(Y ), and (n/m)p times amongst the roots z1, . . . , zn of f(Y ). HINT. See L5§5(Q27)(C41) and L5§5(Q32)(T142).

EXERCISE (E10). [Behaviour Under Finite Algebraic Field Extensions]. In (D6) show that if L0 is any field between K and L with z ∈ L0 then, upon letting δ = n/n0 with n0 = [L0 : K], for the field polynomial f 0(Y ) of z relative to L0/K we have f(Y ) = f 0(Y )δ and for the norms and traces we have δ NL/K (z) = (NL0/K (z)) and TL/K (z) = δTL0/K (z).

EXERCISE (E11). [Norm Giving Field Polynomial]. In (D6) we defined the norm in terms of the field polynomial. Show that conversely the field polynomial f(Y ) of any element z ∈ L relative to any finite algebraic field extension L/K can be expressed in terms of a norm by the formula

f(Y ) = NL(Y )/K(Y )(Y − z) where we note that L(Y )/K(Y ) is clearly a field extension whose field degree is equal to the field degree n of L/K [cf.L5§6(E26)]. HINT. Taking an indeterminate T over L(Y ) and letting m eb(Y ) = (−1) e(T − Y ) we see that eb(Y ) is the minimal polynomial of T − z over K(T ) and hence m NK(T )(T −z)/K(T )(T − z) = (−1) eb(0) = e(T ) and therefore replacing T by Y we get

e(Y ) = NK(z)(Y )/K(Y )(Y − z). In view of the norm portion of (E10), our assertion follows by raising both sides of the above equation to their δ-th powers.

DEFINITION (D8). [Field Polynomial as Characteristic Polynomial]. Noting that in (D6), L is a K-vector space of dimension n, multiplication by z gives the K-linear transformation (= map) τz : L → L defined by τz(x) = zx for all x ∈ L. In (E12) below we shall show that the field polynomial f(Y ) equals the characteristic polynomial of any matrix of τz as defined in (D9) below.

DEFINITION (D9). [Spur and Characteristic Matrix]. Let K be a field and let L be a K-vector space of dimension n ∈ N+. Let v = (v1, . . . , vn) be a basis 42 LECTURE L6: PAUSE AND REFRESH of L. The characteristic matrix cmat(A) and likewise the characteristic polynomial cpol(A) of an n × n matrix A = (Aij) over K are defined by putting

cmat(A) = YIn − A and cpol(A) = det(YIn − A) where In is the n × n identity matrix and cmat(A) is an n × n matrix over K[Y ]. Note that then n X n−1 cpol(A) = Y + baiY with bai ∈ K. 1≤i≤n Also we define the spur of A by putting X spur(A) = Aii. 1≤i≤n We are particularly interested in these quantities when A is the matrix of a K-linear transformation (= map) τ : L → L relative to v, i.e., when we have X τ(vj) = Aijvi for 1 ≤ j ≤ n. 1≤i≤n Moreover we are also interested in the matrix B of τ relative to any other basis w = (w1, . . . , wn) of L.

EXERCISE (E12). [Properties of Norms and Traces]. In the situation of (D9) show that (1) cpol(B) = cpol(A) with n (2) det(B) = det(A) = (−1) ban and spur(B) = spur(A) = −ba1. Also show that if τ = τz with τz as in (D8) then (3) f(Y ) = cpol(A) with

(4) NL/K (z) = det(A) and TL/K (z) = spur(A). Moreover show that, in the situation of (D6), for any z∗ ∈ L we have ∗ ∗ (5) NL/K (zz ) = NL/K (z)NL/K (z ) and ∗ ∗ (6) TL/K (z + z ) = TL/K (z) + TL/K (z ). and for any κ ∈ K we have n (7) NL/K (κ) = κ and TL/K (κ) = nκ and n (8) NL/K (κz) = κ NL/K (z) and TL/K (κz) = κTL/K (z). Finally show that in the situation of (D6) we have n/m (9) NL/K (z) = (y1 . . . ym) and

(10) TL/K (z) = (n/m)(y1 + ··· + ym) §6: DEFINITIONS AND EXERCISES 43 and if ch(K) = p 6= 0 then in the situation of (D7) and (E9) we have

(n/m)p (11) NL/K (z) = (x1 . . . xµ) and

 (12) TL/K (z) = (n/m)p (x1 + ··· + xµ). HINT. Since v and w are both bases of L, we have B = CAC−1 for some C ∈ GL(n, K). Since the matrix YIn commutes with every n × n matrix, we get cmat(B) = Ccmat(B)C−1, and hence by taking determinants we obtain (1). Now B = CAC−1 implies det(B) = det(A) and spur(B) = spur(A), and an easy n calculation with the determinant cpol(A) yields the equations det(A) = (−1) cn and spur(A) = −c1, which proves (2). Now let the situation be as in (D8), and let τ = τz. Before turning to (3), let L0 = K(z) and δ = n/n0 with n0 = m = [L0 : K], let us consider the K-basis m−1 0 0 0 0 0 u = (u1, u2, . . . , um) = (1, z, . . . , z ) of L , let τ = τz : L → L be the K-linear 0 0 0 0 transformation defined by taking τz(x) = zx for all x ∈ L , and let A = (Aij) be 0 0 the m × m matrix of τ relative to u. Then Amj = bm−j, and for 1 ≤ i ≤ m − 1 we 0 have Aij = 0 or 1 according as j 6= i + 1 or j = i + 1. Therefore by expanding in terms of the last row we see that (30) e(Y ) = cpol(A0).

0 Let t1, t2, . . . , tδ be an L -basis of L. For the alternative basis w we may assume that wm(i−1)+j = tiuj for 1 ≤ i ≤ δ and 1 ≤ j ≤ m. Now B is a diagonal block matrix with A0 repeated δ times along the principal diagonal. Therefore cpol(B) = (cpol(A0))δ, and hence by (1) and (30) we get (3). By (2) and (3) we get (4). ∗ ∗ ∗ For any z ∈ L let A be the matrix of τz∗ relative to v. Then clearly AA and ∗ A + A are the respective matrices of τzz∗ and τz+z∗ relative to v. Therefore (5) and (6) follow from (4). For any κ ∈ K, by (D6) we see that NK/K (κ) = TK/K (κ) = κ and hence by (E10) we get (7). By (6) and (7) we get the first part of (8). The second part of (8) follows from (4) by noting that for the matrix A of τκz we clearly have A = κA. Finally (9) to (12) follow from the definition (D6).

EXERCISE (E13). [Condition For Inseparable Element]. In (D6) show that if z is inseparable over K then TL/K (z) = 0. HINT. See part (12) of (E12).

EXERCISE (E14). [Extending Derivations and Separable Extensions]. Referring to L1§12(N1) for the definition of derivations, and mimicking the calculus method of implicit differentiation discussed in L3§2, show that, given any (not necessarily finite) separable algebraic field extension L/K, every D ∈ Der(K,K) has a unique extension E ∈ Der(L, L), i.e, for any derivation D of K there is a unique derivation E of L such that for all x ∈ K we have E(x) = D(x). HINT. For showing uniqueness, let E ∈ Der(L, L) be any extension of any given D ∈ Der(K,K). For any y ∈ L, let

n n−1 f(Y ) = Y + a1Y + ··· + an 44 LECTURE L6: PAUSE AND REFRESH be its minimal polynomial over K. Then in view of L1§12(E12) we see that

E(f(y)) = fY (y)E(y) + fD(y) where n−1 X n−i−1 fY (Y ) = nY + iaiY 1≤i≤n−1 and X n−i fD(Y ) = D(ai)Y . 1≤i≤n

Since f(y) = 0, we also have E(f(y)) = 0. By L5§5(Q27)(C41) we get fY (y) 6= 0, and hence E(y) = −fD(y)/fY (y) which proves uniqueness. Turning to existence, given any D ∈ Der(K,K), for every y ∈ L we define E(y) ∈ K(y) by the above formula. Now clearly E(x) = D(x) for all x ∈ K. To show that E ∈ Der(L, L), in view of the primitive element theorem L5§5(Q32)(T140), it suffices to prove this in the case when L = K(y) for some y. In view of the uniqueness proved above, this case is a consequence of the |I| = 1 case of the following Exercise (E15).

DEFINITION (D10). [Polynomials in a Family and Pure Transcendental Extensions]. As in L4§10(E17) we can consider the polynomial ring

RI = K[Y ] in a family of (distinct) indeterminates

Y = (Yi)i∈I over a field K where I is any indexing set which need not be finite. This makes sense for any ring K, but here we continue to assume K to be field. Given any family of elements y = (yi)i∈I in an overfield L of K, as in L1§6 we have the substitution map

ΦI : RI → L which is the unique K-homomorphism of rings such that

ΦI (Yi) = yi for all i ∈ I. Again L could be an overring of a ring K, but here we assume both to be fields. Let there be given any D ∈ Der(K,L) and let

u = (ui)i∈I be any other family of elements in L. Note that y is a transcendence basis of L/K means ker(ΦI ) = 0 and L/K(y) is algebraic. If L = K(y) for some transcendence basis y of L/K then we say that L/K is pure transcendental. If ker(ΦI ) = 0 and L/K(y) is separable algebraic then we call y a separating transcendence basis of L/K. If L/K has a separating transcendence basis then we say that L/K is separably generated. In accordance with L5L5(Q32)we say that L/K is finitely separably generated if L/K is finitely generated as well as separably generated. §6: DEFINITIONS AND EXERCISES 45

Let Ω be the set of all finite subsets J of I, and let us label the elements of any such J as J(i)1≤i≤|J| and let us put m = |J| with (Zi, zi, vi) = (YJ(i), yJ(i), uJ(i)) for 1 ≤ i ≤ m where the reference to J is invisible in m, Zi, zi, vi. Then clearly [ ker(ΦI ) = ker(ΨJ ) J∈Ω where ΨJ : SJ = K[Z1,...,Zn] → L is the substitution map sending Zi to zi for 1 ≤ i ≤ m. For any J ∈ Ω clearly there is a unique homomorphism ΨD,J : SJ → L of underlying additive groups such that for all

X i1 im f = f(Z1,...,Zm) = ai1...im Z1 ...Zm ∈ SJ i1,...,im in N with ai1...im ∈ K we have X ΨD,J (f) = fD(z1, . . . , zm) + fZi (z1, . . . , zm)vi 1≤i≤m where X i1 im fD(Z1,...,Zm) = D(ai1...im )Z1 ...Zm i1,...,im in N and

fZi (Z1,...,Zm) = the (Zi)-partial derivative of f(Z1,...,Zm).

Also clearly there is a unique homomorphism ΦD,I : RI → L of underling additive groups such that for all J ∈ Ω and f ∈ RI , regarding SJ as a subring of RI , we have ΦD,I (f) = ΨD,J (f). Moreover clearly [ ker(ΦD,I ) = ker(ΨD,J ). J∈Ω EXERCISE (E15). [Criterion for Extensions of Derivations]. In the above situation of (D10) we have  (1) ker(ΦD,I ) contains a set of generators of ker(ΦI )  ⇒ (2) ker(Φ ) ⊂ ker(Φ )  I D,I ⇒ (3) D can be extended to some E ∈ Der(K(y),L)   with E(yi) = ui for all i ∈ I  ⇒ (4) ker(ΦI ) ⊂ ker(ΦD,I ) and moreover: (2) ⇒ the E of (3) is unique. HINT. By L1§12(E12) we see that (3) ⇒ (4). By the product rule for derivations we have (1) ⇒ (2). Now assuming (2) we get a unique homomorphism F : K[y] → L of underlying additive groups such that for all f = f(Y ) ∈ RI we have

F (f(y)) = ΦD,I (f). Clearly F ∈ Der(K[y],L) and F is an extension of D. By L1§12(E12) F can be uniquely extended to E ∈ Der(K(y),L). 46 LECTURE L6: PAUSE AND REFRESH

EXERCISE (E16). [Derivations and Purely Inseparable Extensions]. Let L be an overfield of a field K of characteristic p 6= 0 and let y ∈ L be such that pe pe−1 y ∈ K but y 6∈ K for some e ∈ +. Let there be given any D ∈ Der(K,L). N e e Then D has an extension E ∈ Der(K(y),L) iff D(yp ) = 0. Moreover, if D(yp ) = 0 and u is any element of L then D has a unique extension E ∈ Der(K(y),L) with E(y) = u. e HINT. For any derivation E we have E(yp ) = 0, and by L5§5(Q27)(C41) we see e e that Y p − yp is the minimal polynomial of y over K. Hence we are again reduced to the |I| = 1 case of the above Exercise (E15).

EXERCISE (E17). [More About Purely Inseparable Extensions]. Let L be an overfield of a field K of characteristic p 6= 0 such that Lp ⊂ K (where Lp = {xp : x ∈ L}), let S ⊂ L be such that L = K(S), and let D ∈ Der(K,L) be such that D(xp) = 0 for all x ∈ S. Show that then D has an extension E ∈ Der(L, L). HINT. Let W be the set of all pairs (L0,E0) where L0 is a field between K and L, and E0 ∈ Der(L0,L0) is an extension of D. In W define (L0,E0) ≤ (L00,E00) to mean that L0 ⊂ L00 and E00 is an extension of E0. The W is a partially ordered set having the Zorn property, and hence by Zorn’s Lemma it has a maximal element (L0,E0). By (E16) we must have E0 = E.

EXERCISE (E18). [Derivations and Separably Generated Extensions]. Let L/K be a separably generated field extension, let y = (yi)i∈I be a separating transcendence basis of L/K, and let u = (ui)i∈I be a family of elements in L. Then D has a unique extension E ∈ Der(L, L) with E(yi) = ui for all i ∈ I. [The assumptions say that L is an overfield of a field K having a transcendence basis y = (yi)i∈I such that L is a separable algebraic field extension of K(y), and u = (ui)i∈I is any family of elements in L; in particular we could have a pure transcendental field extension L = K(y)]. HINT. Follows from (E14) and (E16).

EXERCISE (E19). [Criterion for Separable Algebraic Extension]. Let L be a finitely generated field extension of a field K, i.e., L = K(y1, . . . , ym) where y1, . . . , ym are elements in L with m ∈ N. Show that then L is separable algebraic over K iff DerK (L, L) = 0, i.e., iff for every E ∈ DerK (L, L) we have E(x) = 0 for all x ∈ L. HINT. The “only if” part follows from (E14). Conversely, suppose if possible that DerK (L, L) = 0 but yj is not separable algebraic over H = K(y1, . . . , yj−1) for some j ∈ {1, . . . , m}, and let j be the largest such. Then yj is either transcendental or inseparable over H. In the first case by (E18), and in the second case by (D7), × (E9), and (E16), we can find F ∈ DerH (H(xj),H(xj)) with F (yj) = u ∈ H(yj) and then by (E14) we can extend F to some E ∈ Der(L, L), which is a contradiction.

DEFINITION (D11). [Jacobian Matrix and Jacobian]. For polynomials f1, . . . , fm in indeterminates Y1,...,Ym over a field K with m ∈ N+, we define the jacobian matrix of f1, . . . , fm relative to Y1,...,Ym by putting

∂(f , . . . , f )  ∂f  1 m = i ∂(Y1,...,Ym) ∂Yj §6: DEFINITIONS AND EXERCISES 47 i.e., the m × m matrix whose (i, j)-th entry is the partial derivative

∂fi ∈ R = K[Y1,...,Ym]. ∂Yj

Its determinant is called the jacobian of f1, . . . , fm relative to Y1,...,Ym and we denote it by J(f , . . . , f )  ∂f  1 m = det i . J(Y1,...,Ym) ∂Yj Given any K-homomorphism φ : R → L where L is any overfield of K, let

yi = φ(Yi) for 1 ≤ i ≤ m. We write   ∂(f1, . . . , fm) ∂fi (y1, . . . , ym) or (y1, . . . , ym) ∂(Y1,...,Ym) ∂Yj for the m × m matrix obtained by substituting (y1, . . . , ym) for (Y1,...,Ym) in the above matrix, and we write   J(f1, . . . , fm) ∂fi (y1, . . . , ym) or det (y1, . . . , ym) J(Y1,...,Ym) ∂Yj for the corresponding determinant.

EXERCISE (E20). [Jacobian Criterion of Separability]. In (D11) assume that L = K(y1, . . . , ym). Show that then: L/K is separable algebraic iff   ∂fi det (y1, . . . , ym) 6= 0 ∂Yj for some f1, . . . , fm in ker(φ). HINT. If the above determinant is nonzero then for any K-derivation D of L P ∂fi we have the m homogeneous linear equations (y1, . . . , ym)D(yj) = 0 1≤j≤m ∂Yj for 1 ≤ i ≤ m whose determinant is nonzero, and hence D(y1) = ··· = D(ym) = 0, and therefore D = 0; consequently by (E19) we see that L/K is separable algebraic. Conversely, assuming L/K to be separable algebraic, by (E19) we see that DerK (L, L) = 0; consequently by (E15) we see that the system of homogeneous P linear equations 1≤j≤m fYj (y1, . . . , yj)uj = 0 in u1, . . . , um, with f varying over m ker(φ), has u1 = ··· = um = 0 as the only solution in L , and hence the above determinant is nonzero for some f1, . . . , fm in ker(φ).

EXERCISE (E21). [Lagrange’s Theorem]. Show that: (1) if G, H, I are nonempty finite sets together with a surjection φ : G → I as well as a bijection −1 φx : H → φ (φ(x)) for every x ∈ G, then |G| = |H| × |I|. From (1) deduce that: (2) if H is any subgroup of any finite group G then for I = |G/H| (= the set of all left cosets of H in G) we have |G| = |H| × |I| and hence in particular the order of H divides the order of G. From (2) deduce that: (3) the order of any element of a finite group divides the order of the group, and: (4) every group of prime order is a simple group as well as a cyclic group. HINT. (1) is essentially the definition of multiplication in N+. To deduce (2) let φ and φx be defined by φ(x) = xH and φx(y) = xy. 48 LECTURE L6: PAUSE AND REFRESH

DEFINITION (D12). [Action of a Group, Orbit, and Stabilizer]. By an action of a group G on a set W we mean a (group) homomorphism θ : G → Sym(W ), where we recall that Sym(W ) is the group of all bijections W → W ; the elements of W may be called points of W . Note that if H is any subgroup of G then θ|(H,W ) : H → W is an action of H on W . For any u ∈ W , we may write g(u) instead of θ(g)(u), i.e., we need not mention θ explicitly; we may simply say that G acts on W , or some such thing; this is similar to the nonmention of the underlying (ring) homomorphism φ : R → S of an algebra in L5§5(Q18). For any u ∈ W we define the G-orbit of u (in W or on W ) by putting

orbG(u) = {g(u): g ∈ G}. and we define the G-stabilizer of u, or the stabilizer of u in G, by putting

stabG(u) = {g ∈ G : g(u) = u} and we note that this is a subgroup of G. For any U ⊂ W we define the G-stabilizer of U, or the stabilizer of U in G, by putting

stabG(U) = {g ∈ G : g(U) = U} and we note that this is a subgroup of G, and we define the elementwise G-stabilizer of U, or the elementwise stabilizer of U in G, by putting

estabG(U) = {g ∈ G : g(u) = u for all u ∈ U} and we note that this is a subgroup of G. More precisely, we could call these the θ-orbit of u (in W ), the θ-stabilizer of u (in G), the θ-stabilizer of U (in G), and the elementwise θ-stabilizer of U (in G), and denote them by orbθ(u), stabθ(u), stabθ(U), and estabθ(U) respectively. If G ≤ Sym(W ) then, unless otherwise stated, we take the action θ to be the subset injection. Returning to any group G acting on any set W : We generalize orbG(u) by putting orbH (u) = {h(u): h ∈ H} for any u ∈ W and H ⊂ G and calling it the H-orbit of u (in W ). We say that h ∈ G stabilizes u ∈ W to mean that h ∈ stabG(u). We say that H ⊂ G stabilizes u ∈ W to mean that H ⊂ stabG(u). We say that h ∈ G stabilizes U ⊂ W to mean that h ∈ stabG(U). We say that H ⊂ G stabilizes U ⊂ W to mean that H ⊂ stabG(U). We say that h ∈ G stabilizes U ⊂ W elementwise to mean that h ∈ estabG(U). We say that H ⊂ G stabilizes U ⊂ W elementwise to mean that H ⊂ estabG(U).

EXERCISE (E22). [Orbit-Stabilizer Lemma]. Show that for any finite group G acting on a finite set W , and any u ∈ W , we have

|orbG(u)| = |G|/|stabG(u)| i.e., |orbG(u)| = [G : stabG(u)] or in words, the orbit size equals the index of the stabilizer. HINT. For any u in W , and g, h in G, we have −1 −1 g(u) = h(u) ⇔ (h g)(u) = u ⇔ h g ∈ stabG(u) and hence g(u) 7→ g stabG(u) gives a bijection orbG(u) → G/stabG(u). §6: DEFINITIONS AND EXERCISES 49

DEFINITION (D13). [Orbit Set and Fixed Points]. Let G be a group acting on a set W . By an orbit of H ⊂ G (in W ) we mean a subset of W which is of the form orbH (u) for some u ∈ W . We denote the set of all H-orbits in W by orbsetH (W ) and call it the orbit set (or orbset) of H (in W ). For any g ∈ G we denote the set of all fixed points of g (in W ) by fixW (g) (or fix(g)), i.e., fixW (g) = {u ∈ W : g(u) = u}, and call it the fixed point set of g (in W ). For any H ⊂ G we denote the set of all fixed points of H (in W ) by fixW (H) (or fix(H)), i.e., fixW (H) = ∩g∈H fixW (g), and call it the fixed point set of H (in W ). Note that this agrees with (E1).

EXERCISE (E23). [Orbit-Counting Lemma]. Show that the number of orbits of a finite group G acting on a finite set W equals the average number of fixed points, i.e., P |fixW (g)| |orbset (W )| = g∈G . G |G| HINT. Any orbit V of G on W is a nonempty finite subset of W , and for any nonempty finite set V we have P (1/|V |) = 1, and hence in our case we get P u∈V P u∈V (|G|/|V |) = |G|, and therefore by E(21) we see that u∈V |stabG(u)| = |G|. Summing the last equation over orbsetG(W ) we get " # X X X |stabG(u)| = |stabG(u)| = |orbsetG(W )| × |G|.

u∈W V ∈orbsetG(W ) u∈V Considering the set P of all pairs (g, u) with g ∈ G and u ∈ W such that g(u) = u, and expressing it as a disjoint union in two different ways we see that ` ` P P u∈W stabG(u) = P = g∈G fixW (g). Hence u∈W |stabG(u)| = g∈G |fixW (g)| and therefore by the above displayed equation we get the desired equation.

DEFINITION (D14). [Conjugation Action and Conjugacy Classes]. Let G be any group. For any g ∈ G, by the g-conjugate of h ∈ G (or the conjugate of h by g) we mean the element ghg−1, and by the g-conjugate of H ⊂ G (or the conjugate of H by g) we mean the set gHg−1 = {ghg−1 : h ∈ H}; note that if H is a subgroup of G then so is gHg−1, and recall that H is a normal subgroup of G means H is a subgroup of G such that for all g ∈ G we have gHg−1 = H. For any g ∈ G, clearly h 7→ ghg−1 gives an automorphism of G. By taking g(h) = ghg−1 for all g, h in G, we get an action of G on G which we call the conjugation action. The orbits of G under this action are called conjugacy classes of G. These classes clearly form a partition of G, and hence if G is finite then their sizes add up to the size of G. In other words, labelling the distinct conjugacy classes of a finite group G as H1,...,Hs we have a G = Hi 1≤i≤s P and hence |G| = 1≤i≤s |Hi|. Alternatively, letting h1, . . . , hs be representatives of the distinct conjugacy classes of G, i.e., picking hi ∈ Hi, we have orbG(hi) = Hi 50 LECTURE L6: PAUSE AND REFRESH for 1 ≤ i ≤ s, and hence by (E22) we get X (1) |G| = [G : stabG(hi)]. 1≤i≤s

More generally let G be any group acting on any finite set W . Let V1,...,Vb be the distinct orbits of G on W , and let v1, . . . , vb be their representatives, i.e, pick some vi ∈ Vi for 1 ≤ i ≤ b. Then a W = Vi with Vi = orbG(vi) 1≤i≤b and hence by (E22) we get X (2) |W | = [G : stabG(vi)]. 1≤i≤b

DEFINITION (D15). [Normalizer, Centralizer, and Center of a Group]. Let any group G act on itself by the conjugation action explained above. Now the G-stabilizer of h ∈ G under this action is denoted by CG(h) and is called the G-centralizer of h or the centralizers of h (in G). Similarly the G-stabilizer and the elementwise G-stabilizer of H ⊂ G under the said action are denoted by NG(H) and CG(H) and are called the G-normalizer and the G-centralizer of H or the normalizer and the centralizer of H (in G) respectively. We put Z(G) = CG(G) and we call Z(G) the center of G. We say that k ∈ G centralizes h ∈ G to mean that k ∈ CG(h); note that this is so iff k commutes with h, i.e., iff kh = hk. We say that K ⊂ G centralizes h ∈ G to mean that K ⊂ CG(h); note that this is so iff every k ∈ K commutes with h. We say that k ∈ G centralizes H ⊂ G to mean that k ∈ CG(H); note that this is so iff k commutes with every h ∈ H. We say that K ⊂ G centralizes H ⊂ G to mean that K ⊂ CG(H); note that this is so iff every k ∈ K commutes with every h ∈ H. We say that k ∈ G normalizes H ⊂ G to mean that k ∈ NG(H); note that this is so iff kHk−1 = H. We say that K ⊂ G normalizes H ⊂ G to mean that −1 K ⊂ NG(H); note that this is so iff for every k ∈ K we have kHk = H. The above three paragraphs give a direct characterization of normalizers and centralizers, without talking about actions of groups on sets. The resulting direct characterization of the center Z(G) says that it is the set of all elements of G which commute with every element of G. The following properties of the above concepts are easy to establish. (1) For any h ∈ G and H ⊂ G we have CG(h) ≤ G and CG(H) ≤ G. (2) Z(G) /G. Moreover: G is abelian ⇔ G ⊂ Z(G) ⇔ G = Z(G). (3) The normalizer of any subgroup H of G is the largest group between H and G in which H is normal, i.e., H/NG(H) and for every K ≤ G with H/K we have K ≤ NG(H).

EXERCISE (E24). [Class Equation]. Show that for any finite group G, upon letting h1, . . . , hs be representatives of the distinct conjugacy classes of G, we have X (1) |G| = [G : CG(hi)] 1≤i≤s §6: DEFINITIONS AND EXERCISES 51 and upon letting g1, . . . , gr be representatives of the distinct conjugacy classes of G not contained in the center Z(G) of G, we have X (2) |G| = |Z(G)| + [G : CG(gi)]. 1≤i≤r More generally show that for any finite group G acting on any finite set W , upon letting v1, . . . , vb be representatives of the distinct orbits of G on W , we have X (3) |W | = [G : stabG(vi)] 1≤i≤b and upon letting u1, . . . , ua be representatives of the distinct orbits of G on W of size > 1, we have X (4) |W | = |fixW (G)| + [G : stabG(ui)]. 1≤i≤a

HINT. (1) follows from (D14)(1), and (2) follows from (1) by picking h1, . . . , hs so that hi = gi for 1 ≤ i ≤ r and hj ∈ Z(G) for r + 1 ≤ j ≤ s. (3) follows from (D14)(2), and (4) follows from (3) by picking v1, . . . , vb so that vi = ui for 1 ≤ i ≤ a and vj ∈ fixW (G) for a + 1 ≤ j ≤ b.

DEFINITION (D16). [Prime Power Group and Prime Power Subgroup]. Let p be any prime. By a p-group we mean a finite group H such that |H| is a power of p, i.e., |H| = pe for some e ∈ N. By a p-subgroup of a finite group G we mean a subgroup H of G such that H is a p-group. By Subp(G) we denote the set of all p-subgroups of G.

EXERCISE (E25). [Action of Prime Power Group]. Let G be a finite group acting on a finite set W , let p be a prime number, and let H be a p-subgroup of G. Show that then

|H| − |fixW (H)| ∈ pZ. HINT. Obvious if |H| = 1, Otherwise apply (E24)(4) with G = H.

EXERCISE (E26). [Cauchy’s Theorem]. Show that for any finite group G 6= 1 and any prime number p which divides the order of G we have the following. (1) G has elements of order p. (2) If G is a p-group then Z(G) 6= 1. (3) G is a p-group iff every element of G is of p-power order. HINT. Let h be a generator of a cyclic group H of order p, and consider the p finite set W = {w = (w1, . . . , wp) ∈ G : w1 . . . wp = 1}. Then H acts on W z by taking h (w) = (wz+1, . . . , zp, zp + 1, . . . , wz) for all z ∈ {0, . . . , p − 1} and p p w ∈ W . Clearly fixW (H) = {(g, . . . , g) ∈ G : g = 1} and hence |fixW (H)| ≥ 1 because (1,..., 1) ∈ fixW (H). But by (E25) we have |fixW (H)| ∈ pZ, and hence p |fixW (H)| > 1. Therefore there exists g ∈ G \{1} with g = 1. It follows that the order of g is p. This proves (1). In (E24)(2) we clearly have [G : CG(gi)] ∈ pZ for 1 ≤ i ≤ r, and hence |Z(G)| ∈ pZ, and therefore Z(G) 6= 1, which proves (2). By (1) and (E21) we get (3). 52 LECTURE L6: PAUSE AND REFRESH

EXERCISE (E27). [Normalizer of Prime Power Subgroup]. Let G be a finite group, let p be a prime number, and let H be a p-subgroup of G. Show that

(1) [G : H] − [NG(H): H] ∈ pZ and

(2) [G : H] ∈ pZ ⇒ NG(H) 6= H. HINT. Let G act on W = G/H by taking g(uH) = guH for all g, u in G. Then for any u ∈ G we have  uH ∈ fixW (H)  ⇔ huH = uH for all h ∈ H  ⇔ u−1huH = H for all h ∈ H  −1 ⇔ u hu ∈ H for all h ∈ H  ⇔ u ∈ NG(H) for all h ∈ H.

Therefore |fixW (H)| = |NG(H)| and hence by (E25) we get (1). By (1) we get (2).

DEFINITION (D17). [Transitive Action and Transitive Group]. Let a group G act on a set W . We say that the action of G on W is transitive, or G acts transitively on W , or G is transitive on W , if W = orbG(u) for some u ∈ W . Note that then H ≤ G acts on V ⊂ W means V = ∪u∈U orbH (u) for some U ⊂ V , and H ≤ G acts transitively on V ⊂ W means V = orbH (u) for some u ∈ V . By a transitive (permutation) group we mean a subgroup G of Sym(W ), where W is some set, such that G acts transitively on W under the subset injection as action.

DEFINITION (D18). [Sylow Subgroup and Prime Power Orbit]. By |G|p we denote the highest power of a prime p which divides the size |G| of a nonempty finite set G and, in case G is a finite group, by a p-Sylow subgroup of G we mean e a subgroup P of G whose order equals |G|p, i.e., P ≤ G such that |P | = p = |G|p e and |G| = πp with e ∈ N and π ∈ N+ \ pZ; by Sylp(G) we denote the set of all p-Sylow subgroups of G; note that then Sylp(G) ⊂ Subp(G). Assuming G to be a finite group acting on a finite set W , by a p-power G-orbit (in W ) we mean a nonempty subset of W whose size is a power of p and which is of the form orbG(u) for some u ∈ W .

EXERCISE (E28). [Sylow Transitivity]. Let G be a finite group acting on a finite set W . Using the notation of (D18) above, prove the following five assertions where in (1) to (4) u is any point of W and in (2) to (5) p is any prime. (1) |G|/|stabG(u)| = |orbG(u)|. (2) |G|p/|stabG(u)|p = |orbG(u)|p. (3) Let H ≤ G be such that |H|p = |G|p (for instance H could be a p-Sylow subgroup of G). Then |orbH (u)|p ≥ |orbG(u)|p. (4) In the situation of (3) assume that |orbG(u)| = a power of p. Then H acts transitively on orbG(u), i.e., orbG(u) = orbH (u). (5) Every p-Sylow subgroup of G acts transitively on every p-power G-orbit. HINT. (1) is a restatement of (E22). To prove (2) note that for any H ≤ G we clearly have |G|p/|H|p = the highest power of p which divides |G|/|H|, and §6: DEFINITIONS AND EXERCISES 53 now apply (1) with H = stabG(u). To prove (3) note that obviously we have stabH (u) = H ∩ stabG(u), and hence |stabH (u)| divides |stabG(u)|, and therefore |stabH (u)|p divides |stabG(u)|p, and hence our assertion follows from (2). To prove (4) note that

|orbH (u)| ≥ |orbH (u)|p ≥ |orbG(u)|p = |orbG(u)| where the first inequality is obvious, the second inequality is (3), and the third equation is equivalent to the assumption of |orbG(u)| being a power of p; therefore |orbH (u)| ≥ |orbG(u)|; but obviously |orbH (u)| ≤ |orbG(u)|; therefore we have |orbH (u)| = |orbG(u)| and hence orbH (u) = orbG(u). Finally (5) follows from (4) by taking H to be the given p-Sylow subgroup of G, and orbG(u) to be the given p-power G-orbit.

DEFINITION (D19). [Complete Set of Conjugates]. For any group G, by a complete set of G-conjugates in G we mean a G-orbit under the conjugation action, i.e., a subset ω of G such that ω = {ghg−1 : g ∈ G} for some h ∈ G. Likewise, by a complete set of G-conjugates in the set of subsets of G we mean a set Ω of subsets of G such that that Ω = {gHg−1 : g ∈ G} for some H ⊂ G. In the above phrases the reference(s) to G may be dropped when it is clear from the context; for instance see (E29)(2) below.

EXERCISE (E29). [Sylow’s Theorem]. Show that for any finite group G and prime p, with notation as in (D18), we have the following. (1) Given any H ∈ Subp(G) \ Sylp(G) there exists K ∈ Subp(G) such that H/K with [K : H] = p. Hence given any H ∈ Subp(G) there exists P ∈ Sylp(G) with H ≤ P . In particular Sylp(G) 6= ∅. (2) Given any H ∈ Subp(G) and P ∈ Sylp(G) there exists g ∈ G such that −1 H ≤ gP g . In particular Sylp(G) is a complete set of conjugate subgroups of G, −1 i.e., Sylp(G) = {gP g : g ∈ G} for some P ≤ G. (3) |Sylp(G)| − 1 ∈ pZ and |Sylp(G)| = [G : NG(P )] for every P ∈ Sylp(G). (4) If P ∈ Sylp(G) then NG(NG(P )) = NG(P ). HINT. The first sentence of (1) clearly implies the other two; to prove the first, by (D15)(3) and (E27)(2) we have H/NG(H) with H 6= NG(H), and hence by (E26)(1) we can find the desired K. The first sentence of (2) clearly implies the second; to prove the first we may assume that H 6= 1 and then letting G act on W = G/P by taking u(gP ) = ugP for all u, g in G, by (E25) we can find g ∈ G with gP ∈ fixW (H), and now it suffices to note that for any g ∈ G we have  gP ∈ fixW (H)  ⇔ hgP = gP for all h ∈ H  ⇔ g−1hgP = P for all h ∈ H  ⇔ g−1Hg ≤ P  ⇔ H ≤ gP g−1.

To prove (3) we can take P ∈ Sylp(G); the assertion being obvious when P = 1, assume that P 6= 1; now P acts on W = Sylp(G) by conjugation and clearly fixW (P ) = {P } and hence |fixW (P )| = 1; consequently by (E24)(4) we see that |Sylp(G)| − 1 ∈ pZ because for 1 ≤ i ≤ a we clearly have [P : stabP (ui)] ∈ pZ; by (3) we know that, by conjugation, G acts transitively on W and by definition we 54 LECTURE L6: PAUSE AND REFRESH have stabG(P ) = NG(P ), and hence by (E22) we get |Sylp(G)| = [G : NG(P )]. To prove (4) let Q = NG(P ); since P/Q, by (2) we get Sylp(Q) = {P }; now it suffices to note that  g ∈ NG(Q)  −1 −1 −1 ⇒ gP g ∈ Sylp(gQg ) and gQg = Q −1 ⇒ gP g = P  ⇒ g ∈ Q.

EXERCISE (E30). [Existence of Prime Power Subgroups]. Reprove most of (E29)(1) along the following lines. For any finite group G and prime number d+e p, let us write |G| = mp with d, e in N and m in N+ \ pZ. Let W be the set of all subsets of G of size pe, and let U be the set of all members of W which are subgroups of G. Let G act on W by taking g(u) = {gx : x ∈ u} for all g ∈ G and d+1 u ∈ W . Let V = {v ∈ W : |orbG(v)| 6∈ p Z}. Show that then d+1 |W | 6∈ p Z and from this deduce that ∅= 6 U ⊂ V and

v 7→ stabG(v) gives a surjection V → U. HINT. Now  d+e d+e mp d Y Ri = p − i |W | = e = mp e p Si = p − i 1≤i

e ei and for 1 ≤ i < p , upon letting i = mip with ei ∈ N and mi ∈ N+ \ pZ, we ei ei d+1 get Ri = Aip and Si = Bip with Ai,Bi in N+ \ pZ, and hence |W | 6∈ p Z. For every u ∈ U we clearly have orbG(u) = G/u and stabG(u) = u. Consequently U ⊂ V , and it suffices to show that for any v ∈ V we have |P | = pe where e P = stabG(v). By (E22) we have |P | = |G|/|orbG(v)| and hence |P |/p ∈ N+ and therefore |P | ≥ pe. For every g ∈ P we have g(v) = v and hence by choosing any x ∈ v we see that g 7→ gx gives an injection P → v and hence |P | ≤ |v| = pe. Consequently |P | = pe.

DEFINITION (D20). [Exponential Notation and Subscript Notation]. According to the exponential notation for action, given a group G acting on a set W : for any g ∈ G and u ∈ W we put ug = g(u) for u ∈ W , for any g ∈ G and g H U ⊂ W we put U = g(U), and for any H ⊂ G and u ∈ W we put u = orbH (u). According to the exponential notation for conjugation, given a group G: for any g ∈ G and h ∈ G we put hg = ghg−1, and for any g ∈ G and H ⊂ G we put Hg = gHg−1. According to the subscript notation for action, given a group G acting on a set W : for any u ∈ W we put Gu = stabG(u), and for any U ⊂ W we put GU = stabG(U) and G[U] = estabG(U).

EXERCISE (E31). [Conjugation Rule]. Let G ≤ Sym(W ) where W is a finite set of size n, and let us use the exponential notation for conjugation. Show that §6: DEFINITIONS AND EXERCISES 55 then, for all h, g in G, in the usual permutation notation we have u . . . u  g(u ) . . . g(u ) h = 1 n ⇒ hg = 1 n . v1 . . . vn g(v1) . . . g(vn) i.e., we have hg(g(u)) = g(h(u)) for all u ∈ W.

DEFINITION (D21). [Transportation and Permutation Isomorphisms]. A bijection f : W → W [ of sets induces the isomorphism f ] : Sym(W ) → Sym(W [) given by f ](h)(f(u)) = f(h(u)) for all h ∈ Sym(W ) and u ∈ W ; this may be called transportation isomorphism since it transports the group structure from Sym(W ) to Sym(W [); in some sense, the above Conjugation Rule is also a transportation phenomenon; to bring out the analogy between these two, we note that in effect f ](h) = fhf −1. In turn, for G ≤ Sym(W ), f ] induces the (restriction) isomorphism ] [ [ ] ] fG : G → G where G = f (G); we call fG a permutation-isomorphism, and we say that G and G[ are permutation-isomorphic. If permutation groups G and G[ are permutation-isomorphic then clearly: G is transitive ⇔ G[ is transitive, G is semi-regular ⇔ G[ is semi-regular, G is regular ⇔ G[ is regular, and so on. Upon taking W [ = W and f ∈ G, by applying the above Conjugation Rule (with f replacing g) we get f ](h) = hf for all h ∈ G and hence, in the subscript notation ] of (D20), for any u ∈ W we have f (Gu) = Gf(u) and the induced isomorphism Gu → Gf(u) makes Gu permutation-isomorphic to Gf(u). This provides a HINT for the following Exercise.

EXERCISE (E32). [Conjugacy Lemma]. Show that if W = {w1, . . . , wn} is a

finte set of size n and G ≤ Sym(W ) is transitive, then Gw1 ,...,Gwn are mutually permutation-isomorphic and they form a complete conjugacy class of subgroups of

G. Also show that the said permutation-isomorphism Gwi → Gwj is induced by a bijection W → W which sends wi to wj.

DEFINITION (D22). [Regular and Semi-regular Permutation Groups]. Let G be a permutation group of degree n, i.e., G ≤ Sym(W ) where W is a finite set of size n. G is semi-regular means that only the identity of G has a fixed point, i.e., fixW (g) = ∅ for all g ∈ G \{1}. G is regular means that G is both transitive and semi-regular. Notice that if G is regular then |G| = n; namely, taking any u ∈ W , we get a bijection G → W given by g 7→ g(u).

DEFINITION (D23). [Left and Right Regular Representations]. For any finite group X, the left and right multiplications in X correspond to the (group) monomorphisms

LX : X → Sym(X) and RX : X → Sym(X) −1 given by letting LX (x)(y) = xy and RX (x)(y) = yx for all x, y in X. Their images LX (X) and RX (X) are clearly regular subgroups of Sym(X) and they may respectively be called the left and right regular permutation representations of X. Clearly LX (X) and RX (X) centralize each other.

EXERCISE (E33). [Cayley’s Theorem]. As indicated in (D23), any finite group G is isomorphic to the regular permutation groups LX (X) and RX (X) which 56 LECTURE L6: PAUSE AND REFRESH centralize each other. Show that they are actually the full centralizers of each other in Sym(X). Also show that they are permutation-isomorphic to each other.

NOTE. [Hint to Cayley and Preamble to Burnside]. Sometimes only the first sentence of (E33) is called Cayley’s Theorem. The other two sentences are proved in (E34) below. Burnside’s Theorem (E42) below may be viewed as further embelishment of Cayley’s Theorem. From (E34) to (E42) our main aim is to reproduce the proof of Burnside’s Theorem given in my 2002 paper [?]. In the second 1911 edition of his famous book [?], Burnside gave two proofs of his theorem, one of which was taken from the first 1897 edition of [?]. Henceforth we shall tacitly use the notation of (D20). The HINTS (= Proofs) being rather long, we shall leave blank lines before them for better display.

EXERCISE (E34). [Centralizer Lemma]. Show that, given any G ≤ Sym(W ) where W is any finite set, for the centralizer G0 of G in Sym(W ), we have the following: (1) If G is transitive then G0 is semi-regular. [It follows that if a transitive subgroup G of Sym(W ) is centralized by a subgroup G00 of Sym(W ) then G00 is semi-regular.] (2) If G and G0 are transitive then G and G0 are regular, G is the centralizer of G0 in Sym(W ), and G0 (resp: G) is the only transitive subgroup of Sym(W ) which centralizes G (resp: G0). [It follows that if a transitive subgroup G of Sym(W ) is centralized by a transitive subgroup G00 of Sym(W ), then G and G00 are the full centralizers of each other in Sym(W ), and G00 (resp: G) is the only transitive subgroup of Sym(W ) which centralizes G (resp: G00).] (3) If G is regular then G0 is also regular, and by taking any w ∈ W and letting f : W → G be the bijection whose inverse is given by g 7→ g(w), we ] ] 0 ] have f (G) = LG(G) and f (G ) = RG(G) where f : Sym(W ) → Sym(G) is the isomorphism induced by f and LG(G) and RG(G) are the left and right permutation representations of G respectively, and hence G are G0 are permutation-isomorphic. [By (2) and (3) it follows that if G is regular then G0 (resp: G) is the unique transitive subgroup of Sym(W ) which centralizes G (resp: G0).]

HINT. To prove (1) assume that G transitive. If G0 is not semi-regular then we can find u 6= v in W and g ∈ G0 with ug = u and vg 6= v. Since G is transitive, we can find h ∈ G with h(u) = v. Then hg(ug) = vg by the conjugation rule, and hg = h because h and g commute. Thus v = h(u) = hg(ug) = vg 6= v which is a contradiction. Therefore G0 is semi-regular. To prove (2) assume that G and G0 are transitive, and let n be the size of W . Then by (1) they must be regular, and hence |G| = n = |G0|. If G00 ≤ Sym(W ) centralizes G then G00 ≤ G0, and if G00 is also transitive then |G00| ≥ n and hence G00 = G0. Let G† be the centralizer of G0 in Sym(W ). Since G is transitive and centralizes G0, it follows that the G ≤ G† and G† is transitive. Therefore by reversing the roles of G0 and G† in what we have already proved, it follows that G = G† and G is the only transitive subgroup of Sym(W ) which centralizes G0. To prove (3) assume that G is regular. Take w ∈ W , let f : W → G be the bijection whose inverse is given by g 7→ g(w), let f ] : Sym(W ) → Sym(G) be §6: DEFINITIONS AND EXERCISES 57 the isomorphism induced by f, and let LG(G) and RG(G) be the left and right ] permutation representations of G respectively. Then f (G) = LG(G), and hence ‡ ]−1 ‡ upon letting G = f (RG(G)) we see that G is a regular subgroup of Sym(W ) which centralizes G and is permutation-isomorphic to it. By (2) we get G‡ = G0.

DEFINITION (D24). [Multitransitive and Antitransitive Groups]. Let G ≤ Sym(W ) where W is a finite set of size n. Given t ∈ N, we say that G is t-transitive if t ≤ n and any t points of W can be mapped to any other t points of W by an element of G (with prescribed order of the points), and we say that G is t-antitransitive if t ≤ n and the identity is the only element of G having t fixed points. G is (t, τ) means G is t-transitive and τ-antitransitive. Note that then: (0, 0) = every group, (0, 1) = semi-regular, (1, 0) = transitive, and (1, 1) = regular. By a sharply t-transitive group we mean a (t, t) group. In the above type definitions, when necessary, reference to W may be made explicit. For instance, instead of saying that G is t-transitive we may say that G is t-transitive on W . Note that Cayley’s Theorem (E33) says that every finite group is isomorphic to a regular permutation group, and the Centralizer Lemma (E34) says that in this manner we get all the regular permutation groups. We call G semi-transitive if all the G-orbits have the same size, and we note: (1) the Obvious Lemma which says that if G is semi-transitive then, upon letting m to be the common size of all the G-orbits and µ to be their number, we have n = µm with m = |uG| for all u ∈ W . If n ≥ 2 then, for any u ∈ W , Gu may be viewed as a subgroup of Sym(W \{u}). We call G sesqui-transitive if G is transitive on W and, for every u ∈ W , Gu is semi-transitive on W \{u}. As a consequence of the Conjugacy Lemma (E32) we see that:  G is two-transitive  (2) ⇔ G is transitive with n ≥ 2 and   Gu is transitive on W \{u} for some u ∈ W and ( G is regular (3) ⇔ G is transitive and Gu = 1 for some u ∈ W and so on. As another obvious fact we note that: (4) if G is regular then, by fixing any w ∈ W , we get a bijection f : W → G whose inverse is given by g 7→ g(w). The significance of this bijection f : W → G and the transportation isomorphism f ] : Sym(W ) → Sym(G) induced by it is exemplified by the Centralizer Lemma (E34) above and the Normality Lemma (E38) below.

DEFINITION (D25). [Frobenius and Semi-Frobenius Groups]. Again let G ≤ Sym(W ) where W is a finite set of size n. G is semi-Frobenius means that G is transitive, n is at least 2, and only the identity of G has 2 fixed points. G is Frobenius means that G is semi-Frobenius but is not regular. Note that then: (1, 2) = semi-Frobenius, and (1, 2)\(1, 1) = Frobenius. A sharply 2-transitive group is also called sharp-Frobenius. 58 LECTURE L6: PAUSE AND REFRESH

EXERCISE (E35). [Order Lemma].For any G ≤ Sym(W ) where W is a finite set of size n, we have the following: (1) G semi-transitive ⇒ |Gu| = |Gv| for all u, v in W . (2) G semi-regular ⇒ |uG| = |G| for all u ∈ W . (3) G semi-regular ⇒ G semi-transitive and |G| divides n. (4) G transitive ⇒ |G| = n|Gu| for all u ∈ W . (5) G regular ⇔ G semi-regular and |G| = n ⇔ G transitive and |G| = n. (6) G semi-Frobenius ⇒ |G| divides n(n − 1). (7) G sesqui-transitive and |G| divides n2 ⇒ |G| = n.

HINT. Namely, (1) follows from (E22) by noting that if G is semi-transitive then for all u, v in W we have |Gu| = |G|/m = |Gv| where m is the common size of all the G-orbits. (2) follows from (E22) by noting that if G is semi-regular then Gu = 1 for all u ∈ W . (3) follows from (D24)(1) and (2). (4) follows from (E22) by noting that if G is transitive then |uG| = n for all u ∈ W . (5) follows from (2) and (4) by noting that G is transitive ⇔ |uG| = n for all u ∈ W . (6) follows by noting that if G is semi-Frobenius then by (4) we get |G| = n|Gu| for all u ∈ W , and by taking (Gu,W \{u}) for (G, W ) in (3) we see that |Gu| divides n − 1. Finally, to prove (7) assume that G is sesqui-transitive and |G| divides n2; then fixing any u ∈ W , by (4) we have |G| = n|Gu| and hence |Gu| divides n and we want to show that |Gu| = 1, i.e., Gu = 1; now Gu is semi-transitive on W \{u}, and hence by taking 0 0 0 (Gu,W \{u}) for (G, W ) in (D24)(1) we get n − 1 = µ m where µ is the number 0 of orbits of Gu on W \{u} and m is their common size; by taking (Gu,W \{u}) 0 for (G, W ) in (E22) we also see that m divides |Gu| which we know divides n; thus m0 divides n as well as n − 1 and hence we must have m0 = 1, i.e., every orbit of Gu on W \{u} has size 1; therefore Gu = 1.

DEFINITION (D26). [Invariant Partition and System of Imprimitivity]. Let G ≤ Sym(W ) where W is a finite set of size n. By a G-invariant partition (of W ) we mean a partition T = (Ti)1≤i≤r of W into pairwise disjoint nonempty sets T1,...,Tr (with W = T1 ∪· · ·∪Tr) such that for every g ∈ G and i ∈ {1, . . . , r} we have g(Ti) = Tj for some j ∈ {1, . . . , r} which may or may not be equal to i. The above partition T is trivial means either r ≤ 1 or |Ti| = 1 for 1 ≤ i ≤ r.A G-system of imprimitivity is a nontrivial G-invariant partition. By a G-block we mean a nonempty subset U of W such that for every g ∈ G we have either U g = U or U g ∩ U = ∅; we call U a trivial block if either U = W or |U| = 1. Note that if G is transitive then n and r are in N+ and there exists s ∈ N+ with n = rs such that |Ti| = s for 1 ≤ i ≤ r; in this case we (r, s) the type of T . The following Exercise relates blocks with orbits and partitions.

EXERCISE (E36). [Block Lemma]. Let G ≤ Sym(W ) where W is a finite set of size n. Show that we have the following: g (1) If u ∈ W and H ≤ G then (uH )g = (ug)H for all g ∈ G. H H (2) If u ∈ W and Gu ≤ H ≤ G then u is a G-block with |u | = [H : Gu]. (3) If U is a G-block and g, h are elements in G then U g and U h are G-blocks such that either U g = U h or U g ∩ U h = ∅. (4) If T = (Ti)1≤i≤r is a G-invariant partition then each Ti is a G-block. §6: DEFINITIONS AND EXERCISES 59

(5) If G is transitive and U is a G-block then, upon letting T = (Ti)1≤i≤r to be the family of all distinct sets of the form U g as g varies over G (we then say that T is generated by U), we have that r = n/|U| and T is a G-invariant partition of type (r, |U|), and hence in particular |U| divides n. (6) Conversely, if G is transitive and T = (Ti)1≤i≤r is a G-invariant partition then, for 1 ≤ i ≤ r, Ti is a G-block which generates T . Moreover, if G is transitive then: a G-block is trivial ⇔ the partition it generates is trivial. (7) If G is transitive and N/G then the set of all N-orbits is a G-invariant partition, and hence N is semi-transitive and the size of any N-orbit divides n.

HINT. To prove (1) let u ∈ W and H ≤ G. Then for any g ∈ G, we have g h(u)g = hg(ug) for every h ∈ H, and hence (uH )g = (ug)H . To prove (2) let u ∈ W and Gu ≤ H ≤ G. Then for any g ∈ G we have: H g H −1 (u ) ∩ u 6= ∅ ⇒ g(f(u)) = h(u) for some f, h ∈ H ⇒ h gf ∈ Gu ⊂ H ⇒ g ∈ H ⇒ (uH )g = uH . Therefore uH is a G-block. Moreover, for any f, h ∈ H we have −1 H f(u) = h(u) ⇔ h f ∈ Gu, and hence |u | = [H : Gu]. To prove (3) let U be a G-block and let g, h be elements in G. Then for any f ∈ G we have: (U g)f ∩ U g 6= ∅ ⇒ (fg)(u) = g(v) for some u, v ∈ U ⇒ v ∈ U e ∩ U with e = g−1fg ∈ G ⇒ (because U is a block) e(U) = U ⇒ (by applying g to both sides) (fg)(U) = g(U) ⇒ (U g)f = U g. Therefore U g is a block, and similarly so is U h. Moreover: U g ∩ U h 6= ∅ ⇒ g(u) = h(v) for some u, v ∈ W ⇒ v ∈ U e ∩ U with e = h−1g ∈ G ⇒ (because U is a block) e(U) = U ⇒ (by applying h to both sides) g(U) = h(U) ⇒ U g = U h. (4) is obvious, and (5) follows from (3). Moreover, (6) follows from (4) and (5). Finally, to prove (7) assume that G is transitive and let N/G. Then for any g ∈ G we have N g = N and hence by taking N for H in (1) we get (uN )g = (ug)N for all u ∈ W . Therefore by taking uN for U in (5) we see that the set of all N-orbits is a G-invariant partition, and hence N is semi-transitive and the size of any N-orbit divides n.

DEFINITION (D27). [Maximal and Minimal Normal Subgroups]. Define a maximal subgroup of a group to be a subgroup which is not contained in any subgroup other than itself or the whole group; note that every nonidentity finite group has maximal subgroups. Define a minimal normal subgroup of a group to be a minimal element in the set of all nonidentity normal subgroups; note that every nonidentity finite group has minimal normal subgroups.

DEFINITION (D28). [Primitive and Imprimitive Groups]. Now we turn to the concept of primitivity which, like sesqui-transitivity, is between transitivity and two-transitivity. So let G ≤ Sym(W ) where W is a finite set. Define G to be imprimitive if it is transitive and has a system of imprimitivity. Define G to be primitive if it is transitive but not imprimitive.

EXERCISE (E37). [Primitivity Lemma]. Let G ≤ Sym(W ) where W is a finite set of size n. Show that have the following: (1) G primitive ⇔ G transitive and has no nontrivial block ⇔ G transitive and Gu maximal in G for every u ∈ W . [In view of (E32), the phrase “for every u ∈ W ” may be replaced by the phrase “for some u ∈ W ”]. 60 LECTURE L6: PAUSE AND REFRESH

(2) G two-transitive ⇒ G primitive. (3) G sesqui-transitive and imprimitive ⇒ G semi-Frobenius.

HINT. The first implication in (1) follows from (5.7.5) and (5.7.6). To prove the second implication, first assume that G is transitive but has no nontrivial block, H and let Gu ≤ H ≤ G with u ∈ W ; then by (5.7.2) we see that u is a G-block with H |u | = [H : Gu], and hence [H : Gu] = 1 or n; the transitivity of G tells us that [G : Gu] = n, and therefore we must have H = Gu or G. Conversely, suppose that G is transitive and has a nontrivial block U; then we can find u 6= v in U, and w g in W \ U; now upon letting H = stabG(U) = {g ∈ G : U = U} we have H ≤ G, and by the blockness of U we get Gu ≤ H; since u 6= v in U, by the transitivity of G we find h ∈ G with h(u) = v and then h∈ / Gu but by the blockness of U we get h U = U and hence h ∈ H and therefore Gu 6= H; since u 6= w in W with u ∈ U and w∈ / U, by the transitivity of G we find f ∈ G with f(u) = w and then f∈ / H and hence H 6= G; thus Gu is not maximal in G. To prove (2) assume that G is two-transitive. Let if possible, U be a nontrivial G-block. Then first by the nontriviality we can find u 6= v in U and w ∈ W \U, and then by the two-transitivity we can find g ∈ G such that g(u) = u and g(v) = w. Now U g ∩ U 6= ∅ because u ∈ U g ∩ U, and U g 6= U because w ∈ U g \ U. This contradicts the blockness of U. Therefore G is primitive by (1). Finally, to prove (3) assume that G is sesqui-transitive and imprimitive. Then there is a G-invariant partition T = (Ti)1≤i≤r of W of type (r, s) with r > 1 and s > 1 and Ti = {wi1, . . . , wis} where wi1, . . . , wis are distinct points of W . Given any u 6= v in W , upon letting H = Gu, we want to show that Hv = 1. By relabelling the Ti and the wij we may assume that u = w11. Since G is sesqui-transitive, all the H-orbits on W \{u} have the same size, say m. Since W \{u} is a disjoint union of the H-orbits on it, m divides |W \{u}| = rs − 1. H Therefore GCD(m, s) = 1. Fix any i with 1 < i ≤ r. Then clearly u∈ / (Ti) H h H H and ((Ti) ) = (Ti) for all h ∈ H, and hence (Ti) is a disjoint union of some H H-orbits on W \{u} and therefore m divides |(Ti) |. Since T is a G-invariant H H partition, (Ti) is also the disjoint union of some of the Tj, and hence |(Ti) | is H divisible by s. Since GCD(m, s) = 1, it follows that ms divides |(Ti) |. Now we H H H H H have the union (Ti) = (wi1) ∪ · · · ∪ (wis) with |(wi1) | = ··· = |(wis) | = m H H H and hence |(Ti) | ≤ ms; since ms divides |(Ti) |, we must have |(Ti) | = ms and the said union must be disjoint. Since Ti = {wi1, . . . , wis} and the H-orbits H H H (wi1) ,..., (wis) are pairwise disjoint, we must have: (*) (wij) ∩ Ti = {wij} for 1 ≤ j ≤ s. If v ∈ Ti and h ∈ Hv, then the G-blockness of Ti tells us that h H h H (Ti) = Ti and, for 1 ≤ j ≤ s, we clearly have ((wij) ) = (wij) , and hence by h H h H h h H (*) we get (wij) = ((wij) ∩ Ti) = ((wij) ) ∩ (Ti) = (wij) ∩ Ti = {wij}; since i was any integer with 1 < i ≤ r and h was any element of Hv, we conclude that: (**) v ∈ Ti with i ∈ {2, . . . , r} ⇒ Hv ≤ GTi where we recall that GTi = g {g ∈ G : t = t for all t ∈ Ti}. Clearly Hv = (Gv)u, and hence by interchanging ∗ u and v in (**) we get: (1 ) v ∈ Ti with i ∈ {2, . . . , r} ⇒ Hv ≤ GT1 . If v ∈ Ti ∗ then, for 2 ≤ j ≤ s, obviously GT1 ≤ Hw1j and hence by (1 ) we get Hv ≤ Hw1j ; ∗ again since i was any integer with 1 < i ≤ r, we conclude that: (2 ) v ∈ Ti with ∗ i ∈ {2, . . . , r} ⇒ Hv = Hw12 = ··· = Hw1s . The only property of v used in (2 ) was ∗ that v∈ / T1; consequently: (3 ) Hw12 = ··· = Hw1s = Hw for all w ∈ W \ T1. By ∗ 0 ∗ 0 (3 ) we see that: (1 ) Hw1j = 1 for 2 ≤ j ≤ s. By (2 ) and (1 ) we also see that: §6: DEFINITIONS AND EXERCISES 61

0 0 0 (2 ) v ∈ Ti with i ∈ {2, . . . , r} ⇒ Hv = 1. Finally by (1 ) and (2 ) we conclude that we always have Hv = 1.

DEFINITION (D29). [Characteristic Subgroup]. Define a characteristic subgroup of a group X to be a subgroup which is mapped onto itself by every automorphism of X. Clearly all “group theoretically well-defined” subgroups are characteristic subgroups. For instance: (1) the center of X is a characteristic subgroup of X; similarly: (2) if X is finite and has a unique p-Sylow subgroup for some prime divisor p of |X| then it is also a characteristic subgroup of X; more generally: (3) if X is finite and p is any prime then the subgroup of G generated by all of its p-Sylow subgroups (which is denoted by p(G) and which may be defined as the subgroup of G generated by all of its elements of p-power order) is a characteristic subgroup of X; likewise: (4) if X has a unique minimal normal subgroup then it is a characteristic sub- group of X; more generally: (5) the subgroup of X generated by all of its minimal normal subgroups is a characteristic subgroup of X; moreover: (6) if Y is a minimal normal subgroup of X, and Z is a nonidentity characteristic subgroup of Y , then (clearly Z is normal in X and hence by minimality) we must have Y = Z; finally: (7) if Y is a minimal normal subgroup of a finite group X, and Y has a unique minimal normal subgroup Z, then (by (4) and (6) we get Y = Z and by minimality Z is contained in every nonidentity normal subgroup of Y and hence) Y must be a simple group.

DEFINITION (D30). [Elementary Abelian Group]. By an additive ele- mentary abelian group we mean an additive abelian group which is isomorphic to GF(q)+ for some prime power q = pe with prime p and positive integer e. Equiva- lently, it is the direct sum of a finite number of copies of an additive cyclic group of prime order. By an elementary abelian group we mean the multiplicative version of an additive elementary abelian group.

EXERCISE (E38). [Normality Lemma]. Let G ≤ Sym(W ) where W is a finite set of size n. Show that for any 1 6= N/G we have the following: (1) G primitive ⇒ N transitive. (2) G two-transitive ⇒ N sesqui-transitive. (3) G two-transitive and N regular ⇒ N elementary abelian.

HINT. To prove (1), note that if N/G with G primitive then, by (E36)(7), either all the N-orbits are singletons or W is the only N-orbit; if N 6= 1 then the first alternative is not possible and hence W is the only N-orbit, i.e., N is transitive. To prove (2) assume that 1 6= N/G with G two-transitive. Then N is transitive by (E27(2) and (1). For any u ∈ W we have Nu = N ∩ Gu /Gu with Gu transitive on W \{u}, and hence by (E36)(7) we see that Nu is semi-transitive on W \{u}. Therefore N is sesqui-transitive. Finally, to prove (3), let N/G be such that N is regular. Fixing any w ∈ W , let f : W → N be the bijection whose inverse is given by g 7→ g(w), and let 62 LECTURE L6: PAUSE AND REFRESH

] f : Sym(W ) → Sym(N) be the isomorphism induced by f. Then, for any h ∈ Gw and g ∈ N, upon letting u = g(w) ∈ W and v = h(u) ∈ W and g∗ = f(v) ∈ N, by the definition of f we have f(u) = g and g∗(w) = v, and hence by the definition of f ] we get f ](h)(g) ∈ N with (f ](h)(g))(w) = v, and by taking (g, h, w) for (h, g, u) in the Conjugation Rule (E31) we also get gh ∈ N with gh(w) = v; since g 7→ g(w) ] h gives a bijection N → W , we must have f (h)(g) = g . Thus, for every h ∈ Gw, the permutation f ](h): N → N coincides with the conjugation automorphism given by g 7→ gh and hence the order of any g ∈ N equals the order of f ](h)(g) ∈ N. For any g 6= 1 6= g0 in N we have g(w) 6= w 6= g0(w) and, assuming G to be two-transitive, 0 ] 0 we can find h ∈ Gw with h(g(w)) = g (w), and then clearly f (h)(g) = g , and hence the order of g equals the order of g0. Let p > 1 be the common order of all elements of N \{1}. If p = rs with r > 1 and s > 1 then taking 1 6= g ∈ N we get 1 6= gr ∈ N and the order of gr is s with 1 < s < p which is a contradiction. Therefore p must be prime. If the order of N were not a power of p then by Cauchy’s Theorem (E26) N would contain an element whose order is a prime different from p which would be a contradiction. Therefore the order of N must be a power of p, and hence N has a nontrivial center M; obviously M is a characteristic subgroup of N and hence an automorphism of N cannot send an element in M to one not in M; but we have just shown that any two nonidentity elements of N are conjugate in G; therefore we must have M = N, i.e., N is abelian. Since the order of every nonidentity element of N is p, it follows that N is elementary abelian.

EXERCISE (E39). [Fixed Point Lemma]. Let G ≤ Sym(W ) where W is a finite set of size n. Assume that G is semi-Frobenius, and let N = {1} ∪ N 0 where N 0 is the set of all fixed point free elements of G. Show that we have the following: (1) |N| = n, and for each prime divisor p of n we have that N has an element of order p, and every element of G whose order is a power of p belongs to N. (2) If all elements of N 0 have equal order, then that order is a prime p, n is a power of p, N is the unique p-Sylow subgroup of G, N is a characteristic subgroup of G, and N is regular. (3) If G/G∗ ≤ Sym(W ) with G∗ two-transitive, then all elements of N 0 have equal order which is a prime p, n is a power of p, N is the unique p-Sylow subgroup of G, N is a characteristic subgroup of G, N is regular, and N is elementary abelian. (4) If G/G∗ ≤ Sym(W ) with G∗ two-transitive and G minimal normal in G∗, then G is regular and G = N = an elementary abelian group.

HINT. To prove this, first note that, assuming G to be semi-Frobenius, and letting w1, . . . , wn be the distinct elements of W , by (E35)(4) and (E35)(6) we have |G| = mn where m is a divisor of n − 1 with |Gwi | = m for 1 ≤ i ≤ n.

Since G is semi-Frobenius, we have Gwi ∩ Gwj = {1} for all i 6= j, and hence 0 0 | ∪1≤i≤n Gwi | = 1 + (m − 1)n. Upon letting N = {1} ∪ N where N is the set of 0 all fixed point free elements of G, we clearly have N = G \ ∪1≤i≤nGwi , and hence |N 0| = mn − [1 + (m − 1)n], and therefore |N| = 1 + |N 0| = n. Since the order of

Gwi divides n − 1, the order of any nonidentity element of Gwi must divide n − 1 0 and hence it cannot divide n; since N = G\∪1≤i≤nGwi , every nonidentity element of G whose order divides n must belong to N 0; therefore N contains every element of G whose order divides n. Since n divides |G|, by Sylow’s Theorem G does have elements of every prime order dividing n. Consequently, for each prime divisor p of §6: DEFINITIONS AND EXERCISES 63 n we have that N has an element of order p, and every element of G whose order is a power of p belongs to N. This completes the proof of (1). To prove (2) assume that all elements of N 0 have the same order, say p. Then by (1), p must be prime and n must be a power of p. Since |G| = mn with GCD(m, n) = 1, n must be the highest power of p dividing |G|. By (1), |N| = n and N contains every element of G whose order is a power of p. Therefore N must be the unique p-Sylow subgroup of G. Consequently N is a characteristic subgroup of G. Since G transitive, W is an orbit of G; since |W | = a power of p, and N is a p-Sylow subgroup of G, by Sylow Transitivity (E28)(5) we see that N is transitive on W ; therefore, since |N| = n, by (E35)(5) we conclude that N is regular. This completes the proof of (2). To prove (3) assume that G/G∗ ≤ Sym(W ) with G∗ two-transitive. Let A = {(h, u, v) ∈ N 0 × W × W : h(u) = v}. Then (h, u, v) 7→ (h, u) gives a bijection A → N 0 ×W , and |W | = n and by (1) we have |N 0| = n−1; therefore |A| = (n−1)n. Let us fix some (h, u, v) ∈ A. Upon letting B = {(u0, v0) ∈ W × W : u0 6= v0}, given any (u0, v0) ∈ B, by the two-transitivity of G∗ we can find g ∈ G∗ with g(u) = u0 and g(v) = v0, and now by the Conjugation Rule (5.4) and the normality of G we get (hg, u0, v0) ∈ A; for each (u0, v0) ∈ B we fix such g ∈ G∗ and then (u0, v0) 7→ (hg, u0, v0) gives an injection B → A; but |B| = (n − 1)n = |A| and hence the said injection must be a bijection. Therefore {hg : g ∈ G∗} = N 0. Since conjugate elements have the same order, it follows that all elements of N 0 have equal order, say p. Now by (2) it follows that p is prime, n is a power of p, N is the unique p-Sylow subgroup of G, N is a characteristic subgroup of G, and N is regular. Since N is a regular normal subgroup of the two-transitive group G∗, by (E38)(3) we see that N is elementary abelian. This completes the proof of (3). Finally, if G/G∗ ≤ Sym(W ) with G∗ two-transitive and G minimal normal in G∗, then by (3) we see that N is regular, elementary abelian, and a characteristic subgroup of G. Hence by (D29)(6) we get G = N. This completes the proof of (4).

DEFINITION (D31). [Direct Product]. The direct product of a finite number number of groups X1,...,Xm is defined to be their cartesian product

X = X1 × · · · × Xm

equipped with componentwise multiplication, i.e., for any y = (y1, . . . , ym) and z = (z1, . . . , zm) in X1 ×· · ·×Xm we have yz = (y1z1, . . . , ymzm). To distinguished this from the mere cartesian product we may say something like “where the product is direct.”

EXERCISE (E40). [Double Normality Lemma]. Let G ≤ Sym(W ) where W is a finite set of size n. Assume that G has two distinct minimal normal subgroups N and N 0. Show that we have the following: (1) N ∩ N 0 = 1, N and N 0 centralize each other, and NN 0 = N × N 0 /G where the product is direct. (2) If G is primitive, then N and N 0 are the centralizers of each other in Sym(W ), N and N 0 are regular, N and N 0 are nonabelian groups, G has no minimal normal subgroup other than N and N 0, and NN 0 is a characteristic subgroup of G with |NN 0| = n2. 64 LECTURE L6: PAUSE AND REFRESH

[Now by (E34)(3) it follows that N and N 0 are permutation-isomorphic and, in the sense explained there, they are the left and right regular representations of each other]. (3) G cannot be two-transitive. (4) If G/G∗ ≤ Sym(W ) with G primitive and minimal normal in G∗, then G∗ cannot be two-transitive.

HINT. To prove this assume that G has two distinct minimal normal subgroups N and N 0. Then obviously N ∩ N 0 /G with and hence by the minimality of N and N 0 we get N ∩ N 0 = 1, and therefore NN 0 = N × N 0 /G where the product is direct. Now the commutator group [N,N 0] is the subgroup of G generated by all the commutators ghg−1h−1 with g ∈ N and h ∈ N 0; clearly gh = hg ⇔ ghg−1h−1 = 1, and hence: [N,N 0] = 1 ⇔ N and N 0 centralize each other. For all g ∈ N and h ∈ N 0 by the normality of N 0 we have ghg−1 ∈ N 0 and hence ghg−1h−1 ∈ N 0, and similarly by the normality of N we get ghg−1h−1 ∈ N; therefore [N,N 0] ≤ N ∩ N 0. Consequently [N,N 0] = 1. Hence N and N 0 centralize each other. This proves (1). To prove (2) assume that G is primitive. Then by (E38)(1) we know that N and N 0 are transitive, and hence by (E34)(2) we see that N and N 0 are the centralizers of each other in Sym(W ), N and N 0 are regular, and N 0 is the only transitive subgroup of Sym(W ) which centralizes N. Since N ∩ N 0 = 1 and N and N 0 are the centralizers of each other, they must be nonabelian. As we have just seen, by (E38)(1) and (E34)(2) it follows that if N 00 is any minimal normal subgroup of G with N 00 6= N such that N 00 centralizes N then N 00 is the only transitive subgroup of Sym(W ) which centralizes N, and since (as said above) N 0 is also the only transitive subgroup of Sym(W ) which centralizes N, we must have N 00 = N. Thus G has no minimal normal subgroups other than N and N 0, and hence NN 0 is a characteristic subgroup of G. Since N and N 0 are regular, by (E35)(5) we get |N| = |N 0| = n; since NN 0 is the direct product of N and N 0, we must have |NN 0| = n2. This completes the proof of (2), and the bracketed remark at the end of it follows from (E34)(3). If G were two-transitive, then by (E37)(2) G would be primitive, and hence by (2) N would be a nonabelian regular normal subgroup of G which would contradict (E38)(3). This proves (3). To prove (4) let G/G∗ ≤ Sym(W ) with G primitive and minimal normal in G∗. Then by (2) we know that NN 0 is a characteristic subgroup of G, and hence by (D29)(2) we get NN 0 = G. By (2) we have |NN 0| = n2, and hence by (E35)(7) we see that G cannot be sesqui-transitive, and therefore by (E38)(2) we conclude that G∗ cannot be two-transitive.

EXERCISE (E41). [Burnside’s Lemma]. Let G ≤ Sym(W ) where W is a finite set of size n. Assuming G to be two-transitive, show that for any minimal normal subgroup N of G we have the following. (5.14.1) N is primitive or semi-Frobenius. (5.14.2) If N is semi-Frobenius then it is regular. (5.14.3) If N is regular then it is elementary abelian. (5.14.4) If N is primitive abelian simple then it is regular. (5.14.5) If N is primitive then it is simple. §6: DEFINITIONS AND EXERCISES 65

HINT. To prove this assume that G is two-transitive and let N be a minimal normal subgroup of G. Then by (5.10.2) N is sesqui-transitive, and hence by (5.8.3) it is primitive or semi-Frobenius, which proves (5.14.1). (5.14.2) follows by taking (N,G) for (G, G∗) in (5.12.4). (5.14.3) follows from (5.10.3). (5.14.4) follows by noting that if N is primitive then N is transitive and by (5.3.4) and (5.8.1) we see that for any u ∈ W we have Nu 6= N (because n > 1 by two-transitivity of G) and there is no group H with Nu ≤ H ≤ N and Nu 6= H 6= N, and hence if N is also abelian simple (i.e., of prime order) then we must have Nu = 1, i.e., N is regular. Finally, (5.14.5) follows by noting that if N is primitive then by taking (N,G) for (G, G∗) in (E40)(4) we see that N has a unique minimal normal subgroup, and hence by (5.9.7) we conclude that N is simple.

EXERCISE (E42). [Burnside’s Theorem]. Given any permutation group G (acting on a finite set), show that if G is two-transitive then it has a unique minimal normal subgroup N. Also show that if the said subgroup N is regular then it is elementary abelian, and if it is nonregular then it is primitive as well as nonabelian simple.

HINT. Namely, if G is two-transitive then by (E40)(3) we see that G has a unique minimal normal subgroup N. The rest follows from (E41).

NOTE. Let N = {1} ∪ N 0 where N 0 is the set of all fixed point free elements of G. Recall that G is semi-Frobenius means G is transitive and has no nonidentity element fixing two points, and G is Frobenius means G is semi-Frobenius but not regular. In the Fixed Point Lemma (E39)(4) we have proved that if G is semi- Frobenius and G/G∗ ≤ Sym(W ) with G∗ two-transitive and G minimal normal in G∗, then G is regular and G = N = an elementary abelian group. This motivates Frobenius’ Theorem, proved in his 1901 paper, which says that if G is Frobenius then N is a regular normal subgroup of G. It is clear that Frobenius’ Theorem implies the Fixed Point Lemma (E39)(4), and this gave rise to the second proof of Burnside’s Theorem which, unlike his first proof, uses Frobenius’ Theorem, and which he gave in the second edition of his book [?].

EXERCISE (E43). [Simplicity of Alternating Groups]. Show that the alternating group An is simple for n ≥ 5.

HINT. Now An ≤ Sn = Sym(W ) with W = {1, . . . , n}, and we want to show that, for any 1 6= H/An we have H = An. If H contains a 3-cycle then relabelling 1, . . . , n we may assume that (123) ∈ H; now τ = (213) = (123)2 ∈ H and for −1 any m ∈ {4, . . . , n} we have σ = (12)(3m) ∈ An and hence (12m) = στσ ∈ H; consequently by L1§11(E3) we get H = An. Thus it suffices to show that H contains a 3-cycle. Let d = min{|V (h)| : 1 6= h ∈ H} where V (h) = {u ∈ W : h(u) 6= u}, and take h ∈ H with |V (h)| = d. We want to show that d = 3. Suppose if possible that d ≥ 4. Then either (1) h is a product of disjoint transpositions which upon relabelling 1, . . . , n looks like h = (12)(34) ... with V (h) = {1, . . . , d}, or (2) in the contrary case upon relabelling 1, . . . , n we have V (h) = {1, . . . , d} with d ≥ 5 and as a product of disjoint cycles of nonincreasing length h = (123 ... ) ... . Now 0 −1 0 g = (345) ∈ An and hence upon letting h = ghg we have h ∈ H. Referring to 66 LECTURE L6: PAUSE AND REFRESH

(E31), in case (1) we get h0 = (12)(45) ... and in case (2) we get h0 = (124 ... ) ... ; consequently in both the cases, upon letting h∗ = h−1h0, we have 1 6= h∗ ∈ H. But in case (1) we see that V (h∗) ⊂ {3, . . . , d} ∪ {5} and in case (2) we see that V (h∗) ⊂ {2, . . . , d}. This contradicts the minimality of d.

EXERCISE (E44). [Unit Ideals in Polynomial Rings]. Give a detailed ` proof of L2§5(T5). In other words, show that if L = H∈Ω H is a partition of a nonempty set L (where Ω is a set of pairwise disjoint nonempty subsets of S), if RL = R[{Xl : l ∈ L}] is the polynomial ring in indeterminates {Xl : l ∈ L} over a field R, if for each H ∈ Ω we are given a nonunit ideal JH in the polynomial ring RH = R[{Xh : h ∈ H}], and if JL is the ideal in RL generated by all the ideals JH as H varies over Ω, then JL is a nonunit ideal in RL. Also show that this is not true if R is assumed to be a domain instead of a field.

HINT. For the negative assertion take RL = R[X,Y ] with R = Z, and take JX and JY to be the ideals in RX = R[X] and RY = R[Y ] generated by (2,X) and (3,Y ) respectively; then these are clearly nonunit ideals but the ideal generated by them in RL is the unit ideal. Turning to the positive assertion, first considering the case when L is finite, suppose if possible that JL is the unit ideal. We can label the distinct members of Ω as H(i)1≤i≤n with n ∈ N+ and we can label the distinct elements of H(i) as (ij)1≤j≤m(i) with m(i) ∈ N+. We can take an indeterminate Y over RL, and let us do our work in finite algebraic field extensions of the field

K = R(Y,X11,...,X1m(1),...,Xn1,...,Xnm(n)).

Note that now JL is an ideal in the subring RL of K given by

RL = R[X11,...,X1m(1),...,Xn1,...,Xnm(n)] and JH(i) is a nonunit ideal in the subring RH(i) of RL given by

RH(i) = R[Xi1,...,Xim(i)]. By our supposition we can write X 1 = aibi 1≤i≤n where for 1 ≤ i ≤ n we have × ai = ai(Xi1,...,Xim(i)) ∈ RH(i) \ R and

bi = bi(X11,...,X1m(1),...,Xn1,...,Xnm(n)) ∈ RL.

Let S0 = R(Y ) and K0 = K. For 1 ≤ i ≤ n we shall find elements (yij)1≤j≤m(i) in a finite algebraic field extension Si of Si−1 inside an overfield Ki of Ki−1 with Ki = Ki−1(Si) such that ai(yi1, . . . , yim(i)) = 0. In view of an obvious induction, it suffices to do this for n = 1. If a1 = 0 then we can take S1 = S0 with K1 = K0 and y1j = 0 for 1 ≤ j ≤ m(1). If a1 6= 0, then upon relabelling X11 ...,X1m(1) ∗ we may assume that a1 6∈ R = R[X12,...,X1m(1)]. By applying L3§12(E6) to the ∗ product of the nonzero coefficients of a1 as a polynomial in X11 over R we can ∗ find (y1j)2≤j≤m(1) in S0 such that for the polynomial a (X11) ∈ S0[X11] obtained ∗ by substituting (y1j)2≤j≤m(1) for (X1j)2≤j≤m(1) in a1 we have a (X11) 6∈ S0. Now §6: DEFINITIONS AND EXERCISES 67 letting K1 to be an overfield of K0 such that K1 = K0(S1) where S1 is a splitting ∗ ∗ field of a over S0, we can find y11 ∈ S1 with a (y11) = 0. By substituting Xij = yij for 1 ≤ i ≤ n and 1 ≤ j ≤ m(i) in the equation P 1 = 1≤i≤n aibi we get the contradiction 1 = 0. Now, without assuming L to be finite, suppose if possible that JL is the unit ideal. Then we can find a finite number of distinct members H(1),...,H(n) of Ω P with n ∈ N+ such that 1 = 1≤i≤n aibi where for 1 ≤ i ≤ n we have ai ∈ JH(i) and bi ∈ RH(1)∪···∪H(n). Clearly for 1 ≤ i ≤ n we can find a finite nonempty subset L(i) of H(i) such that for 1 ≤ i ≤ n we have ai ∈ RH(i) and bi ∈ RL(1)∪···∪L(n). Thus we are reduced to the case when L is finite.

EXERCISE (E45). [Generalized Power Series]. Give details of the proof of L2§7(E13). In other words, given a field K and an ordered abelian group G, show that the set G K((X))G = {A ∈ K : Supp(A) is well ordered} is a field where addition is componentwise and multiplication is Cauchy multipli- cation. Concretely speaking, a typical member of K((X))G can be written as X i A(X) = AiX i∈G where we are writing Ai for the previous A(i) [cf. L2§3]. For proving that K((X))G is a field, the fact that multiplication makes sense is to be established by showing that  for any A, B in K((X)) and g ∈ G,  G (V4) {(i, j) ∈ G2 : i + j = g with i ∈Supp(A) and j ∈Supp(B)} is a finite set and ( for any A, B in K((X)) , (V5) G Supp(AB) is well ordered. and the existence of inverse is to be established by showing that ( 0 6= A ∈ K((X))G (V6) 0 0 ⇒ AA = 1 for some A ∈ K((X))G.

HINT. To prove (V4) let W be the set of all (i, j) ∈ G2 such that i + j = g for some i ∈ Supp(A) and j ∈ Supp(B). If W is empty then it is finite. So assume that W is nonempty. Then U1 = {i ∈ Supp(A): i + j = g for some j ∈ Supp(B)} is a nonempty subset of (the well ordered set) Supp(A) and hence it has a smallest element i1. If U2 = {i ∈ Supp(A) \{i1} : i + j = g for some j ∈ Supp(B)} is nonempty then (because it is a subset of Supp(A)) it has a smallest element i2 and clearly i1 < i2. If U3 = {i ∈ Supp(A) \{i1, i2} : i + j = g for some j ∈ Supp(B)} is nonempty then (because it is a subset of Supp(A)) it has a smallest element i3 and clearly i1 < i2 < i3. If this process continues indefinitely then we would have found a strictly decreasing infinite sequence g − i1 > g − i2 > g − i3 > . . . in Supp(B) which would contradict the well orderedness of Supp(B). Therefore Un+1 must be empty for some n > 1 in N. It follows that W is the finite set

{(i1, g − i1),..., (in, g − in)}. 68 LECTURE L6: PAUSE AND REFRESH

To prove (V5) let W be the set of all g ∈ G such that g = i + j for some i ∈ Supp(A) and j ∈ Supp(B). Clearly Supp(AB) ⊂ W and hence it suffices to show that W is well ordered, because every subset of a well ordered set is well ordered. So given any nonempty subset V of W , we want show that V has a smallest element. Let U1 be the set of all i ∈ Supp(A) such that i + j ∈ V for some j ∈ Supp(B). For every i ∈ U1 let J(i) be the set of all j ∈ Supp(B) such that i + j ∈ V . Then J(i) is nonempty subset of Supp(B) and hence it has a smallest element j(i). Now U1 is a nonempty subset of Supp(A) and hence it has a smallest element i1. If U2 = {i ∈ U1 : j(i) < j(i1)} is nonempty then it has a smallest element i2 because it is subset of Supp(A). If U3 = {i ∈ U1 : j(i1) < j(i2)} is nonempty then it has a smallest element i3 because it is subset of Supp(A). If this process continues indefinitely then we would have found a strictly decreasing infinite sequence j(i1) > j(i2) > j(i3) > . . . in Supp(B) which would contradict the well orderedness of Supp(B). Therefore Un+1 must be empty for some n > 1 in N. Now clearly U1 % U2 % U3 ··· % Un % Un+1 = ∅. Also for any m ∈ {1, . . . , n} and any i ∈ Um \ Um+1 we clearly have i ≥ im with j(i) ≥ j(im) and hence i + j(i) ≥ im + j(im). It follows that min{i1 + j(i1), . . . , in + j(in)} is the smallest element of V . To prove (V6) we can write A = αXf (1 − B) where α ∈ K× with f ∈ G and 2 3 B ∈ K((X))G with Supp(B) ⊂ G+. Now it suffices to show that 1+B+B +B +... makes sense as an element of K((X))G and there it is the inverse of 1 − B. By taking H = Supp(B), and noting that any subset of a finite (resp: well ordered) set is finite (resp: well ordered), this follows from the following claims (1) to (4) which we shall prove in a moment. Let H be a well ordered subset of G+. For any m ∈ N+ let (H, m) be the set of all g ∈ G+ such that g = h1 + ··· + hm for some elements h1, ··· + hm in H, and for any g ∈ (H, m) let g(H, m) be the set of all sequences (h1, . . . , hm) of elements in H such that g = h1 + ··· + hm. Also let (H, ∞) = ∪m∈N+ (H, m), and for any g ∈ (H, ∞) let g(H) be the set of all m ∈ N+ such that g ∈ (H, m). We claim that then

(1) for any m ∈ N+ and g ∈ (H, m) the set g(H, m) is finite and

(2) for any m ∈ N+ the set (H, m) is a well ordered subset of G+ and

(3) the set (H, ∞) is a well ordered subset of G+ and (4) for any g ∈ (H, ∞) the set g(H) is finite.

Namely, for any well ordered subset L of G+, let (H,L) be the set of all g ∈ G+ such that g = h + l for some h ∈ H and l ∈ L, and for any g ∈ (H,L) let g(H,L) be the set of all (h, l) ∈ H × L such that g = h + l. Then the above proof of (V4) subsumes a proof of (1*) which says that for any g ∈ (H,L) the set g(H,L) is finite and, the above proof of (V5) subsumes a proof of (2*) which says that (H,L) is a well ordered subset of G+. Now by induction on m, (1) follows from (1*), and (2) follows from (2*). §6: DEFINITIONS AND EXERCISES 69

In proving (3) and (4) we shall use the following easy to prove criterion of well orderedness which was implicitly used in the above proofs of (V4) and (V5) and which says that

I ⊂ G is well ordered  ⇔ every infinite sequence g , g ,... in I has a nondecreasing (5) 1 2  subsequence gµ(1) ≤ gµ(2) ≤ ... with 1 ≤ µ(1) < µ(2) < . . .  ⇔ I has NO strictly decreasing infinite sequence g1 > g2 > . . . .

To give a cyclical proof of (5), first assume that I is well ordered, and let g1, g2,... be an infinite sequence in I. Then, by applying the well orderedness to the nonempty subset {g1, g2,... } of I, we find µ(1) ∈ N+ such that gµ(1) ≤ gi for all i > µ(1), and by applying the well orderedness to the nonempty subset {gµ(1)+1, gµ(1)+2,... } of I, we find µ(1) < µ(2) ∈ N+ such that gµ(2) ≤ gi for all i > µ(2), and so on. Next assume that every infinite sequence in I has a nondecreasing subsequence, and if possible let g1 > g2 > . . . be a strictly decreasing infinite sequence in I. But then clearly the infinite sequence g1, g2,... in I cannot have a nondecreasing subsequence. Finally assume that I has no strictly decreasing infinite sequence, and if possible let J be a nonempty subset of I having no smallest element. Then taking any g1 ∈ J, it is not the smallest element in J and hence g1 > g2 for some g2 ∈ I. Again g2 is not the smallest element in I and hence g2 > g3 for some g3 ∈ I, and so on. To prove (3) suppose if possible that (H, ∞) is not well ordered. Then by (5) it has a strictly decreasing infinite sequence h1 > h2 > . . . . Now for every i ∈ N+ we can write hi = hi1 + ··· + him(i) with m(i) ∈ N+ and hij ∈ H for 1 ≤ j ≤ m(i). Upon relabelling hi1, hi2, . . . , him(i) we may assume that hi1 ≤ hi2 ≤ · · · ≤ him(i). By (5) we can find a nondecreasing subsequence hµ(1)m(µ(1)) ≤ hµ(2)m(µ(2)) ≤ ... . Upon replacing h1 > h2 > . . . by hµ(1) > hµ(2) > . . . we may also assume that h1m(1) ≤ h2m(2) ≤ ... . Now letting h = (hi, hij)1≤i<∞,1≤j≤m(i) we have h ∈ Γ 0 0 0 0 0 where Γ is the set of all systems h = (hi, hij)1≤i<∞,1≤j≤m0(i) of elements hi, hij in 0 0 0 0 0 0 0 H with m (i) ∈ N+ such that hi = hi1 + ··· + him0(i) with hi1 ≤ hi2 ≤ · · · ≤ him0(i) 0 0 and h1m0(1) ≤ h2m0(2) ≤ ... . ∗ ∗ 0 0 Since H is well ordered, there is h ∈ Γ such that h1m∗(1) ≤ h1m0(1) for all h ∈ Γ. 0 ∗ ∗ 0 0 Let us call g , g in G+ archimedeanly comparable to mean that g ≤ n g and 0 ∗ ∗ 0 ∗ ∗ 0 g ≤ n g for some n and n in N+. Let Γ be the set of all h ∈ Γ such that 0 ∗ h1m0(1) and h1m∗(1) are archimedeanly comparable. Without loss of generality we may assume that h ∈ Γ∗. Since H is well ordered, there is g∗ ∈ H such that g∗ is the smallest element in the set of all elements in H which are archimedeanly ∗ comparable to h1,m∗(1). 0 ∗ 0 0 0 ∗ Now for every h ∈ Γ there is a unique p(h ) ∈ N+ with h1 ≤ p(h )g such that 0 0 ∗ p(h ) ≤ q for all q ∈ N+ for which h1 ≤ qg . Let p ∈ N+ be the smallest amongst p(h0) as h0 varies over Γ∗. Let Γ0 be the set of all h0 ∈ Γ∗ with p(h0) = p. Without loss of generality we may assume that h ∈ Γ0. If m(i) = 1 for infinitely many values of i then by arranging these values in a strictly increasing sequence i1 < i2 < . . . of positive integers we get a strictly 70 LECTURE L6: PAUSE AND REFRESH

decreasing infinite sequence hi1 > hi2 > . . . of elements in H contradicting the well orderedness of H. Therefore there is n ∈ N such that m(i) > 1 for all i > n in N. By (5) there exists a strictly increasing sequence of integers n < l1 < l2 < . . . such that hl1,m(l1)−1 ≤ † † † † hl2,m(l2)−1 ≤ ... . For all i ∈ N+ let m (i) = m(li) − 1 with hi = hi1 + ··· + him†(i) † † † † † where hij = hli,j for 1 ≤ j ≤ m (i). Also let h = (hi , hij)1≤i<∞,1≤j≤m†(i). Then clearly h† ∈ Γ∗ but p(h†) < p(h) which is a contradiction. Thus (3) has been established. To prove (4) let L be the set of all g ∈ (H, ∞) for which g(H) is infinite, and suppose if possible that L is nonempty. Then by (3) L has a smallest element h and, by (1), for all i ∈ N+ we can write h = hi1 + ··· + him(i) with hij ∈ H and 1 < m(1) < m(2) ... in N. By (5), upon replacing 1, 2,... by a suitable subsequence we may assume that h11 ≥ h21 ≥ ... . Let gi = hi2 + ··· + him(i). Then g1 ≤ g2 ≤ ... in (H, ∞). By (3) and (5), we can find n ∈ N and g ∈ (H, ∞) such that gi = g for all n < i ∈ N. Clearly g ∈ L but g < h which is a contradiction. Thus (4) has been established.

EXERCISE (E46). [Simplicity of Projective Special Linear Group]. Prove the simplicity of PSL(n, q) for n ≥ 2 with (n, q) 6= (2, 2), (2, 3) as asserted in L3§1.

HINT. In view of (E37)(2), this follows from (E47) to (E50) below.

EXERCISE (E47). [Group Action and Iwasawa’s Simplicity Criterion]. Let a group G act primitively on a finite set W , i.e., let the action θ : G → Sym(W ) be such that θ(G) is a primitive subgroup of Sym(W ). Assume that G is generated by all the conjugates of an abelian normal subgroup H of Gu for some u ∈ W . Let G0 be the commutator subgroup of G. Show that then, using the subscript notation introduced in (D20) above, we have the following: 0 (1) N/G ⇒ either N ≤ G[W ] or G ≤ N. 0 (2) G[W ] 6= G = G ⇒ G/G[W ] is simple.

NOTE. Note that for any action θ : G → Sym(W ) of any group G on any set W we have ker(θ) = G[W ]. The action is faithful means ker(θ) = 1. Moreover, as above, if the group θ(G) has a certain property P then we may say that the group G has property P on W or that the action θ has property P. For instance, in case of a finite set W , we may that G is two-transitive (resp: regular, semi-regular, etc.) on W or G acts two-transitively (resp: regularly, semi-regularly, etc.) on W , to mean that θ(G) is two-transitive (resp: regular, semi-regular, etc.).

HINT. Clearly (2) follows from (1). To prove (1), for any N/G with N 6≤ G[W ], by (E38)(1) we see that N is transitive on W . Consequently, given any γ ∈ G there −1 −1 −1 exists ν ∈ N such that γ(u) = ν(u); now ν γ ∈ Gu and hence ν γHγ ν = H because H/Gu; therefore γHγ−1 = ν(ν−1γHγ−1ν)ν−1 = νHν−1 ≤ G∗ where G∗ is the subgroup of G generated by H and N. Since by assumption G is generated by the conjugates γHγ−1 as γ varies over G, we conclude that G = G∗. Now the residue class epimorphism G∗ → G∗/N induces an isomorphism of G∗/N §6: DEFINITIONS AND EXERCISES 71 onto H/(H ∩ N) where the later is abelian because H is abelian. Therefore G∗/N, i.e., G/N is abelian, and hence G0 ≤ N.

EXERCISE (E48). [Two-Transitive Action on Projective Space]. Let n > 1 be an integer, and let k be a field. Let V be the vector space of all column vectors of size n over k, and let W be the associated projective space, i.e., the set of all one-dimensional subspaces of V . Consider the action G = SL(n, k) on W obtained by putting A(kv) = k(Av) for all A ∈ G and 0 6= v ∈ V . Also consider the induced action of G = PSL(n, k) on W . Show that the action of G on W is two-transitive, and hence so is the action of G.

NOTE. The definitions of t-transitive group, τ-antitransitive group, (t, τ) group, and sharp t-transitive group, introduced in (E24) are applicable without the set W being finite. For instance, G ≤ Sym(W ) is t-transitive means |W | ≥ t and any t points of W can be mapped to any other t points of W by an element of G. Similarly for t-transitive action, τ-antitransitive action, (t, τ) action, and sharp t-transitive action.

HINT. Given any nonzero elements v1, v2, w1, w2 in V such that kv1 6= kv2 and kw1 6= kw2, we can find nonzero elements v3, . . . , vn, w3, . . . , wn in V such that v1, . . . , vn and w1, . . . , wn are k-bases of V . Now we can find C in GL(n, k) such that Cvi = wi for 1 ≤ i ≤ n. Let A be obtained by multiplying the first column of −1 C by µ = det(C) . Then A ∈ G, and for 1 ≤ i ≤ n we have Avi = µwi and hence A(kvi) = kwi.

EXERCISE (E49). [Transvection Matrix and Dilatation Matrix]. Let the notation be as in (E48). In L5§5(Q31)(C51) we have defined the transvection matrix Tij(n, λ) and the dilatation matrix Tii(n, λ). In L5§5(Q31)(T121) we have shown that ( G is generated by the matrices T (n, λ) as (1) ij i 6= j vary over {1, . . . , n} and λ varies over k.

Let H denote the subgroup of G generated by the matrices T1j(n, λ) as j varies over {2, . . . , n} and λ varies over k. Let ei ∈ V be the column vector with 1 in the i-th spot and zeroes elsewhere. Let u ∈ W consist of all multiples of e1. Show that then H is an abelian normal subgroup of Gu, and G is generated by all the conjugates of H.

HINT. In L5§5(Q31)(C55) we have defined the permutation matrix T (n, σ) for any σ ∈ Sn and, in view of (1), the generation of G by all the conjugates of H follows by noting that −1 (2) T (n, σ)Tij(n, λ)T (n, σ) = Tσ(i)σ(j)(n, λ) and hence −1 (3) Sii(n, σ)Tij(n, λ)Sii(n, σ) = Tσ(i)σ(j)(n, sgn(σ)λ) where

(4) Sii(n, σ) = T (n, σ)Tii(n, sgn(σ)) ∈ G 72 LECTURE L6: PAUSE AND REFRESH and where (2) ⇒ (3) because for all l ∈ {1, . . . , n} and µ ∈ k we have  Tij(n, λ) if i 6= l 6= j −1  (5) Tll(n, µ)Tij(n, λ)Tll(n, µ) = Tij(n, λµ) if i = l  −1 Tij(n, λµ ) if l = j. n−1 For any (λ2, . . . , λn) ∈ k consider the n × n matrix

S(λ2, . . . , λn) = (S(λ2, . . . , λn)ij) where  1 if i = j  S(λ2, . . . , λn)ij = λj if 1 = i 6= j 0 if 1 < i 6= j. Clearly (λ2, . . . , λn) 7→ S(λ2, . . . , λn) gives an isomorphism + n−1 ∗ + n−1 (k ) → H = {S(λ2, . . . , λn):(λ2, . . . , λn) ∈ (k ) } of an additive abelian group onto a multiplicative subgroup of G. Also clearly ∗ H = H ≤ Gu. So it only remains to prove that for every B ∈ Gu we have ∗ −1 ∗ BH B = H . But this follows by noting that for any B = (Bij) ∈ G we have: B ∈ Gu iff Bi1 = 0 for 2 ≤ i ≤ n.

EXERCISE (E50). [Perfect Group]. By a perfect group we mean a group which equals its commutator subgroup. Let the notation be as in (E48) and (E49). Show that if either n > 2 or |k| > 3 then G and G are perfect groups.

HINT. In view of (E49)(1) it suffices to note that in case of n > 2 by taking any l ∈ {1, . . . , n}\{i, j} we have −1 −1 (1) Til(n, 1)Tlj(n, λ)Til(n, 1) Tlj(n, λ) = Tij(n, λ) and in case of |k| > 3, by taking any κ ∈ k× with κ2 6= 1 and letting −1 2 Sij(n, κ) = Tii(n, κ)Tjj(n, κ ) and δ = λ/(κ − 1) we have Sij(n, κ) ∈ G and, in view of (E49)(5), −1 −1 (2) Sij(n, κ)Tij(n, δ)Sij(n, κ) Tij(n, δ) = Tij(n, λ).

EXERCISE (E51). [Socle-Size]. Complete the proof of L5§5(Q16)(T75.3) by showing that if S is a 1-dimensional CM local ring, and yS and zS are parameter ideals in R then socz(S/(yS)) = socz(S/(zS)). HINT. Now y, z belong to M(S) \ ZS(S) and hence so does yz. Let us consider the residue class epimorphisms α : S → S/(yS) and β : S → S/(yzS), and let γ : α(S) → β(S) be the unique R-module homomorphism such that for all t ∈ R we have γ(α(t)) = β(zt). Clearly γ is injective and

γ((0 : M(α(S)))α(S)) = (0 : M(β(S)))β(S).

Thus we get an R-module isomorphism (0 : M(α(S)))α(S) → (0 : M(β(S)))β(S). Consequently socz(S/(yS)) = socz(S/(yz)S). Now by interchanging y and z we get socz(S/(zS)) = socz(S/(yz)S). Therefore socz(S/(yS)) = socz(S/(zS)). §6: DEFINITIONS AND EXERCISES 73

EXERCISE (E52). [Coprincipal Primes and Isolated Subgroups]. Citing L5§5(Q21) for the definitions of the above two concepts and the concepts of the core C(E) of a loset ( = linearly ordered set) E and the set P (G) of all nonzero principal isolated subgroups of an ordered abelian group G, give more details of the proof of that part of L5§5(Q21)(T96.1) which says that C(S(G)) = P (G) where S(G) is the loset of all nonzero isolated subgroups of G. In other words: (i) given any H ∈ P (G), i.e., given any H in S(G) for which there exists 0 < τ ∈ G such that 0 0 H = ∩H0∈S(G) with τ∈H0 H , show that the set ∪H0∈S(G) with τ6∈H0 H is empty or an immediate predecessor of H in S(G) according as the set {L ∈ S(G): L < H} is empty or not, and hence H ∈ C(S(G)), and conversely: (ii) given any H ∈ C(S(G)), taking any 0 < τ ∈ H such that τ does not belong to the immediate predecessor of 0 H in S(G) in case there is such a predecessor, show that H = ∩H0∈S(G) with τ∈H0 H , and hence H ∈ P (G).

HINT. Both (i) and (ii) can easily be deduced from the fact that S(G) is a loset. This does not directly use L4§12(R7)(7.5). In the proof of (i) and (ii), the reference to L4§12(R7)(7.5) only signifies that, in view of (3) and (9) of L5§5(Q21), by realizing G as the value group Gv of a valuation v : Kb → Gv ∪ {∞} of a field Kb, the claim C(S(G)) = P (G) becomes equivalent to the assertion which ∗ says that: (•) J ∈ spec(Rv) = spec(Rv) \{M(Rv)} is coprincipal ⇔ the set ∗ W (J) = {P ∈ spec(Rv) : J $ P } is either empty or has a member Q for which the ∗ set W (J, Q) = {Q ∈ spec(Rv) : J $ Q $ Q} is empty. Here J is coprincipal means 0 for some 0 6= t ∈ M(Rv) we have J = ∪J 0∈spec(R)∗ with t6∈J 0 J . To establish (•) we tacitly use L4§12(R7)(7.5) which states that spec(Rv) is a loset under inclusion. To prove ⇒, assuming W (J) 6= ∅, clearly t ∈ P for all P ∈ W (J), and it suffices to take Q = ∩P ∈W (J)P . To prove ⇐, if W (J) = ∅ then take any 0 6= t ∈ M(Rv) \ J, whereas if there exists Q ∈ W (J) with W (J, Q) = ∅ then take any 0 6= t ∈ Q \ J. By (E45) we can take Kb = K((X))G with v = ord, or in view of (E53) below, we can take Kb to be the quotient field of finite G K((X))G = {A ∈ K : Supp(A) is finite}.

EXERCISE (E53). [General Valuation Functions]. In L5§5(Q13) we gener- alized the idea of a valuation on a field by defining a valuation function on a ring R to be a map v : R → Z ∪ {∞} such that for all x, y, z in R we have: (i) v(xy) = v(x) + v(y), (ii) v(x + y) ≥ min(v(x), v(y)), and (iii) v(z) = ∞ ⇔ z = 0, with the usual understanding about ∞. We generalize this further by replacing Z by an ordered abelian group G, and calling v a generalized valuation function on the ring R. If v is a generalized valuation function on a nonnull ring R then clearly R is a domain and we get a unique valuation v : QF(R) → G ∪ {∞} such that for all x, y in R× we have v(x/y) = v(x) − v(y); we say that the valuation v is induced by v; usually we denote v also by v. finite In particular, for any field K, taking R to be the domain K((X))G we get a generalized valuation function ord: R → G ∪ {∞}, and taking Lb to be the quotient field of R we get a valuation v : Lb → G ∪ {∞} with Gv = G. 74 LECTURE L6: PAUSE AND REFRESH

DEFINITION (D32). [Lexicographic Products and Sums]. ¿From L5§1 we Q recall that the cartesian product i∈E Wi of a family of sets (Wi)i∈E is the set of all maps φ : E → ∪i∈EWi such that for all i ∈ E we have φ(i) ∈ Wi. Moreover, assuming each Wi to be an additive abelian group, the cartesian product is converted into an additive abelian group by taking componentwise sums, and then it Q is called the direct product which is again denoted by Wi. The direct sum Q i∈E ⊕i∈EWi is the set of all φ in i∈E Wi whose support supp(φ) = {i ∈ E : φ(i) 6= 0} is finite; this is a subgroup of the direct product. Assuming E to be a loset (= linearly ordered set) and referring to L5§5(Q21), Qwo Q the well ordered product i∈E Wi is the set of all φ in i∈E Wi whose support supp(φ) is well ordered; this is a subgroup of the direct product containing the Qwo direct sum. If Wi = W for all i ∈ E then i∈E Wi and ⊕i∈EWi are denoted by E E Wwo and W⊕ and are called the well ordered E-th power and the restricted E-th power of W respectively. Note that these two sets respectively correspond to the sets E E (W )wellord and (W )finite considered in L2§2. Also note that if E is a finite set then the direct product, the well ordered product, and the direct sum, all coincide. Assuming each set Wi to be an ordered abelian group, and by putting the Qwo lexicographic order on i∈E Wi, this becomes a linearly ordered set which we de- Qlex note by i∈E Wi and call it the lexicographic product (of the family); here Qwo lexicographic order means that for φ 6= ψ in i∈E Wi we have: φ < ψ ⇔ φ(j) < ψ(j) where j = min{i ∈ E : φ(i) 6= ψ(i)}. Qwo Correspondingly, the subset ⊕i∈EWi of i∈E Wi also becomes a loset and as such we denote it by
i∈EWi and call it the lexicographic direct sum (of the family). E If Wi = W for all i ∈ E then Wwo becomes a loset and as such we denote it by E [E] Wlex or W and call it the lexicographic E-th power of W , and similarly, as a loset,
i∈EWi is now denoted by W E
and called the lexicographic restricted E-th power of W . In case E is a finite set of cardinality d ∈ N, say the set of integers 1, . . . , d, and we are using the tuple notation instead of the functional notation, we may write W1
···
Wd in place of
i∈EWi and call it the lexicographic direct sum of W1,...,Wd, and if W1 = ··· = Wd = W then we may write d [d] Wlex or W E in place of Wlex and call it the lexicographic d-th power of W . Given any family of sets (Ti)i∈F , by tbi∈F Ti we denote the set of all pairs (i, x) with i ∈ F and x ∈ Ti, and we call this the abstract disjoint union of the family. Given any family of losets (Si)i∈E indexed by the loset E, we convert tbi∈ESi into a loset by defining (i, x) ≤ (i0, x0) ⇔ either (i) i = i0 with x ≤ x0 or (ii) i ≤ i0 and we denote the new loset by ti∈ESi and call it the ordered disjoint union of the family. Recalling that the real rank ρ(Wi) of the ordered abelian group Wi is §6: DEFINITIONS AND EXERCISES 75 the order-type of the loset S(Wi) consisting of all the nonzero isolated subgroups of Wi, we define the lexicographic disjoint sum ] of real ranks by the equation ]i∈Eρ(Wi) = the order-type of ti∈ES(Wi). In case E is a finite set of cardinality d ∈ N, say the set of integers 1, . . . , d, and we are using the tuple notation instead of the functional notation, we may write S1 t · · · t Sd and ρ(W1) ]···] ρ(Wd) in place of ti∈ESi and ]i∈Eρ(Wi) and call these the ordered disjoint union of S1,...,Sd and the lexicographic disjoint sum of ρ(W1), . . . , ρ(Wd) respectively. If d ≥ 1 = |Sd| or d ≥ 1 = |S1| or d > 1 = |S1| = |Sd| then in place of S1 t· · ·tSd we may respectively write S1 t· · ·tSd−1 ∪{∞} or {−∞}∪S2 t· · ·tSd or {−∞} ∪ S2 t · · · t Sd−1 ∪ {∞}. If d ≥ 1 = |S(Wd)| or d ≥ 1 = |S(W1)| or d > 1 = |S(W1)| = |S(Wd)| then in place of ρ(W1) ]···] ρ(Wd) we may respectively write ρ(W1) ]···] ρ(Wd−1) + 1∞ or 1−∞ + ρ(W2) ]···] ρ(Wd) or 1−∞ + ρ(W2) ]···] ρ(Wd−1) + 1∞. Recalling that −E denotes the negative of a loset E, i.e., E with the reverse order, we claim that  if (G ) is a family of nonzero ordered abelian groups  i i∈E (1) indexed by a woset E and G =
j∈EGj then  ρ(G) = ]i∈−Eρ(Gi).

To prove (1) we find an order preserving bijection ti∈−ES(Gi) → S(G) thus. For any (i, H) with i ∈ E and H ∈ S(Gi) we clearly get He ∈ S(G) by putting

He = {φ ∈ G : φ(i) ∈ H and φ(j) = 0 for all j < i}.

Also clearly H 7→ He gives an order preserving map ti∈−ES(Gi) → S(G). Since E is a woset, for any L ∈ S(G), the set {j ∈ E : φ(j) 6= 0 for some φ ∈ L} has a smallest element j(L), and upon letting J(L) = {φ(j(L)) : φ ∈ L} we clearly have

J(L) ∈ S(Gj(L)) with J](L) = L. Moreover, if L = He then we clearly have j(L) = i with J(L) = H. Therefore by L5§6(D2) it follows that H 7→ He gives a bijection ti∈−ES(Gi) → S(G). Now without using (1) we shall generalize it by proving the following claim (2). Note that (2) ⇒ (1) by L5§5(Q21)(T96.6). Recalling that Esc is the loset of all nonempty left segments of a loset E, we claim that  if (Gi)i∈E is a family of nonzero ordered abelian groups  indexed by a loset E and G ≤ G ≤ Qlex G 
i∈−E i i∈−E i  then, regarding G as an ordered abelian group  under the induced order, we have  0 (2) ρ(G) = ]i∈Esc ρ(Gi)  ∗ sc where for every i ∈ E = { ∈ E :  has a max δ() in }  0 we put Gi = Gδ(i)  and for every i ∈ Esc \ E∗   0 we put Gi = a nonzero archimedean ordered abelian group. 76 LECTURE L6: PAUSE AND REFRESH

0 To prove (2) we find an order preserving bijection ti∈Esc S(Gi) → S(G) thus. sc 0 For any (i, H) with i ∈ E and H ∈ S(Gi) we clearly get He ∈ S(G) by putting ( {φ ∈ G : φ(δ(i)) ∈ H and φ(j) = 0 for all j ∈ E \ i} if i ∈ E∗ He = {φ ∈ G : φ(j) = 0 for all j ∈ E \ i} if i ∈ Esc \ E∗

Also clearly H 7→ He gives an order preserving map ti∈ES(Gi) → S(G). For any L ∈ S(G), upon letting j(L) = {l ∈ E : φ(l) 6= 0 for some φ ∈ L} we clearly have j(L) ∈ Esc, and upon letting ( {φ(δ(j(L))) : φ ∈ L} if j(L) ∈ E∗ J(L) = 0 sc ∗ Gj(L) if j(L) ∈ E \ E

0 we clearly have J(L) ∈ S(Gj(L)) with J](L) = L. Moreover, if L = He then we clearly have j(L) = i with J(L) = H. Therefore by L5§6(D2) it follows that 0 H 7→ He gives a bijection ti∈−ES(Gi) → S(G). By (1) we see that  if (G ) is a family of nonzero ordered abelian groups  i 1≤i≤d (3) with d ∈ N then  ρ(G1
···
Gd) = ρ(Gd) ]···] ρ(G1).

Letting N− be the loset of all negative integers, by (1) we also see that  if (Gi)i∈ is a family of  N+ nonzero archimedean ordered abelian groups (4) (for instance 0 6= G ≤ such as G = or or ) then  i R i Z Q R  ρ(
i∈N+ Gi) = order-type of N− and  if (Gi)i∈ ∪{∞} is a family of  N+ nonzero archimedean ordered abelian groups (5) (for instance 0 6= Gi ≤ R such as Gi = Z or Q or R) then  ρ( G ) = order-type of {−∞} ∪ .
i∈N+∪{∞} i N− By (2) we see that  if (Gi)i∈ is a family of  N nonzero archimedean ordered abelian groups (6) (for instance 0 6= G ≤ such as G = or or ) then  i R i Z Q R  ρ(
i∈−NGi) = order-type of N ∪ {∞}. In view of the d = 2 case of (3), by (4) and (6) we see that  if (Gi)i∈ is a family of  Z nonzero archimedean ordered abelian groups (7) (for instance 0 6= G ≤ such as G = or or ) then  i R i Z Q R 
ρ (
i∈−NG−i)
(
i∈N+ Gi) = order-type of Z ∪ {∞} §6: DEFINITIONS AND EXERCISES 77 and by (5) and (6) we see that  if (Gi)i∈ is a family of  Z nonzero archimedean ordered abelian groups (8) (for instance 0 6= Gi ≤ R such as Gi = Z or Q or R) then  ρ ( G ) ( G )
= order-type of {−∞} ∪ ∪ {∞}.
i∈−N −i

i∈N+∪{∞} i Z

EXERCISE (E54). [Possible Real Ranks]. Letting N− be the loset of all negative integers, consider the losets given by:

E3 = {1, . . . , d} with d ∈ N,E4 = N−,E5 = {−∞} ∪ N−, and E6 = N ∪ {∞},E7 = Z ∪ {∞},E8 = {−∞} ∪ Z ∪ {∞}. Given any field K, in view of (E45) and (53) above, by (D32) above we see that for

3 ≤ i ≤ 8 there exists a K-valuation vi : Ki → Gvi ∪ {∞} of an overfield Ki of K whose real rank ρ(vi) is the order-type of Ei. The exercise is to give other proofs of this directly using the characterizations given in L5§5(Q21)(T97).

EXERCISE (E55). [Impossible Real Ranks]. Using the characterizations cited above show that there is no valuation (of any field) whose real rank is the order-type of any one of the losets [cf. L5§5(Q21)(T96.2)]: N, Z, Q, R, Q ∪ {∞}, R ∪ {∞}, and {−∞} ∪ Q ∪ {∞}, {−∞} ∪ R ∪ {∞}.

DEFINITION (D33). [Principal Rank and Real Rank]. In L5§5(Q21) we defined the real rank ρ(G) of an ordered abelian group G to be the order-type of the loset S(G) of all nonzero isolated subgroups of G. We also defined the real rank ρ(v) of a valuation v, as well as the real rank ρ(Rv) of its valuation ring, to be the same as the real rank ρ(Gv) of its value group Gv. Recalling that the set of all nonzero principal isolated subgroups of G is denoted by P (G), let us introduce the principal rank ρ0(G) of G by putting ρ0(G) = the order-type of the loset P (G). Let us also define the principal ranks 0 0 ρ (v) and ρ (Rv) by putting 0 0 0 ρ (v) = ρ (Rv) = ρ (Gv). ∗ Recalling that coprin(Rv) is the set of all nonmaximal coprincipal prime 0 ∗ ideals in Rv, we see that ρ (Rv) is the order-type of coprin(Rv) with reverse containment. In L5§5(Q21)(T97.2) and in the above two Exercises we discussed the problem of characterizing real ranks. Turning to the problem of characterizing principal ranks, it turns out that ρ0(G) can be the order-type of any loset E. Indeed, upon letting [−E] G = R by L5§5(Q21)(T97.1) we see that ρ0(G) is the order-type of E. Moreover, by L5§5(Q21)(T96.1) we also see that E = C(Esc) where the core C(E) of E, and the segment completion Esc of E, are as defined in L5§5(Q21). All this may be compared with the material of the above Definition (D32). 78 LECTURE L6: PAUSE AND REFRESH

EXERCISE (E56). [Compositions of Valuations]. In L5§5(Q22)(7) we have defined the composition of a valuation w by a valuation u which we have denoted by w u. We have put a dot in the circle to distinguish it from ψ ◦ φ which is how the basic composition of maps ψφ defined in L1§2 is denoted by some authors. For clarity, we may call w u the valuation theoretic composition of w by u. The exercise is to give a concrete example of w u. Moreover, referring to (D32) for the definition of lexicographic direct sum G1
G2 = the set of all pairs (g1, g2) ∈ G1 × G2 with componentwise addition and lexicographic order, in the notation of L5§5(Q22)(7) show that there exists a unique order preserving isomorphism

θ : Gw u → Gw
Gu such that for all x ∈ Rw u we have

θ(w u)(x)) = (w(x), u(πw(x))). HINT. The “Moreover” part is straightforward. For the example, let X,Y be indeterminates over a field k, and consider the valuation v of L = k(X,Y ) with value group Z[2] = the set of all lexicographically ordered pairs of integers (m, n), and which is defined by putting v(aXmY n) = (m, n) for all a in k× and m, n in Z. Then v = w u with real discrete valutions w and u.

EXERCISE (E57). [Relabelling Prime Ideals]. Justify the relabelling claimed in the proof of L5§5(Q23)(T98.2). In other words let I,J be ideals in a local ring 2 R such that J = M(R) with (I : J)R = I, let y ∈ R be such that y ∈ J \ J , and let P1,...,Pn be the distinct members of assR(R/I). Show that P1,...,Pn can be relabelled to get r and m in N with m ≤ r ≤ n such that (i) y ∈ Pi for 1 ≤ i ≤ m and y 6∈ Pj for m + 1 ≤ j ≤ r, (ii) Pj 6⊂ Pi for 1 ≤ i ≤ m and m + 1 ≤ j ≤ r, and (iii) ∪1≤i≤rPi = ∪1≤i≤nPi. By an example show that (i) and (ii) cannot always be arranged with r = n.

HINT. Relabelling P1,...,Pn, we can find n ≥ r ∈ N such that (1) for each j with r + 1 ≤ j ≤ n there exists some i with 1 ≤ i ≤ r for which Pj ⊂ Pi, and such that (2) there are no inclusion relations amongst P1,...,Pr, i.e., Pi 6⊂ Pj for all i 6= j in {1, . . . , r}. Then ∪1≤i≤rPi = ∪1≤i≤nPi. By further relabelling P1,...,Pr, we can find r ≥ m ∈ N such that y ∈ Pi for 1 ≤ i ≤ m and y 6∈ Pj for m + 1 ≤ j ≤ r. For the desired example, in the trivariate power series ring R = k[[t, y, z]] over a 2 field k, take J = (t, y, z)R and I = (t , ty)R. Then (I : J)R = I, and the distinct members of assR(R/I) are P1 = (t, y)R and P2 = (t)R with n = 2. Now if we take r = n then the only choice of relabelling and m satisfying (i) is the given labelling and m = 1, but (ii) does not hold for this choice because P2 ⊂ P1.

EXERCISE (E58). [Suslin Localization]. (1) Note that in the last sentence of the proof of L5§5(Q31)(128.4) we have A(cX)A(dX)−1 ∈ GL(n, [R,XR]) because by putting X = 0 in this matrix it is is obviously reduced to the n × n identity matrix. (2) Show that the proof of L5§5(Q31)(128.4) remains valid if the substitution (Y,Z) = (c, λ) is changed to the substitution (X,Y,Z) = (1, cX, λX). §6: DEFINITIONS AND EXERCISES 79

(3) Also show that the proof works if the phrase “Consider the trivariate poly- nomial ring R = K[X,Y,Z] as an overring of R,” to the phrase “Consider the bivariate polynomial ring R = K[Y,Z],” if the reference to X is dropped every- where else, and if the substitution (Y,Z) = (c, λ) is changed to the substitution (Y,Z) = (cX, λX). BIBLIOGRAPHY 701

References [A01] S. S. Abhyankar, Ramification Theoretic Methods In Algebraic Geometry, Princeton Uni- versity Press, Princeton, 1959. [A02] S. S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, American Mathematical Monthly, vol. 83 (1976), pages 409-448. Kyoto 1977, pages 240-414. [A03] S. S. Abhyankar, On the semigroup of a meromorphic curve, Part I, Proceedings of the International Symposium on Algebraic geometry, Kyoto 1977, pages 240-414. [A04] S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathemati- cal Society, 1990. [A05] S. S. Abhyankar, Resolution of Singularities of Embedded Algebraic Surfaces, Springer, 1998. [A06] S. S. Abhyankar, Local Analytic Geometry, World Scientific Publishing Company, Singa- pore, 2001. [A07] S. S. Abhyankar, Two step descent in modular Galois theory, theorems of Burnside and Cayley, and Hilbert’s thirteenth problem, Proceedings of the Saskatoon Valuation The- ory Conference of August 1999, Fields Institute Communications, American Mathematical Society, vol. 32 (2002), pages 1-31. [Ask] E. H. Askwith, Pure Geometry, Cambridge University Press, 1921. [AuB] M. Auslander and D. Buchsbaum, Homological dimension in local rings, Transactions of the American Mathematical Society, vol. 85 (1957), pages 390-405. pages 2941-2969. [BRJ] R. J. T. Bell, An Elementary Treatise on Coordinate Geometry of Three Dimensions, Cambridge University Press, 1920. [Bez] E.´ B´ezout, Th´eorieg´en´erale des ´equationesalg´ebriques Paris, 1770. [Bha] Bhaskaracharya, Beejganit, Ujjain, India, 1150. [Bur] W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, First Edition 1897 and Second Edition 1911. [Chr] G. Chrystal, Textbook of Algebra, Parts I and II, Originally Published by A. C. Black, Ltd, Edinburgh (1986-1989), Many Chelsea Reprints. [CPM] P. M. Cohn, On the structure of the GL2 of a ring, I. H. E. S., vol. 30 (1966), pages 5-53. [Dic] L. E. Dickson, Linear Groups, Teubner, 1901. [Hi1] D. Hilbert Uber¨ die Theorie der algebraischen Formen, Mathematische Annalen, vol. 36 (1890), 473-534. [Gal] E. Galois, Oeuvres Math´ematique, Journal de Math´ematiqesPures et Appl., vol. 11 (1846), pages 381-444. [Jac] N. Jacobson, Basic Algebra I and II, Freeman, 1980. [Jor] C. Jordan, Trait´edes Substitutions et des Equations´ Alg´ebriques, Gauthier-Villa, 1870. [Kle] F. Klein, Entwicklung der Mathematik im neunzehnten Jahrhundert, Berlin, 1926. [KLi] P. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge University Press, 1990. [Kr1] W. Krull, Elementare und klassische Algebra vom moderne Standpunkt, Parts I and II, De Gruyter, Berlin, 1952-1959. [Kr2] W. Krull, Idealtheorie, Springer-Verlag, Berlin, 1935. [Mat] H. Matsumura, Commutative Algebra, Benjamin, 1989, [Mo1] E. H. Moore, A doubly-infinite system of simple groups, Proceedings of the Chicago Worlds Fair, 1893, pages 208-242. [Mo2] E. H. Moore, A Two-fold generalization of Fermat’s theorem, Bulletin of the American Mathematical Society, vol. 2 (1896), pages 189-199. [Nag] M. Nagata, Local Rings, John Wiley, 1962 [PaR] K. H. Parshall and D. E. Rowe, The Emergence of the American Mathematical Research Community, American Mathematical Society, 1994. [Qui] D. Quillen, Projective modules over polynomial rings, Inventiones Mathematicae, vol. 36 (1976). [Sal] G. Salmon, Higher Plane Curves, Dublin, 1852. [Se1] J.-P. Serre, Al`gbre Local–Multiplicit´e, Springer Lecture Notes in Mathematics, vol. 7 (1958). [Se2] J.-P. Serre, Sue les modules projective, S´eminairesDubreil-Pisot, (1960). [Ste] E. Steinitz, Algebraische Theorie der K¨oper, Chelsea Publishing Company, 1950. 702 BIBLIOGRAPHY

[Su1] A. Suslin, Projective modules over polynomial rings, Dokl. Akad. Nauk S.S.R., vol. 26 (1976). [Su2] A. Suslin, On the structure of the special linear group over polynomial rings, Izv. Akad. Nauk S.S.R. Ser. Math., vol. 41 (1977), pages 235-252. [Syl] J. J. Sylvester, On a general method of determining by mere inspection the derivations from two equations of any degree, Philosophical Magazine, vol. 16 (1840), pages 132–135. [Sy2] J. J. Sylvester, On a theory of syzygetic relations of two rational integral functions, com- prising an application to the theory of Sturm’s functions, and that of the greatest common measure, Philosophical Transactions of the Royal Society of London, vol. 143 (1853), pages 407-458. [VYo] O. Veblen and J. T. Young Projective Geometry I–II, Ginn and Company, 1910-1918. [Wey] H. Weyl, Classical Groups, Princeton University Press, 1939. [ZaS] O. Zariski and P. Samuel, Commutative Algebra I and II, Springer-Verlag, Berlin, 1986 DETAILED CONTENTS 703

DETAILED CONTENTS C = Comment, D = Definition, E = Exercise, MU = Mute = No Label, N = Note, O = Observation, P = Problem, Q= Quest, R = Remark, T = Theorem or Lemma, X = Example

Title page

Lecture L1: QUADRATIC EQUATIONS 1 §1: Word Problems 1

§2: Sets and Maps 2

§3: Groups and Fields 3

§4: Rings and Ideals 5

§5: Modules and Vector Spaces 7

§6: Polynomials and Rational Functions 8

§7: Euclidean Domains and Principal Ideal Domains 11

§8: Root Fields and Splitting Fields 12 (T1) FUNDAMENTAL THEOREM OF GALOIS THEORY 13 (T2) LAGRANGE RESOLVENT THEOREM 13 (T3) SOLVABILITY THEOREM 13 (T4) NEWTON’S SYMMETRIC FUNCTION THEOREM 13 (T5) GALOIS’ SYMMETRIC GROUP THEOREM 13 (T6) UNSOLVABILITY THEOREM 14

§9: Advice to the Reader 14

§10: Definitions and Remarks 14 (D1) Divisibility and Prime Ideals 14 (R1) Division Algorithm 15 (R2) Long Division or Euclidean Algorithm 15 (R3) ED ⇒ PID 16 (R4) ED ⇒ UFD 16 (R5) Factorization of Integers 17 (R6) Factorization of Univariate Polynomials 17 (D2) Characteristic 18 (D3) Minimal Polynomial 18 (D4) Least Common Multiple 18 (R7) Relative Algebraic Closure 19 (R8) Uniqueness of Splitting Field 19

§11: Examples and Exercises 20 (X1) Special Permutations 20 704 DETAILED CONTENTS

(X2) General Permutations 21 (E1) (modified discriminant) 21 (E2) to (E6) (symmetric and alternating groups) 22 (E7) (division algorithm) 22 (E8) to (E10) (euclidean domains and rationalization of surds) 23

§12: Notes 23 (N1) Derivations 23 (E11) to (E12) (derivations) 24 (N2) Very Long Proofs 24

§13: Concluding Note 24

Lecture L2: CURVES AND SURFACES 25 §1: Multivariable Word Problems 25

§2: Power Series and Meromorphic Series 29

§3: Valuations 33

(T1) NEWTON’S THEOREM 36 (T2) GENERALIZED NEWTON’S THEOREM 36

§4: Advice to the Reader 36

§5: Zorn’s Lemma and Well Ordering 37 (T3) LEMMA (preparation for equivalence theorem) 37 (T4) EQUIVALENCE THEOREM 37 (R1) (variation of Zorn’s lemma) 38 (R2) (consequence of Zorn’s lemma) 38 (R3) (cofinal sets) 39 (R4) Existence of Algebraic Closure by Well Ordering 39 (R5) Existence of Maximal Ideals by Zorn’s Lemma 39 (R6) Existence of Algebraic Closure by Zorn’s Lemma 39 (T5) LEMMA (polynomials in infinitely many variables) 40 (R7) Uniqueness of Algebraic Closure by Zorn’s Lemma 40 (R8) Existence of Vector Space Bases and Transc. Bases by Zorn’s Lemma 41 (R9) Linear Order on Cardinals by Zorn’s Lemma 42 (R10) Well Order on Ordinals by Zorn’s Lemma 42

§6: Utilitarian Summary 44

§7: Definitions and Exercises 44 (D1) Real and Complex Numbers 44 (D2) Ordered Fields 45 (D3) Torsion Subgroups and Divisible groups 45 (E1) to (E2) (ordered abelian groups) 45 DETAILED CONTENTS 705

(D4) Rational and Real Completions 46 (E3) to (E4) (induced relation) 46 (E5) to (E6) (archimedean order) 47 (E7) to (E8) (real numbers) 47 (D5) Rational and Real Ranks 47 (E9) to (E12) (ordered abelian groups) 48 (D6) Dedekind Cuts 48 (E13) (geometric series identity) 49 (E14) (vector space bases and transcendence bases) 49 (D7) Approximate Roots 49 (E15) to (E17) (properties of approximate roots) 50

§8: Notes 50 (N1) Algebraic Closedness of Complex Numbers 50 (N2) Integers and Rational Numbers 50

§9: Concluding Note 50

Lecture L3: TANGENTS AND POLARS 51 §1: Simple Groups 51

§2: Quadrics 52 (H1) HOMEWORK PROBLEM 52 (H2) GENERALIZED HOMEWORK PROBLEM 53

§3: Hypersurfaces 54

§4: Homogeneous Coordinates 56

§5: Singularities 58 (H3) HOMEWORK PROBLEM 59 (T1) (Binomial Theorem Power Series Identity) 60

§6: Hensel’s Lemma and Newton’s Theorem 60 (T2) BASIC HENSEL’S LEMMA 61 (MU) COMPLETING THE POWER 62 (T3) LEMMA (univariate polynomial to have two different roots) 63 (MU) PROCESSES ON ROOTS 63 (T4) BASIC NEWTON’S THEOREM 64 (MU) SHREEDHARACHARYA’S PROOF OF NEWTON’S THEOREM 64

§7: Integral Dependence 65

§8: Unique Factorization Domains 67 (MU) GAUSS LEMMA 68

§9: Remarks 68 706 DETAILED CONTENTS

(R1) Factorization of Univariate Power Series 68 (R2) Kronecker’s Theorem 69

§10: Advice to the Reader 69

§11: Hensel and Weierstrass 70 (T5) SUPPLEMENTED WEIERSTRASS PREPARATION THEOREM 70 (T6) ABSTRACT WEIERSTRASS PREPARATION THEOREM 72 (T7) LEMMA (preparation for abstract Weierstrass) 73 (T8) ABSTRACT HENSEL’S LEMMA 74

§12: Definitions and Exercises 75 (D1) Minors and Cofactors 75 (E1) to (E2) (determinants) 75 (D2) Transpose and Adjoint 75 (E3) to (E7) (scalar matrices, nonzero polynomials, quasilocal rings) 76 (D3) Matrices and Linear Maps 76 (D4) Linear and Polynomial Automorphisms 76 (E8) to (E11) (linear automorphisms and Hensel’s lemma) 78 (D5) Hausdorff Modules and Bilinear Maps 78 (E12) Completeness Lemma 79 (E13) Hensel’s Sublemma 79 (E14) Hensel’s Lemma 80 (E15) (derivations) 80 (O1) (derivation of a power) 80 (E16) (binomial theorem) 80 (E17) (binomial coefficients) 81 (O2) (Newton’s binomial theorems) 81 (E18) (univariate polynomial ring over a UFD is a UFD) 81 (O3) (what remains to be done) 81

§13: Notes 81 (N1) Preparation for WPT 81 (N2) Abhyankar’s Proof of Newton’s Theorem 82

§14: Concluding Note 82

Lecture L4: VARIETIES AND MODELS 83 §1: Resultants and Discriminants 83 (T1) BASIC FACT 83 (T2) COROLLARY (basic property of discriminant) 84 (O1) Resultant and Projection 84 (O2) Discriminant and Projection 84 (O3) Isobaric Property 84 (O4) Plane B´ezout 84 (O5) Hyperspatial B´ezout 85 (X1) Resultant and Discriminant in terms of Roots 85 DETAILED CONTENTS 707

(X2) Quadratic Resultant 85 (X3) Quadratic Discriminant 85 (X4) Cubic Discriminant 86 (X5) Special Quartic Discriminant 86 (X6) General Quartic Discriminant 86 (O6) Future Plans 86

§2: Varieties 86

§3: Noetherian Rings 87 (T3) HILBERT BASIS THEOREM 87 (T4) POWER SERIES VERSION OF HILBERT BASIS THEOREM 88

§4: Advice to the Reader 89

§5: Ideals and Modules 89 (O7) Primary for a Prime Ideal 91 (O8) Colons and Intersections of Primary Ideals 91 (O9) Primaries and Powers of Primes 91 (X7) (Primary for a maximal ideal which is not a power of it) 91 (X8) (Nonmaximal prime with a nonprimary power with same radical) 91 (O10) Factor Modules and Isomorphism Theorems 91 (O11) Abelian Groups and Isomorphism Theorems 93 (O12) Annihilators and Colons 93 (O13) Primary Submodules 93 (O14) Noetherian Modules with ACC and MXC 94 (O15) Irreducible and Primary Submodules 94 (X9) (Primary nonirreducible ideal in a noetherian ring) 95 (O16) Spectrum and Irredundant Decomposition 95 (O17) Associator and Tight Associator 97 (O18) Tacit Use of PIC and PMC 98 (O19) Multiplicative Sets and Isolated Components 98 (O20) Zerodivisors and Quasiprimary Decomposition 99 (O21) Length of a Module 103 (O22) Heights and Depths of Ideals, and Dimensions of Rings 105 (X10) (Height and Depth in a Polynomial Ring) 106 (O23) Modular Law 106 (O24) Generalities on Prime Ideals 106 (O25) Artinian Modules with DCC and MNC 107 (O26) Comaximal Ideals 110 (O27) Affine Rings and Hilbert Basis Theorem 111

§6: Primary Decomposition 111 (T5) PRIMARY DECOMPOSITION THEOREM FOR IDEALS 112 (T6) PRIME UNIQUENESS THEOREM FOR IDEALS 112 (T7) PRIMARY UNIQUENESS THEOREM FOR IDEALS 112 (T8) QUASIPRIMARY DECOMPOSITION THEOREM FOR IDEALS 113 708 DETAILED CONTENTS

§6.1: Primary Decomposition for Modules 113 (T50) PRIMARY DECOMPOSITION THEOREM FOR MODULES 114 (T60) PRIME UNIQUENESS THEOREM FOR MODULES 114 (T70) PRIMARY UNIQUENESS THEOREM FOR MODULES 114 (T80) QUASIPRIMARY DECOMPOSITION THEOREM FOR MODULES 114

§7: Localization 114 (T9) CHARACTERIZATION THEOREM FOR LOCALIZATION 114 (T10) TRANSITIVITY OF LOCALIZATION 116 (T11) PERMUTABILITY OF LOCALIZATION AND SURJECTION 116 (T12) IDEAL CORRESPONDENCE FOR LOCALIZATION 116 (T13) LEMMA ON CONTRACTED AND EXTENDED IDEALS 118

§7.1: Localization at a Prime Ideal 114 (T14) CHARACTERIZATION THEOREM FOR PRIME LOCALIZATION 120 (T15) TRANSITIVITY OF PRIME LOCALIZATION 120 (T16) PERMUTABILITY OF PRIME LOCALIZATION AND SURJECTION 120 (T17) IDEAL CORRESPONDENCE FOR PRIME LOCALIZATION 120 (T18) LOCAL RING CONSTRUCTION THEOREM 121

§8: Affine Varieties 122 (MU) GEOMETRIC MOTIVATION 124 (MU) INTUITIVE DEFINITION 124 (T19) DIMENSION THEOREM 124 (T20) PRIMITIVE ELEMENT THEOREM 125 (T21) INCLUSION RELATIONS THEOREM 125 (T22) HILBERT NULLSTELLENSATZ 126

§8.1: Spectral Affine Space 126

§8.2: Modelic Spec and Modelic Affine Space 127

§8.3: Simple Points and Regular Local Rings 127 (T23) DIM–EMDIM THEOREM 127 (T24) ORD VALUATION THEOREM 128 (X11) (Simple points and regular local rings) 128

§9: Models 128 (T25) VALUATION MAXIMALITY THEOREM 129 (T26) VALUATION EXISTENCE THEOREM 129 (T27) VALUATION EXTENSION THEOREM 129 (T28) VALUATION CHARACTERIZATION THEOREM 130

§9.1: Modelic Proj and Modelic Projective Space 130 (T29) MODELIC PROJ THEOREM 131

§9.2: Modelic Blowup 132 (T30) MODELIC BLOWUP THEOREM 133 DETAILED CONTENTS 709

§9.3: Blowup of Singularities 133 (X12) Nodal and Cuspidal Cubics 133 (X13) Higher Cusps 134 (X14) Circular Cones and Fermat Cones 134

§10: Examples and Exercises 134 (E1) Algebraic Closure 134 (E2) Integral Closure 135 (E3) Conditions of Integral Dependence 136 (X15) Nonarchimedean Valuations 136 (E4) Cramer’s Rule for Linear Equations 137 (E5) Solutions of Homogeneous Linear Equations 137 (E6) Conditions for a Common Factor 138 (E7) Sylvester Resultant 138 (E8) Elementary Properties of Determinants 139 (E9) Blowup of a Point in the Plane and Three Space 139 (X16) Blowup of a Line in Three Space 139 (X17) Tight Associators and Monomial Ideals 140 (X18) Isolated Components of a Module 142 (X19) Annihilator of a Primary Module 142

§11: Problems 142 (P1) Explicit Equations of Integral Dependence 143 (P2) Uniformization 143 (P3) Resolution 143

§12: Remarks 143 (R1) Laplace Development 143 (R2) Block Matrices 144 (R3) Products of Determinants 145 (R4) Solutions of Nonhomogeneous Linear Equations 145 (R5) The How and Why of Cramer and Sylvester 146 (R6) Product Formula for the Resultant Matrix 148 (1) Relation between Polynomial Addition and Matrix Addition 150 (2) Relation between Polynomial Multiplication and Matrix Multiplication 150 (9) Permuting Rule for Resultant 150 (10) Universality Property of Resultant 150 (11) Diagonal Property of Resultant 150 (12) First Homogeneity Property of Resultant 150 (13) Second Homogeneity Property of Resultant 150 (14) Weight Property of Resultant 150 (15) Special Property of Resultant 151 (16) First Sum Rule for Resultant 151 (17) Second Sum Rule for Resultant 151 (18) First Shift Rule for Resultant 152 (19) Second Shift Rule for Resultant 152 (20) Relations 152 710 DETAILED CONTENTS

(MU) ENLARGED RESULTANT MATRIX 152 (MU) SKEWSHIFTED RESULTANT MATRIX 153 (190) Skewshift Rule for Resultant 153 (MU) ENLARGED SKEWSHIFTED RESULTANT MATRIX 153 (23) Product Rule for Resultant matrix 153 (230) Skew Product Rule for Resultant matrix 154 (27) Product Rule for Resultant 156 (28) First Special Product Rule for Resultant 156 (29) Second Special Product Rule for Resultant 156 (30) Third Special Product Rule for Resultant 156 (31) Tiny Resultant theorem 156 (320 Small Resultant theorem 156 (33) Little Resultant theorem 157 (34) First Version of Resultant theorem 157 (35) Second Version of Resultant theorem 157 (R7) Intval Domains, Divisibility Rings, Maximality Rings, and Eq Vals 157 (R8) Modelic Proj 160 (R9) Modelic Blowup 161

§13: Definitions and Exercises 163 (E10) Hilbert Basis Theorem 163 (E11) Primary Modules 163 (D1) Support of a Module 165 (E12) Support and Annihilator 165 (E13) Support and Quasiprimary Decomposition 165 (D2) Homogenization and Dehomogenization 166 (E14) Embedding Projective Space Into Projective Model 166

§14: Notes 167 (N1) Princeton Book 167 (N2) Resolution Book 167 (N3) Kyoto Paper 167 (N4) Valuations and L’Hospital’s Rule 167 (N5) What Remains To Be Done 168

§15: Concluding Note 168

Lecture L5: PROJECTIVE VARIETIES 169 §1: Direct Sums of Modules 169 (X1) Annihilators and Direct Sums 172

§2: Grades Rings and Homogeneous Ideals 172 (C1) Alternative Definition of Graded Rings 174 (C2) Cancellative Monoids and Ordered Monoids 174 (C3) Integrally or Nonnegatively or Naturally Graded Rings 175

§3: Ideal Theory in Graded Rings 175 DETAILED CONTENTS 711

(C4) Positive Portion and Irrelevant Ideals 178 (C5) Relevant Portion and Isolated System of Prime Ideals 178 (C6) Homogeneous Rings and Semihomogeneous Rings 178

§4: Advice to the Reader 180

§5: More about Ideals and Modules 181 (Q1) Nilpotents and Zerodivisors in Noetherian Rings 181 (T1) ZERODIVISOR THEOREM 181 (T2) COROLLARY (property of zerodivisors) 182 (C7) Maximality and Conditions for Prime Ideals 182 (C8) Embedded Prime and Primary Components 182

(Q2) Faithful Modules and Noetherian Conditions 183

(Q3) Jacobson Radical, Zariski Ring, and Nakayama Lemma 183 (C9) Cramer’s Rule for Modules 184 (T3) NAKAYAMA LEMMA 184

(Q4) Krull Intersection Theorem and Artin-Rees Lemma 184 (T4) ARTIN-REES LEMMA 185 (T5) ARTIN-REES COROLLARY 185 (T6) KRULL INTERSECTION LEMMA 185 (T7) KRULL INTERSECTION THEOREM 186 (T8) CLOSEDNESS COROLLARY 186 (T9) CLOSURE COROLLARY 186 (T10) DOMAINIZED KRULL INTERSECTION THEOREM 187 (T11) INTERSECTION OF PRIMARIES THEOREM 187 (C10) Power of Krull Intersection 187 (T12) EXISTENCE OF EMBEDDED PRIMES THEOREM 187 (T13) NONUNIQUENESS OF EMBEDDED PRIMARIES THEOREM 188

(Q5) Nagata’s Principle of Idealization 188 (T14) MODULIZED ARTIN-REES LEMMA 189 (T15) MODULIZED ARTIN-REES COROLLARY 189 (T16) MODULIZED KRULL INTERSECTION LEMMA 189 (T17) MODULIZED KRULL INTERSECTION THEOREM 190 (T18) MODULIZED CLOSEDNESS COROLLARY 190 (T19) MODULIZED CLOSURE COROLLARY 190 (T20) CHARACTERIZATION THEOREM FOR ZARISKI RINGS 191

(Q6) Cohen’s and Eakin’s Noetherian Theorems 191 (T21) LEMMA (on finitely generated ideals) 192 (T22) COHEN’S NOETHERIAN THEOREM 192 (T23) THEOREM (of Eakin on noetherian rings) 192

(Q7) Principal Ideal Theorems 193 (T24) PRINCIPAL IDEAL THEOREM 193 712 DETAILED CONTENTS

(C11) Minimal Primes 193 (T25) COROLLARY (consequence of Principal Ideal Theorem) 194 (T26) COROLLARY (preparing for Generalized Principal Ideal Theorem) 194 (T27) GENERALIZED PRINCIPAL IDEAL THEOREM 194 (T28) HEIGHT THEOREM 195 (T29) EXTENDED DIM–EMDIM THEOREM 195 (T30) DIMENSION LEMMA 195 (C12) Multivariate Ideal Extensions 197 (C13) Univariate Ideal Extensions 197 (C14) Module Generation For Local Rings 198

(Q8) Relative Independence and Analytic Independence 198 (T31) LEMMA (conditions of relative of independence) 199 (T32) LEMMA (independence over univariate polynomial ring) 200 (T33) BLOWUP LEMMA 200 (T34) RELATIVE INDEPENDENCE THEOREM 202 (T35) ANALYTIC INDEPENDENCE THEOREM 202

(Q9) Going Up and Going Down Theorems 202 (T36) LOCALIZATION OF NORMALITY LEMMA 203 (T37) PRESERVATION OF INTEGRAL DEPENDENCE LEMMA 203 (T38) PRESERVATION OF FIELDS AND MAXIMAL IDEALS LEMMA 204 (T39) LYING BELOW LEMMA 204 (T40) LYING ABOVE THEOREM 204 (T41) GOING UP THEOREM 205 (T42) PROPER CONTAINMENT LEMMA 205 (T43) RADICAL DESCRIPTION LEMMA 205 (T44) GOING DOWN THEOREM 206 (T45) DIMENSION COROLLARY 207

(Q10) Normalization Theorem and Regular Polynomials 208 (T46) NOETHER NORMALIZATION THEOREM 208 (T47) EXTENDED DIMENSION THEOREM 210 (C15) Ideals Under Field Extensions 217 (C16) Supplement to Hilbert Nullstellensatz 218 (C17) Decomposition of Ideals and Varieties 218

(Q11) Nilradical, Jacobson Spectrum, and Jacobson Ring 219 (T48) NILRADICAL THEOREM 220 (T49) SPECTRAL RELATIONS THEOREM 220 (T50) SPECTRAL NULLSTELLENSATZ 222 (T51) MINIMAL PRIMES THEOREM 223 (T52) FIRST SUPPLEMENTARY DIMENSION THEOREM 223 (T53) SECOND SUPPLEMENTARY DIMENSION THEOREM 224 (C18) Again Decomposition of Ideals and Varieties 224

(Q12) Catenarian Rings and Dimension Formula 225 (T54) SIMPLE RING EXTENSION LEMMA 225 DETAILED CONTENTS 713

(T55) MULTIPLE RING EXTENSION LEMMA 226 (T56) CATENARIAN CONDITION LEMMA 226 (T57) CATENARIAN DOMAIN THEOREM 227 (T58) CATENARIAN RING THEOREM 227

(Q13) Associated Graded Rings and Leading Ideals 228 (T59) LEMMA (on graded ring to be domain) 230 (T60) THEOREM (on valuation function) 231 (T61) THEOREM (on induced graded ring homomorphism) 231 (T62) COROLLARY (on grad of regular local ring) 232 (C19) Graded Rings of Polynomial Rings 232

(Q14) Completely Normal Domains 232 (T63) COMPLETE NORMALITY LEMMA 233 (T64) ORD VALUATION LEMMA 234

(Q15) Regular Sequences and Cohen-Macaulay Rings 235 (T65) LOCALIZATION LEMMA 236 (T66) REGULAR SEQUENCE LEMMA 237 (T67) INDEPENDENCE VERSUS REGULARITY THEOREM 240 (C20) Binomial Coefficients 243 (C21) Localization of Regular Sequences 243 (T68) COROLLARY 244 (T69) COHEN-MACAULAY LEMMA 244 (T70) COHEN-MACAULAY THEOREM 248 (T71) ANOTHER ORD VALUATION LEMMA 251 (C22) History 251 (C23) Annihilators of Finitely Generated Modules 251 (C24) Univariate Polynomial and Power Series Rings 251

(Q16) Complete Intersections and Gorenstein Rings 252 (T72) LARGEST SUBIDEAL CHARACTERIZATION LEMMA 253 (T73) ZERO DIMENSIONAL GST RING CHARACTERIZATION LEMMA 254 (T74) IRREDUCIBLE IDEAL CHARACTERIZATION LEMMA 255 (T75) SOCLE SIZE LEMMA 256 (T76) GST RING CHARACTERIZATION THEOREM 257 (T77) COMPLETE INTERSECTION THEOREM 258 (T78) ONE DIMENSIONAL GST RING CHARACTERIZATION THEOREM 258 (C25) Inverse Modules and Fractional Ideals 258 (C26) One Dimensional CM Local Rings 259 (C27) One Dimensional Special GST Rings 260 (C28) Conductor 260 (T79) SOMETHING IS TWICE SOMETHING THEOREM 261

(Q17) Projective Resolutions of Finite Modules 261 (T80) PROJECTIVE RESOLUTION LEMMA 263 (T80.8) THE COMPARISON LEMMA 263 (T80.9) THE SNAKE LEMMA 263 714 DETAILED CONTENTS

(C29) Residual dimension and gnb over Quasilocal Rings 275 (T81) DIM–PDIM THEOREM 276 (T81.1) AUSLANDER–BUCHSBAUM 276 (T81.6) SERRE 277 (C30) Localization of a Module 277 (C31) Prime Avoidance 285 (C31.1) PRIME AVOIDANCE LEMMA 285 (C31.2) PRIME AVOIDANCE COROLLARY 285 (C32) History 286

(Q18) Direct Sums of Algebras, Reduced Rings, and PIRs 286 (Q18.1) Orthogonal Idempotents and Ideals in a Direct Sum 287 (Q18.2) Localizations of Direct Sums 289 (Q18.3) Comaximal Ideals and Ideal Theoretic Direct Sums 290 (Q18.4) SPIRs = Special Principal Ideal Rings 293

(Q19) Invertible Ideals, Conditions for Normality, and DVRs 298 (MU) SERRE CONDITION (Rn) 298 (MU) SERRE CONDITION (Sn) 298 (T82) KRULL NORMALITY LEMMA 299 (T83) KRULL NORMALITY THEOREM 300 (T84) NORMAL LOCAL RING LEMMA 301 (T85) CONDITIONS FOR DVR 302 (T86) CONDITIONS FOR CMR 302 (T87) CONDITIONS FOR RNR 304 (T88) SERRE CRITERION 304 (C33) Local Analytic Geometry Book 306

(Q20) Dedekind Domains and Chinese Remainder Theorem 306 (T89) CHARACTERIZATION OF DD 307 (T90) LEMMA ON COMMUTATIVE MONOIDS 308 (T91) PROPERTIES OF DD 310 (T92) CHINESE REMAINDER LEMMA 311 (T93) CHINESE REMAINDER THEOREM 312 (C34) Indian Kuttak Method and the CRT 312

(Q21) Real Ranks of Valuations and Segment Completions 312 (T95) CHARACTERIZATION THM FOR SEGMENT COMPLETIONS 317 (T96) CONDITIONS FOR A LOSET TO BE SEGMENT FULL 318 (T97) CHARACTERIZATION THEOREM FOR REAL RANKS 319 (C35) min, max, lub, glb 320

(Q22) Specializations and Compositions of Valuations 320

(Q23) UFD Property of Regular Local Domains 323 (T98) PARENTHETICAL COLON LEMMA 323 (T99) UFD LEMMA 324 (T100) SNAKE SUBLEMMA 326 DETAILED CONTENTS 715

(T101) SUPPLEMENTARY PDM LEMMA 327 (T102) UFD THEOREM 329 (C36) Finite Modules Over Local Rings 330 (C37) Unique Factorization of Power Series 330

(Q24) Graded Modules and Hilbert Polynomials 330 (T103) HILBERT FUNCTION THEOREM 332

(Q25) Hilbert Polynomial of a Hypersurfaces 333

(Q26) Homogeneous Submodules of Graded Modules 335 (C38) Positive Portion and Irrelevant Submodules 336 (C39) Relevant Portion and Isolated System of Prime Ideals 336

(Q27) Homogeneous Normalization 337 (T104) IRRELEVANT IDEAL LEMMA 337 (T105) HOMOGENEOUS NORMALIZATION THEOREM 337 (C40) Irrelevant Ideals and Integral Dependence 341 (C41) Multiple Roots and Separable Polynomials 341 (C42) Homogeneous and Nonhomogeneous Normalization Theorems 342 (T106) HOMOGENEOUS RING NORMALIZATION THEOREM 342

(Q28) Alternating Sum of Lengths 343

(Q29) Linear Disjointness and Intersection of Varieties 348 (T107) CATENARIAN DOMAIN COROLLARY 349 (T108) FREE MODULE LEMMA 349 (T109) LINEAR DISJOINTNESS LEMMA 350 (T110) IDEALS UNDER TRANSCENDENTAL EXTENSION THEOREM 350 (T111) CODIMENSION OF PRODUCT THEOREM 350 (T112) DIMENSION OF INTERSECTION THEOREM 351 (C43) Ideals in Polynomial Ring Extensions 355 (C44) Ideals in Rational Function Ring Extensions 358 (C45) Regularity of Localizations of Polynomial Rings 358 (C46) Maximal Ideals in Polynomial Rings 360 (C47) Diagonals of Product Spaces 362 (C48) Intersection in Vector Space and Projective Space 363

(Q30) Syzygies and Homogeneous Resolutions 364 (T113) HOMOGENEOUS RESOLUTION LEMMA 366 (T113.8) GRADED COMPARISON LEMMA 367 (T114) HILBERT SYZYGY THEOREM 370

(Q31) Projective Modules Over Polynomial Rings 371 (T115) QUILLEN-SUSLIN THEOREM 371 (C49) Modules Over Univariate Polynomial Rings 373 (T116) INVARIANCE LEMMA 375 (T117) ADMISSIBILITY LEMMA 380 716 DETAILED CONTENTS

(T118) ALMOST ISOMORPHISM THEOREM 390 (C50) Modules Over PIDs 402 (T119) PID LEMMA 402 (T120) PID THEOREM 403 (C51) Elementary Row and Column Operations 404 (T121) TRANSVECTION THEOREM 405 (C52) Completing Unimodular Rows 406 (T122) PID UNIMODULARITY LEMMA 407 (T123) INVARIANT IDEAL LEMMA 408 (T124) COMPLETING UNIMODULAR ROWS LEMMA 408 (T125) QUASIELEMENTARY GROUP LEMMA 408 (T126) COMPLETING UNIMODULAR ROWS THEOREM 408 (C53) Stably Free Modules 417 (T127) STABLY FREE MODULE THEOREM 417 (C54) Special Linear Groups Over Polynomial Rings 419 (T 128) SUSLIN’S LOCALIZATION LEMMA 421 (T129) MENNIKE SYMBOL LEMMA 421 (T130) GENERALIZED TRANSVECTION THEOREM 422 (T131) COHN’S TWO BY TWO THEOREM 422 (C55) Permutation Matrices 430 (T132) FIRST MATRIX LEMMA 430 (T133) SECOND MATRIX LEMMA 430 (T134) THIRD MATRIX LEMMA 430

(Q32) Separable Extensions and Primitive Elements 431 (T135) VANDERMONDE DETERMINANT THEOREM 432 (T136) DISCRIMINANT INVERTING THEOREM 433 (T137) BASIC ROOTS OF UNITY THEOREM 434 (T138) BASIC FINITE FIELD THEOREM 435 (T139) BASIC PERFECT FIELD THEOREM 435 (T140) BASIC PRIMITIVE ELEMENT THEOREM 436 (T141) FINITENESS OF INTEGRAL CLOSURE THEOREM 437 (T142) SEPARABLE GENERATION THEOREM 437 (T143) SEPARATING NORMALIZATION BASIS THEOREM 441 (T144) SUPPLEMENTED PRIMITIVE ELEMENT THEOREM 443

(Q33) Restricted Domains and Projective Normalization 444 (T145) FINITE MODULE THEOREM 444 (T146) AFFINE NORMALIZATION THEOREM 445 (T147) PROJECTIVE NORMALIZATION THEOREM 446 (T148) AUXILIARY THEOREM 447

(Q34) Homogeneous Localization 447 (Q34.1) Projective Spectrum 447 (Q34.2) Homogeneous Localization 449 (Q34.3) Varieties in Projective Space 453 (T149) PROJECTIVE DIMENSION THEOREM 455 (T150) PROJECTIVE PRIMITIVE ELEMENT THEOREM 455 DETAILED CONTENTS 717

(T151) PROJECTIVE INCLUSION RELATIONS THEOREM 455 (T152) PROJECTIVE NULLSTELLENSATZ 456 (Q34.4) Projective Decomposition of Ideals and Varieties 457 (Q34.5) Modelic and Spectral Projective Spaces 458 (Q34.6) Relation between Affine and Projective Varieties 459 (T153) IDEALS AND HOMOGENEOUS IDEALS RELATIONS THEOREM 461 (T154) AFFINE AND PROJECTIVE VARIETIES RELATIONS THEOREM 462

(Q35) Simplifying Singularities by Blowups 462 (Q35.1) Hypersurface Singularities 462 (T155) HYPERSURFACE SINGULARITY THEOREM 462 (Q35.2) Blowing-up Primary Ideals 463 (T156) PRIMARY IDEAL BLOWUP THEOREM 463 (Q35.3) Residual Properties and Coefficient Sets 464 (Q35.4) Geometrically Blowing-up Simple Centers 465 (Q35.5) Algebraically Blowing-up Simple Centers 468 (T157) PRIME IDEAL BLOWUP THEOREM 469 (T158) SIMPLE CENTER BLOWUP THEOREM 469 (Q35.6) Dominating Modelic Blowup 474 (T159) SUPPLEMENTARY MODELIC BLOWUP THEOREM 474 (Q35.7) Normal Crossings, Equimultiple Locus, and Simple Points 474 (T160) NORMAL CROSSING LEMMA 476 (Q35.8) Quadratic and Monoidal Transformations 476 (T161) FIRST MONOIDAL THEOREM 477 (T162) SECOND MONOIDAL THEOREM 478 (T163) THIRD MONOIDAL THEOREM 478 (T164) FOURTH MONOIDAL THEOREM 478 (T165) FIFTH MONOIDAL THEOREM 478 (T166) SIXTH MONOIDAL THEOREM 479 (T167) SEVENTH MONOIDAL THEOREM 479 (T168) EIGHTH MONOIDAL THEOREM 480 (T169) NINTH MONOIDAL THEOREM 480 (T170) TENTH MONOIDAL THEOREM 480 (T171) ELEVENTH MONOIDAL THEOREM 481 (T172) TWELFTH MONOIDAL THEOREM 481 (Q35.9) Regular Local Rings 483

§6: Definitions and Exercises 483 (E1) Isolated System of Prime Ideals 483 (E2) Finite Generation of Ideals and Modules 484 (E3) Bracketed Colon Operation 484 (E4) Principal Ideal Theorems 485 (E5) Relative Independence of Parameters 485 (E6) Radical Description 486 (D1) Minimal Polynomial 486 (E7) Kronecker Divisibility 486 (E8) Going Down 486 (E9) Nongoing Down for Nonnormal Domains 486 718 DETAILED CONTENTS

(E10) Nongoing Down for Nonzerodivisors 487 (E11) Nongoing Down for Nondomains 487 (E12) Jacobson Rings 488 (E13) Krull’s Struktursatz 489 (D2) Domains, Ranges, Restrictions, and Conditions of Bijection 489 (E14) Spectral Nullstellensatz 489 (E15) Associated Graded Rings and Leading Ideals 490 (D3) Generalized Associated Graded Rings 490 (E16) Strong Relative Independence 491 (E17) Ideals in Polynomial Rings 491 (E18) Triviality of Associated Graded Ring 491 (E19) Generating Number 492 (E20) Regularity and Inverse of the Maximal Ideal 492 (D4) Dimension and Subdimension Formulas 492 (E21) Subdimension Formula 492 (E22) Conditions for DVR 493 (E23) Serre Conditions 493 (E24) Reduced Normality Quasicondition 494 (E25) Serre Quasicondition 494 (E26) Linear Disjointness 495 (E27) Split Monomorphisms 495 (E28) Limitations on Normalization Theorem 495 (E29) Auxiliary Theorem 496 (E30) Homogeneous Function Field 496 (E31) Projective Theorems 497 (D5) Homogeneous Localization 497 (E32) Ideals And Homogeneous Ideals 497 (E33) Blowup of a Line in Three Space 498 (E34) Regular Local Rings 498

§7: Notes 499 (N1) Noncancellative Quasiordered Monoids 499 (N2) Embedding Monoids Into Groups 500

§8: Concluding Note 500

Lecture L6: PAUSE AND REFRESH 501 §1: Summary of Lecture L1 on Quadratic Equations 501

§2: Summary of Lecture L2 on Curves and Surfaces 506

§3: Summary of Lecture L3 on Tangents and Polars 508

§4: Summary of Lecture L4 on Varieties and Models 510

§5: Summary of Lecture L5 on Projective Varieties 513

§6: Definitions and Exercises 533 DETAILED CONTENTS 719

(E1) Fundamental Theorem of Galois Theory 533 (D1) Normal Extensions 533 (E2) Splitting Fields 534 (D2) Lagrange Resolvent 534 (E3) Cyclic Extensions of degree prime to characteristic 535 (D3) Elementary Symmetric Functions 535 (D4) Symmetric Functions 536 (E4) Newton’s Symmetric Function Theorem 537 (E5) More on Newton’s Symmetric Function Theorem 537 (E6) Relations Preserving Permutations 538 (E7) Cyclic Groups 538 (D5) Artin-Schreier Extensions 538 (E8) Cyclic Extensions of degree equal to characteristic 538 (D6) Field Polynomials and Norms and Traces 539 (D7) Degrees of Separability and Inseparability 539 (E9) Exponent of Inseparability 539 (E10) Behaviour Under Finite Algebraic Field Extensions 540 (E11) Norm Giving Field Polynomial 540 (D8) Field Polynomial as Characteristic Polynomial 540 (D9) Spur and Characteristic Matrix 540 (E12) Properties of Norms and Traces 541 (E13) Condition For Inseparable Element 542 (E14) Extending Derivations and Separable Extensions 542 (D10) Polynomials in a Family and Pure Transcendental Extensions 543 (E15) Criterion for Extensions of Derivations 544 (E16) Derivations and Purely Inseparable Extensions 545 (E17) More About Purely Inseparable Extensions 545 (E18) Derivations and Separably Generated Extensions 545 (E19) Criterion for Separable Algebraic Extension 545 (D11) Jacobian Matrix and Jacobian 545 (E20) Jacobian Criterion of Separability 546 (E21) Lagrange’s Theorem 546 (E12) Action of a Group, Orbit, and Stabilizer 547 (E22) Orbit-Stabilizer Lemma 547 (D13) Orbit Set and Fixed Points 548 (E23) Orbit-Counting Lemma 548 (D14) Conjugation Action and Conjugacy Classes 548 (D15) Normalizer, Centralizer, and Center of a Group 549 (E24) Class Equation 549 (D16) Prime Power Group and Prime Power Subgroup 550 (E25) Action of Prime Power Group 550 (E26) Cauchy’s Theorem 550 (E27) Normalizer of Prime Power Subgroup 551 (D17) Transitive Action and Transitive Group 551 (D18) Sylow Subgroup and Prime Power Orbit 551 (E28) Sylow Transitivity 551 (D19) Complete Set of Conjugates 552 (E29) Sylow’s Theorem 552 720 DETAILED CONTENTS

(E30) Existence of Prime Power Subgroups 553 (D20) Exponential Notation and Subscript Notation 553 (E31) Conjugation Rule 553 (D21) Transportation and Permutation Isomorphisms 554 (E32) Conjugacy Lemma 554 (D22) Regular and Semi-regular Permutation Groups 554 (D23) Left and Right Regular Representations 554 (E33) Cayley’s Theorem 554 NOTE Hint to Cayley and Preamble to Burnside 555 (E34) Centralizer Lemma 555 (D24) Multitransitive and Antitransitive Groups 556 (D25) Frobenius and Semi-Frobenius Groups 556 (E35) Order Lemma 557 (D26) Invariant Partition and System of Imprimitivity 557 (E36) Block Lemma 557 (D27) Maximal and Minimal Normal Subgroups 558 (D28) Primitive and Imprimitive Groups 558 (E37) Primitivity Lemma 558 (D29) Characteristic Subgroup 560 (D30) Elementary Abelian Group 560 (E38) Normality Lemma 560 (E39) Fixed Point Lemma 561 (D31) Direct Product 562 (E40) Double Normality Lemma 562 (E41) Burnside’s Lemma 563 (E42) Burnside’s Theorem 564 NOTE Theorems of Burnside and Frobenius 564 (E43) Simplicity of Alternating Groups 564 (E44) Unit Ideals in Polynomial Rings 565 (E45) Generalized Power Series 566 (E46) Simplicity of Projective Special Linear Group 569 (E47) Group Action and Iwasawa’s Simplicity Criterion 569 (E48) Two-Transitive Action on Projective Space 570 NOTE Definitions of Transitivity and Antitransitivity 570 (E49) Transvection Matrix and Dilatation Matrix 570 (E50) Perfect Group 571 (E51) Socle-Size 571 (E52) Coprincipal Primes and Isolated Subgroups 572 (E53) General Valuation Functions 572 (D32) Lexicographic Products and Sums 573 (E54) Possible Real Ranks 576 (E55) Impossible Real Ranks 576 (D33) Principal Rank and Real Rank 576 (E56) Compositions of Valuations 577 (E57) Relabelling Prime Ideals 577 (E58) Suslin Localization 577

BIBLIOGRAPHY DETAILED CONTENTS 721

DETAILED CONTENT

INDEX Index Numbers are page numbers where the item algebra (over a ring); 286, 521 is first defined or recalled with further algebra-theoretic (also called enhancement. Symbols are listed multiplicative) direct sum; 286-287, under NOTATION-SYMBOLS and 521 they are separated from their algebraic closure; 19, 39-40, 134, 505 meanings by a colon. Abbreviations algebraic closure, absolute or relative; 19 are alphabetized under algebraic element; 10, 505 NOTATION-WORDS and they are algebraic (extension of) over 10, 505 separated from their long forms by a algebraic function field; 432, 529 colon; we follow the sequence: ordinary algebraic function field over a domain; 444 letters, blackboard bold letters, script algebraic geometry; 26, 82 letters, Greek letters, German letters. algebraic independence; 10, 505 If an item is claimed to be on page x algebraic points; 127, 459 but not found there, it may be found algebraically blowing-up simple center; on page x ± n with small n. 468-473 algebraically closed; 27, 506 Abel; 2 Almost Isomorphism Theorem; 390-401 abelian; 3, 503 alternating group; 5, 21, 503, 564 Abhyankar’s proof of Newton’s Theorem; alternative function field; 122-124 82 alternative homogeneous function field; absolute algebraic closure; 19 454-455 absolute value; 44-46 alternating sum of lengths; 343-347 absolutely saturated chain of prime ideals; analytic branch; 59 209, 516 analytic hypersurface; 55, 58 abstract disjoint union; 573 analytic independence; 32 action; 547-551 Analytic Independence Theorem; 202, 515 action (conjugation); 548 analytic geometry; 26, 52-54, 82 action of a group on a set; 547 annihilator; 93, 181, 510 action (faithful); 569 annihilator of nonprimary submodule; 142 action of prime power group on a set; 550 annihilators and colons; 93 action (transitive); 551, 570 additive abelian group; 3, 503 annihilators and direct sums; 172 additive abelian groups being modules over annihilators and radicals; 93 the ring of integers, direct sum theory annihilators of finitely generated modules; applies to them; 172 251 adjoint of a homomorphism; 374 Another Ord Valuation Lemma; 251, 518 adjoint of a matrix; 75, 509 antitransitive (permutation) group; 556, Admissibility Lemma; 380-390 570 admissible (pair of homomorphisms); 374 approximate root; 49-50 Aesop’s Fables; 14 archimedean; 45 affine algebraic variety; 124, 512 archimedeanly comparable; 45, 568 affine coordinate ring; 86-87, 122-124 arithmetic genus; 334 affine model; 130, 531 Artin; 6 affine portion (or portion at finite Artin-Rees Lemma; 184-189, 514 distance); 57, 460 Artin-Schreier Extensions; 538 affine semimodel; 130, 531 artinian modules and rings; 107-110, 511 affine ring; 111, 512 artinian ring iff noetherian ring of affine varieties; 122-126 dimension zero; 109 affine variety set; 122 artinian: DCC ⇔ MNC; 107 Affine And (Relevant) Projective Varieties ascending chain condition; 94 Relations Theorem; 462 Askwith; 52, 82 Affine And (Relevant) Projective Varieties assassinator = colorful (like annihilator) Relations Theorem as related to Ideals long form of ass; 98, 181, 511 And Homogeneous Ideals; 497 assigned value group; 34 Affine Normalization Theorem; 445, 529 associate; 11, 505 Algebra; 1, 24, 500 associated graded ring; 228, 517 722 INDEX 723 associated graded ring in connection with a Burnside; 555 Corollary about regular local ring; 232 Burnside’s Lemma; 563 associated graded ring in connection with Burnside’s Theorem; 555, 564 its triviality; 491 associated graded rings and leading ideals; calculus; 53-54 490 cancellative; 5, 174, 503 associated graded rings with their canonical; 6-8, 502-504 generalized versions; 490 canonical basis; 365, 374 associated primes; 96, 511 canonical epimorphism (induced by associator; 98, 181, 511 homogeneous generators); 365 associative or associativity; 3, 502 canonical homomorphism (induced by Auslander-Buchsbaum; 276, 286 homogeneous elements); 365 autoequivalent homomorphisms; 374 canonical monomorphism; 406 automorphism; 6-8, 502-504 canonical valuation; 321 Auxiliary Theorem (about complete or Cardano; 2 normal models or semimodels); 447, cardinal; 28, 42 496 cardinality; 28 axiom of choice; 27-28, 37-44, 506-507 cartesian product; 26, 508 cartesian product of a family; 170 backward shift; 364 Castelnuovo; 82 Basic Finite Field Theorem; 434 catenarian; 225, 517 Basic Perfect Field Theorem; 435 Catenarian Condition Lemma; 226 Basic Primitive Element Theorem; 436 Catenarian Domain Corollary; 349 Basic Roots of Unity Theorem; 433 Catenarian Domain Theorem; 227 basis; 7, 504 Catenarian Ring Theorem; 227, 517 basis of an additive abelian group; 307 Cauchy multiplication; 9, 29, 30 Beejganit; 1, 499 Cauchy relative to; 79, 509 Bell; 54 Cauchy sequence; 44-46, 71-74, 509 B´ezout;25-26, 55, 82-85 Cauchy’s Theorem; 550 Bhaskara; 312, 499 Cayley; 82 Bhaskaracharya (Bhaskara + Acharya); 1 Cayley and Burnside; 555 bijection; 2, 489, 501 Cayley’s Theorem; 554 bijective; 2, 489, 501 center of a group; 549 bilinear map; 78-79 center of a monoidal transformation; 467 binary relation; 27, 31 center of a quadratic transformation; 466 binomial coefficients; 243 center of a quasilocal ring; 130, 531 Binomial Theorem; 81 centralize; 549 bivariate; 15 centralizer; 549 block (in the sense of permutation groups); Centralizer Lemma; 555 557 chain in a poset; 27 Block Lemma; 557 characteristic; 3, 18, 505 block matrices; 75, 144 characteristic function; 141 blowing-up primary ideals; 463 characteristic matrix; 540 blowup of equimultiple simple center does characteristic polynomial; 540 not increase multiplicity (proof in characteristic subgroup; 560 Third Monoidal Theorem); 466-468, characterizations of PIDs, PIRs, SPIRs, 470, 478, 530 and UFDs; 293-297, 521 blowup of points and lines; 139-140 Characterization Theorem for Real Ranks; blowup of singularities; 133-134 319, 523 blowing-up simple center (geometrically or Characterization Theorem for Segment-Full algebraically); 465-473 Losets and Segment-Completions; 317 blowup of a line in three space; 498 Chevalley; 82, 500 Blowup Theorem (relative to Prime Ideal Chicago World’s Fair; 3, 51 or Simple Center); 469 Chinese Remainder Lemma; 311 Bombay; 82 Chinese Remainder Theorem; 307, 312, 522 brim-full; 315, 523 chord; 53 bracketed colon; 98-99, 511 Chrystal; 36 bracketed colon operation; 187-188, 484 circle; 52-53, 89, 124 724 INDEX circular and Fermat cones; 134 complete system of orthogonal class equation; 549 idempotents; 288 classical group; 52 completing the power; 62 Classification Theorem; 4, 24 completing the square; 1, 62 closure and closedness corollaries (of Krull completing the square method; 62 Intersection Theorem) for ideals as completing unimodular rows; 406-417, 528 well as modules; 186, 190 Completing Unimodular Rows Lemma; 408 codimension; 348, 527 Completing Unimodular Rows Theorem; Codimension of Product Theorem; 350, 527 408 coefficient field; 465 completing unimodular tuples; 406-417, 528 coefficient set; 465 completely normal domain; 233, 518 cofactor; 75, 509 complex numbers; 3, 44-50, 502 cofinal; 38-39 components (= homogeneous components) Cohen; 251, 500 of an element in a graded ring; 173 Cohen-Macaulay Lemma; 244 composition; 2, 501 Cohen-Macaulay module; 235, 519 composition of valuations; 320-323, 523, Cohen-Macaulay ring; 236, 519 577 Cohen-Macaulay Rings (Conditions for); composition series; 103 302 compositum; 348, 527 Cohen-Macaulay Theorem; 248, 519 condition for inseparable element; 542 Cohen’s Noetherian Theorem; 191-192 conditions for a common factor or a Cohn’s Two By Two Theorem (or Cohn’s common root of two univariate Example); 422, 528 polynomials; 138 collection of objects; 2 conditions for a loset to be segment-full; College Algebra (of Rings and Ideals); 500 318 colon and nonzerodivisor ideal corollary; Conditions for DVR; 302, 493, 522 182 conductor; 260, 520 colons of ideals or modules; 89-90, 510 conic; 53 column; 51 conjugation action; 548 column rank; 75 conjugacy classes; 548 comaximal ideals; 110-111, 511 Conjugacy Lemma; 554 common divisor; 15 conjugate; 548 common multiple; 18 Conjugation Rule; 553 commutative; 3, 5, 503 content of a polynomial; 68, 508 Commutative Monoids Lemma; 308 contracted and extended ideals; 118 commutative triangle; 371 convergent sequence; 45, 71-74, 509 commutator; 22 coordinate ring; 122-124 commutator subgroup; 22 coprime; 16 Comparison Lemma; 263 coprincipal prime ideal (nonmaximal); 316, complement; 3, 501 572 complementary prospectral variety; 448 core; 315, 523, 572 completable; 406, 528 core-full; 315, 523 complete field; 45 coset; 4, 502 complete intersection, ideal-theoretic; 235, countable; 27 519 counting properly; 26 complete intersection, local ring theoretic; Cramer’s rule; 135-138, 145-147 252, 520 Cramer’s rule for modules; 184 complete intersection, (noetherian ring Cremona; 82 being called a local complete Criterion for Extensions of Derivations; 544 intersection); 252, 520 Criterion for Separable Algebraic (Field) Complete Intersection Theorem; 258, 520 Extensions; 545 complete model; 130-132, 531 Criterion of Separability (Jacobian); 546 Complete Normality Lemma; 233, 518 cubic curve; 89 complete ordered abelian group; 45 cubic discriminant; 86 complete quasilocal ring; 71-74, 509 curve; 25-26, 86, 124-125 complete relative to; 79, 509 cusp; 60, 133-134 complete semimodel; 130, 531 cuspidal cubics; 133-134 complete set of conjugates; 552 cycle; 20 INDEX 725 cyclic extension; 13, 534 determinant of a zero matrix; 52 cyclic extensions of degree equal to determinants and matrices; 51 characteristic; 538 diagonal of product spacess; 362 cyclic extensions of degree prime to diagonal map; 170 characteristic; 535 diagonal product of maps; 171 cyclic group; 4, 502, 538 diagonal sum of maps; 171 cyclic permutation; 20 Dickson; 6, 51 cylinder; 25 dilatation matrix; 404, 570 Dim-Emdim Theorem (also Extended diagonals of product spaces; 362 version); 127, 195 decimal expansion; 47 dimension and subdimension formulas; 492 decomposition of ideals and varieties; 218, dimension formula; 225; 517 224, 457-458 dimension inequality; 225, 517 decomposition of ideals and varieties Dimension Corollary (about behavior under (projective version); 457 integral extensions); 207, 515 Dedekind; 306, 500 Dimension Lemma (about multivariate Dedekind completion; 315 polynomial or power series extensions); Dedekind cut; 48, 315 195-197, 514 Dedekind domain; 306, 522 dimension of a model; 130, 532 Dedekind Domain Characterization and its dimension of a ring; 105, 511 Properties; 307, 310, 522 dimension of a variety; 26, 86, 89, 122-124 Dedekind map (closed or open); 315 Dimension of Intersection Theorem; 351, Dedekind map (of a pair of losets); 315 527 degree of a polynomial; 8-9 dimension of vector space; 8, 89 degree of a curve or surface; 26 Dimension Theorem (also Extended version degree of a homogeneous component; 173 with First and Second Supplementary degree of a homogeneous element; 173 versions); 124, 210-217, 223-224 degree of a hypersurface; 26 Dimension Theorem (its Projective degree of inseparability; 539 version); 455 degree of separability; 539 dimensionality; 124 degree of a variety; 26 Dim-Pdim Theorem; 276, 521 degree form; 78 direct product of groups; 562 dehomogenization; 166 dehomogenization map and the direct product of maps; 170 epimorphism induced by it; 449-452 direct product of modules; 170, 513 dehomogenization map in its operational direct summand; 262 incarnation; 460 direct summand (graded); 365 dehomogenization of an element or a direct sum of algebras ; 286-293, 521 polynomial; 449-451 direct sum of algebras and its use in dehomogenize; 56 describing total quotient rings of depth of an ideal; 106, 124, 511 reduced noetherian rings and their derivation; 23-24, 505 normalizations; 286-293 derivations (extensions of derivations, direct sum of maps; 170 especially to separable algebric, direct sum of modules (external unless separably generated, and purely stated otherwise); 169-172, 513 inseparable, field extensions); 542, 544, direct sum of modules (graded); 364 545 direct sum of rings ; 286-293, 521 Derivations and Purely Inseparable direct sum of rings and its use in describing Extensions; 543 total quotient rings of reduced Derivations and Separable Extensions; 542 noetherian rings and their Derivations and Separably Generated normalizations; 286-293 Extensions; 543 direct sum theory applies to additive derivative; 9-10, 23-24, 30 abelian groups by regarding them as Descartes; 25-26, 82 modules over the ring of integers; 172 descending chain condition; 107 direction cosines; 54 determinant map; 51 Discrete Valuation Rings (Conditions for); determinant of a homomorphism; 374 302, 493, 522 determinant of a matrix; 51, 139, 143-145 discriminant; 83-86, 510 726 INDEX discriminant (in its modified form); 22, 431, equimultiple simple center blowup does not 433, 505 increase multiplicity (proved in Third Discriminant Inverting Theorem; 432 Monoidal Theorem); 466-468, 470, 478, disjoint cycles; 20 530 disjoint union; 59, 548, 573 equivalence relation; 31 distinguished polynomial; 70, 508 equivalence relation (for syzygies); 364-366 distributive; 3, 6, 503 equivalence class; 28, 31 divisible group; 45 equivalent or (in greater detail) divisibility; 14, 505 autoequivalent homomorphisms; 374 divisibility group; 157-158 equivalent normal series; 103 divisibility ring; 157-158 equivalent valuations; 158 divisibility ring of a field; 158 essentially equal; 315 divisibility valuation; 157-158, 321, 523 euphony; 432 division algorithm; 15, 23 Existence of Prime Power Subgroups; 553 domain (= nonnull ring having no nonzero exponent of inseparability; 539 zerodivisor); 5, 90, 503 Euler’s Theorem concerning Homogeneous domain of a map (see map); 489, 501 Polynomials; 57, 76 domain with factorization of ideals; 306, euclidean algorithm; 15, 23 522 euclidean domain; 11, 505 domain with group factorization of ideals; even permutation; 5, 21, 502 306, 522 exact sequence; 261, 521 domain with prime factorization of ideals; exact sequence (graded); 365 307 exactness; 261 domain with unique factorization of ideals; exceptional line; 133-134, 466 306, 522 exceptional hyperplane; 465, 468 exponential notation; 553 domains, ranges, restrictions, and conditions of bijection; 489 Extended Dim-Emdim Theorem; 195, 514 Extended Dimension Theorem; 210-212, dominated by; 129-130, 530 516 dominates; 129-130, 530-531 Extending Derivations and Separable dominating modelic blowup; 474, 530 Extensions; 542 domination and subgroups; 532 extensions of derivations; 542-545 domination of quasilocal rings; 129-130, 530 external direct sum of modules; 170, 513 Double Normality Lemma; 562 double point; 55 factor group; 4, 502 factorization; 17 Eakin’s Noetherian Theorem; 191-193 faithful action; 569 element; 2, 501 faithful modules; 183, 514 elementary abelian group; 560 family; 31 elementary row and column operations; 404 Fermat cones; 134 elementary symmetric functions; 535 Ferrari; 2 elementwise stabilizer; 547 field; 3, 503 ellipse; 53 field degree; 8, 504 ellipsoid; 53 fixed field; 13, 505 elliptical cylinder; 25 fixed points; 548 embedded prime and primary components; field generators; 11 182, 187-188 field polynomial; 539 embedding dimension; 127, 512 field polynomial (as norm); 540 embedding monoids into groups; 500 field polynomial (behaviour under finite embedding projective space into projective algebraic field extensions); 540 model; 166-167 field-theoretic compositum; 348, 527 empty set; 3, 501 Field Polynomial as Characteristic Engineering Book; 69, 134 Polynomial; 540 enlargement; 126-127, 458-459 Field Polynomials and Norms and Traces; Enriques; 82 539 epimorphism; 6-8, 502-504 Field Theory; 533 equicharacteristic (quasilocal ring); 465 finite direct sum of rings or ring-isomorphic equimultiple locus; 475, 530 to a direct sum of rings; 287, 521 INDEX 727

Finite Field Theorem (Basic); 434 general elementary group (in a more finite generation; 7, 504 general situation); 420 finite generation of ideals and modules; general linear group; 51; 528 185, 484 general valuation functions; 572 finite free module; 262 generalizations of valuations; 321-323; 523, finite free module (in the context of 572 homogeneously); 365 generalized associated graded rings; 490 finite free resolution; 262, 365, 527 generalized meromorphic series field having finite module; 262, 521 exponents in an ordered abelian group finite graded module; 331, 524 (generalizes the idea of the univariate finite modules over local rings; 330 meromorphic meromorphic series field Finite Module Theorem; 444 over any given field); 35, 506, 566 Finite Module Theorem and Limitations on Generalized Newton’s Theorem; 36 it; 444, 495 generalized power series ring (which is a finite prefree resolution; 262 subring of the generalized finite preprojective resolution; 262 meromorphic series field); 35, 566 finite projective module; 262 Generalized Principal Ideal Theorem; 194, finite projective module (in the context of 485 homogeneously); 365 Generalized Transvection Theorem; 422 finite projective resolution; 262 generating number; 235, 492, 519 Finiteness of Integral Closure Theorem; 437 generating number (homogeneous); 364 fixed field; 13, 533 generating number over quasilocal rings; 275 Fixed Point Lemma; 561 generating set of a (multiplicative) fixed point set; 548 commutative group; 307 fixed points; 548 generators of an ideal; 6 formal power series; 58 geometric series identity; 32 Formanek’s proof of generalized version of geometrically blowing-up simple center; Eakin’s Noetherian Theorem; 191-193 465-468 fractional ideal; 258, 520 geometry; 52-60, 122-134, 444-483, 508, 529 free additive abelian group; 307 geometric motivation; 124-125 free additive abelian monoid; 307 geometrizing project; 529 free (algebra or module); 348-349 glb; 320 free (module); 235, 262, 519 global dimension; 262, 521 Free Module Lemma; 349 God of Learning; 1 free resolution; 262 Going Down Theorem; 206, 515 Frobenius group; 556 Going Down Theorem is true or not under Frobenius’ Theorem; 564 various conditions; 486-487 function field; 86-87, 122-124 Going Up Theorem; 205, 515 functional notation and tuple notation; 171 Gorenstein ring, local or noetherian; 252; Fundamental Theorem of Algebra; 27 520 Fundamental Theorem of Galois Theory; gradation or grading of a graded ring is an 13, 533 indexed family of submodules; 172 gradation or grading of a graded module; Galois; 2, 13, 14, 51 331, 524 Galois extension; 12 Graded Comparison Lemma; 367 Galois field; 3, 504 graded component; 229-230 Galois group; 2, 12, 505 graded direct summand; 365 Galois group as relations preserving graded direct sum of modules; 364 permutations; 13, 538 graded exact sequence; 365 Galois Theory; 13, 533 graded image; 229-230 Galois Theory Theorems; 533 graded map; 229-230 Galois’ Symmetric Group Theorem; 14 graded modules; 330, 524 Ganesh; 1 graded resolution; 365 Ganesh Temple, 1 graded ring and its type; 172, 513 Gauss; 27 graded ring, Lemma for it to be domain; Gauss Lemma; 68 230 general elementary group; 405, 528 graded ring homomorphism; 173, 513 728 INDEX graded ring homomorphism, Theorem on homogeneous and ordinary localizations to induced such; 231, 518 coincide; 453 graded rings; 172-180, 228-232, 513, 517 homogeneous components of a graded graded rings, alternative definition; 174 module; 331, 524 grades rings, integrally or nonnegatively or homogeneous components of a graded ring; naturally; 175, 513 173 graded rings of polynomial rings; 232 homogeneous components of a graded short exact sequence; 365 homogeneous ideal in graded ring; 173 graded short exact sequence splits; 365 homogeneous components of a graded subring; 173 homogeneous submodule of a graded gradient; 54 module; 331, 524 greatest common divisor; 15 homogeneous components of an element in greatest lower bound; 320 a graded module; 331, 524 greatest or largest or maximum element in homogeneous components of an element in a poset; 320 a graded ring; 173 group; 3, 502 homogeneous coordinate ring; 454-455 group action; 547, 548, 550, 551, 569 homogeneous coordinates; 55-58 group generated by; 404 homogeneous dimension, 365, 527 Group Theory; 533 homogeneous element in a graded module ; 331, 524 GST Ring Characterization Theorem; 257, homogeneous element in a graded ring ; 520 172, 513 GST Ring Characterization Theorem, homogeneous elements or generators general as well as details of zero and (canonical homomorphism or one dimensional cases; 254-258 epimorphism induced by); 365 homogeneous function field; 454-455 Harvard; 82 homogeneous generating number; 364 Hausdorff; 71-74, 509 homogeneous function field; 496 Hausdorff relative to; 78-79, 509 homogeneous ideal of a projective variety; height of an ideal; 106, 124, 511 453, 459 Height Theorem; 195 homogeneous ideals, their characterizations Hensel; 60 and some properties (colons, radicals, Hensel’s Lemma; 60-62, 74, 78-80, 508 primary decompositions, and High School Algebra (of Polynomials and associated primes); 173-180; 513 Power Series); 500 homogeneous linear equations; 137-138 higher cusps; 134 homogeneous local ring; 454-455, 459 Hilbert; 86, 364, 527 homogeneous localization; 449, 529 Hilbert Basis Theorem; 86-89, 163 homogeneous localization (generalized Hilbert degree; 333, 525 version); 497 Hilbert function; 330-333, 524, 525 homogeneous localization (or algebraization Hilbert Function Theorem; 332-333, 525 of the geometry of projective Hilbert Function Theorem proof makes use varieties); 447-462, 529 of some basic properties of homogeneous normalization; 337-343, 526 homogeneous submodules together Homogeneous Normalization Theorem; 337, with homogeneous normalization and 526 alternating sum of lengths; 335-347 homogeneous polynomial; 33 Hilbert Nullstellensatz; 126, 218, 516 homogeneous quotient field; 449, 529 Hilbert Nullstellensatz (its Spectral and homogeneous radical ideal set; 448 Projective versions); 222-223, 456 Homogeneous Resolution Lemma; 366-370 Hilbert polynomial; 330-333, 524, 525 homogeneous resolutions; 364-370, 527 Hilbert polynomial of a hypersurface; homogeneous ring; 178, 513 333-334, 525 Homogeneous Ring Normalization Hilbert subdegree; 333, 525 Theorem; 342, 526 Hilbert Syzygy Theorem; 370 homogeneous submodules, Hilbert transcendence; 333, 525 characterizations and some properties Historical Ramblings; 500 (colons, radicals, primary homogeneous and nonhomogeneous decompositions, and associated normalization; 342, 526 primes); 331-332, 335-336, 524 INDEX 729 homogeneous syzygies; 366 ideal-theoretic complete intersection (height homogeneously equivalent; 365 equals generating number); 235, 519 homogeneously finite free graded ideal-theoretic direct sum; 293 resolution; 365 idempotent; 287-289 homogeneously finite free module; 365 identify; 31 homogeneously finite free resolution; 365 identity map; 3 homogeneously finite prefree graded identity matrix; 51 resolution; 365 identity map of a set; 373, 489 homogeneously finite prefree resolution; 365 identity permutation; 21 homogeneously finite preprojective module; image; 4, 501 365 immediate predecessor; 315, 523 homogeneously finite projective graded imperfect field; 431, 529 resolution; 365 implicit differentiation; 53 homogeneously finite projective module; impimitive (permutation) group; 558 365 Implicit Function Theorem; 53 homogeneously finite projective resolution; Impossible Real Ranks; 576 365 Inclusion Relations Theorem; 125, 516 homogeneously minimal free resolution; 365 Inclusion Relations Theorem (Spectral and homogeneously minimal free graded Projective versions); 220-222, 455 resolution; 365 increase the roots; 63 homogeneously minimal prefree graded independence, relative; 198-202, 515 resolution; 365 Independence Versus Regularity Theorem, homogeneously minimal prefree resolution; and its Corollary; 240-244 365 indeterminate; 8-10, 30, 505 homogenize; 56 index; 4, 502 homogenization; 166 indexed family; 89-90 homogenization (minimal); 449-451 indexing set; 39, 90, 501 homomorphism; 6-8, 502-504 induced by (homomorphism); 229-230, 517 initial form; 77 homomorphism induced by; 229-230, 517 injection; 2, 501 homomorphism splits; 373 injective; 2, 489, 501 homothety; 52 infimum; 320 homothety group; 52 infinite chain of prime ideals; 209, 516 hyperbola; 25, 53 infinitely near points; 134 hyperboloid; 53 infinity (hyperplane or portion or points at hyperplane; 53 infinity); 55-57, 460 hyperplane at infinity; 56-57, 460 inflexion; 89 hyperquadric; 53-58 inseparability degree; 539 hypersurface; 26, 53-59, 77-78, 84-86, 124 inseparability eponent; 539 hypersurface singularities; 462 inseparable element; 431, 529 Hypersurface Singularity Theorem; 462 inseparable element (condition for); 542 inseparable extension; 431, 529, 533, 545 ideal; 6, 89, 503 inseparable polynomials; 431, 529 ideal of an affine variety; 122 integral closure; 65, 135-136, 508 ideal set, nonunit ideal set, radical ideal Integral Closure Theorem (Finiteness set; 96, 123, 313, 510 version); 437 ideally closed; 348 integral dependence; 65, 135-136, 508 ideals in a direct sum; 287-289 integral dependence in relation to irrelevant ideals in polynomial ring extensions; 355 ideals; 341, 526 ideals in polynomial rings; 491 integral dependence (satisfaction); 143 ideals in rational function ring extensions; integral element; 65, 135-136, 508 358 integral (extension of) over; 65, 135-136, ideals under field extensions; 217 202, 508, 515 Ideals And Homogeneous Ideals; 497 integral extensions and behavior of height Ideals And Homogeneous Ideals Relations and dimension (Dimension Corollary); Theorem; 461, 497 207, 515 Ideals Under Transcendental Extension integral extensions and preservation of Theorem; 350 fields and maximal ideals; 203 730 INDEX integral extensions and preservation under isolated system of prime ideals of an ideal; epimorphisms and localizations; 203 178, 483 integral extensions and proper isolated system of prime ideals of a containment; 205 submodule; 336 integral extensions and radical description; isomorphism; 6-8, 126, 502-504 205 Isomorphism Theorems and Groups; 92-93 integrally closed; 65, 135-136, 508 Isomorphism Theorems and Modules; 91-92 integrally graded ring; 175 Isomorphism Theorems First to Fifth; iterated monoidal transform; 477 91-93 internal direct sum of modules; 169, 513 Iwasawa’s Simplicity Criterion; 569 internal free additive abelian group; 307 intersection; 3, 501 jacobian; 546 intersection multiplicity; 54-55, 89 jacobian matrix; 545 intersections of ideals or modules; 89-90 Jacobian Criterion of Separability; 546 intersection of varieties; 348-363, 527 Jacobson; 82 Intersection Theorem (Dimension of); 351, Jacobson radical; 183-184, 219, 514 527 Jacobson ring; 219, 517 intersections in vector spaces and Jacobson rings (with examples of rings projective spaces; 363 which are not Jacobson); 488 intuitive definition; 124 Jacobson spectral variety; 219 intval domain; 67, 69, 157-160, 508 Jacobson spectrum; 219, 517 Invariance Lemma; 375-379 Jordan; 51 invariance-set; 374 Jordan-H¨olderTheorem; 4, 103-105 invariant ideal; 407 kernel; 6, 502 Invariant Ideal Lemma; 408 Khahara; 499 invariant partition; 557 Klein; 82 inverse image; 4, 501 Kronecker; 27 inverse module; 258, 520 kroneckerian dimension; 444 inverse modules and fractional ideals; 258 Kronecker divisibility; 486 invertible ideals and modules; 298, 522 Kronecker’s Theorem; 69 involution; 256 Krull; 193, 251, 298, 500, 514 irreducible affine variety set; 122 Krull Intersection Theorem; 182-190, 514 irreducible component; 86, 89, 125, 457-458 Krull Normality Lemma; 299, 522 irreducible ideal decomposition; 95 Krull Normality Theorem; 522 irreducible element; 11, 505 Krull’s Struktursatz (Minimal Primes irreducible ideal or submodule; 90, 510 Theorem); 223, 480, 517 Irreducible Ideal Characterization Lemma; Kuttak Method of Ancient India as 255 reported in Bhaskara’s Beejganit; 312 irreducible ideals and their theory; 252 irreducible projective variety set; 454 Lagrange; 13 irreducible variety; 86-87, 122-123, 448, Lagrange Resolvent (Theorem); 13, 534 454, 459 Lagrange’s Theorem; 546 irreducibility; 124 Laplace development; 75, 143 irredundancy; 125 largest (element of a set of subsets); 253 irredundant premodel; 130, 531 largest (or greatest or maximum) element irredundant primary decomposition of in a poset; 320 ideals and submodules; 95-97, 510 largest proper subideal; 253 irrelevant ideals; 178-179, 457-458 Largest Subideal Characterization Lemma; Irrelevant Ideal Lemma; 337, 526 253 irrelevant ideals in relation to integral leading coefficient; 330, 508 dependence; 341, 526 leading form; 229-230, 517 irrelevant submodule; 336, 526 leading ideal; 229-230, 517 isobaric; 84 leading ideals and associated graded rings; isolated component; 98-99, 142, 511 490 isolated positive upper segment; 313 least common multiple; 18 isolated subgroup; 47, 312-313, 523, 572 least (or smallest or minimum) element in a isolated subgroup (principal); 316 poset; 320 INDEX 731 least upper bound; 320 localization transitivity; 116, 120 left regular representation; 554 localizations commute with epimorphisms, Leibnitz; 49 i.e., permutability of localizations and length of a cycle; 21 surjections; 116, 120 length of a chain; 208, 516 localizations of direct sums; 289-293 length of a module; 89, 103-105, 511 locally free; 371 length of a sequence or series; 103-108 locally projective; 371 length of an exact sequence; 261, 521 logical formality and narrative discourse; lexicographic direct sum; 317, 573 168 lexicographic disjoit sum; 574 long division; 15 lexicographic order; 34, 316, 506, 573 loset; 27, 313, 506 lexicographic power; 317, 573 loset of all nonempty segments; 574 lexicographic product; 316, 573 lower segment (in a poset); 28, 314 lexicographic restricted power; 573 lub; 320 L’Hospital’s Rule; 167-168 Lying Above Theorem; 204, 515 lies above; 202, 515 Lying Below Lemma; 204 lie above; 202, 515 lie below; 202, 515 Macaulay; 251 lies below; 202, 515 map (it associates elements of a set called lifting homomorphism; 371 its domain to elements of a set called lifting property; 371 its range); 2-4, 501 limit; 45, 71-74, 509 mapping (it sends a specific element to its limit relative to; 79, 509 image); 404 Limitations on Normalization Theorem; matrix; 51 495 matrix of a homomorphism; 374 limiting ordinal; 28 Matrix Lemma (First, Second, and Third); linear automorphism; 76-78, 82 430 linear disjointness; 348-363, 527 matrix ring; 51 Linear Disjointness (in various forms); 495 max; 320 Linear Disjointness Lemma; 350 maximal complementary prospectral linear equations; 135-138, 145-148 variety; 448 linear group; 52, 76, 509 maximal condition; 94 linear independence; 7 maximal element of a poset; 27, 320 linear map; 7 maximal ideal; 6, 14, 39, 503 linear order; 27 maximal ideals in polynomial rings; 360 linear transformation; 52 maximal independent set; 41, 507 linearly disjoint; 348, 527 maximal prospectral variety; 448 linearly ordered set; 27, 506 maximal prospectral variety set; 448 lo; 27 maximal prospectrum; 447 Local Analytic Geometry Book; 306 maximal regular sequence; 235, 519 local complete intersection; 252, 520 maximal spectral affine space; 126 local ring; 72, 87, 122-124, 509 maximal spectral projective space; 458-459 Local Ring Construction Theorem; 121 maximal spectral variety; 95, 123, 510 local rings (finite modules over); 330 maximal spectrum; 95, 123, 510 localization; 87, 114-121, 512 maximal subgroup; 558 localization at multiplicative set; 114-120 maximality ring; 157-160 localization at prime ideal; 120-121 maximality ring of a field; 158-160 localization characterization; 115, 120 maximum (or largest or greatest) element localization ideal correspondence; 116, 120 in a poset; 320 localization in the good case is a subring of meet nicely or transversally; 474 the total quotient ring; 115, 120, 202, Mennike symbol; 419, 528 512 Mennike Symbol Lemma; 419 Localization Lemma; 236 meromorphic series; 29, 506 localization of a module; 277-285 meromorphic series field; 31, 317, 506 Localization of Normality Lemma; 202-203 meromorphic series ring; 29, 317, 506 localization of regular sequences; 243 min; 320 localizations (homogeneous and ordinary) minimal assassinator = colorful (like to coincide; 453 annihilator) long form of nass; 96, 181 732 INDEX minimal associator; 96, 181 monomorphism; 6-8, 502-504 minimal complementary prospectral monomorphism splits; 373 variety; 448 Moore; 3, 51 minimal condition; 107 Multiple Ring Extension Lemma; 226, 517 minimal element of a poset; 320 multiple roots and separable polynomials; minimal free resolution; 262 341, 505 minimal generating set; 41, 507 multiplicative; 3, 5, 503 minimal homogenization; 449 multiplicative (also called minimal modelic affine space; 127 algebra-theoretic) direct sum; 286-287, minimal normal subgroup; 558 521 minimal prefree resolution; 262 multiplicative set; 98-99, 511 minimal primes; 182, 193, 223 multiplicity; 54 Minimal Primes Theorem; 223, 489, 517 multiply the roots; 63 minimal polynomial; 18, 486, 505 multitransitive group; 556, 570 minimal spectral variety; 96, 123, 510 multivariable; 25 minimal prospectral variety; 448 multivariate; 86 minimal spectrum; 193, 223, 510 multivariate ideal extensions; 197 minimum (or smallest or least) element in a poset; 320 Nagata; 500 minor; 75, 509 Nagata’s principle of idealization; 188-191, models; 87, 128-134, 530-532 514 modelic affine space; 127 Nakayama Lemma; 184; 514 modelic blowup; 132-134, 198, 512, 530 narrative discourse and logical formality; modelic blowup (dominating); 474, 530 168 Modelic Blowup Theorem; 133 natural epimorphism for homogeneous local Modelic Blowup Theorem (its ring to homogeneous function field; Supplementary portion); 474, 498 454-455 modelic proj; 130-132, 198, 530 natural epimorphism for local ring to modelic proj of a semihomogeneous function field; 123-124 domain; 449, 530 natural injection; 170-171, 513 Modelic Proj Theorem; 131 natural isomorphism for function fields; modelic projective space; 130-132, 452-453, 123-124 458-459 natural isomorphism for homogeneous modelic spec; 127, 512, 530 function fields; 454-455 modified discriminant; 22, 431, 433, 505, natural map; 9 528 natural projection; 170-171, 513 module; 7, 89, 504 naturally graded ring; 175 module generation for local rings; 198 near-field; 6 module theoretic power; 170, 513 neighborhoods of a point (first, second, module theoretic restricted power; 170, 513 etc.); 134 modules over PIDs; 402-403 negative of a loset; 314, 574 modules over univariate polynomial rings; Newton; 13 373-401 Newton’s Symmetric Function Theorem; modulized Hilbert degree; 333, 525 13, 537 modulized Hilbert function; 333, 525 Newton’s Theorem; 36, 64, 507 modulized Hilbert polynomial; 333, 525 nilpotent; 90 modulized Hilbert subdegree; 333, 525 nilpotent set; 181 modulized Hilbert transcendence; 333, 525 nilpotents modulo an ideal; 181 monic (polynomial); 8, 15, 65, 135, 508 nilradical; 219-220, 517 monoid; 5, 503 Nilradical Theorem; 220 Monoidal Theorem (First to Twelfth); nodal and cuspidal cubics; 133-134 477-481 node; 60 monoidal transform; 476-477, 530-531 Noether (Emmy); 86, 207, 500 monoidal transform along a valuation; Noether (Max); 82, 134 476-477, 530-531 Noether Normalization Theorem; 208, 516 monoidal transform centered at; 477 Noether Normalization Theorem and monoidal transformation; 466, 476 homogeneous normalization; 337-343, monomial ideals; 140-142 526 INDEX 733 noetherian conditions; 94, 183 ⇔: if and only if; 3, 501 noetherian model; 130, 532 ⇐: implied by; 3, 501 noetherian module; 94, 108-109, 510 ⇒: implies; 3, 501 noetherian ring; 72, 86-89, 108-110, 509 ∼: equivalence relation; 31 noetherian: ACC ⇔ MXC ⇔ NNC; 94 / ∼: quotient by equivalence relation; 31 nonarchimedean valuations; 136 ∼R: R-equivalent; 364, 527 noncancellative quasiordered monoids; 499 ∼[R]: homogeneously R-equivalent; 366 nonconstant polynomial; 8-9 ∼H : equivalent or completable relative to a nonempty lower segment; 314 subgroup H of the general linear nongoing down for nondomains; 487 group; 406 nongoing down for nonnormal domains; 486 σ ∼ σ0: equivalent homomorphisms or (in nongoing down for nonzerodivisors; 487 greater detail) autoequivalent; 373 nonmaximal coprincipal prime ideal; 316 ≈: isomorphism; 126 . nonnegative portion; 307 =: essentially equal; 315, 523 nonnegatively graded ring; 175 ≤: partial order; 27 nonP (= opposite property); 90, 99 <: (strict) partial order; 27 nonsingular; 58 /: normal subgroup; 4, 502 nonsingular model; 130-132, 532 ≤: subgroup; 4, 502, 532 nonzerodivisor; 90, 99 <: proper subgroup; 4, 502, 532 nonzerodivisor (multiplicative) set; 99 ≥: overgroup; 4 nonzero principal isolated subgroup; 316 >: proper overgroup; 4 norm; 539 <: dominated by; 129-130, 531-532 norm (behaviour under finite algebraic field >: dominates; 129-130, 531-532 extension); 540 &: specializes; 321, 523 norm (properties of); 541 %: generalizes; 321, 523 normal (to); 54 : composition of valuations; 323, 523, 577 normal crossing; 59, 474-476, 532 0 = {0}: zero group or monoid; 8, 504 normal crossings, equimultiple locus and 1 = {1}: identity group or monoid; 4, 8, resolved ideals; 474-476 502 normal domain or ring; 65, 202, 508 0: zero map; 8 normal (field) extension; 533 1: identity map; 3 normal model; 130, 532 0S : zero map S → S; 489 normal ring or domain; 65, 202, 508 1P : identity map P → P ; 373, 489 normal series; 103 0u×v: u × v (u by v) zero matrix; 152 normal subgroup; 4, 502 1n: n × n (n by n) identity matrix; 51 Normal Crossing Lemma; 476 |r|: absolute value; 44-46 Normal Local Ring Lemma; 301, 522 |S|: size of S; 2, 502 Normality Lemma; 560 |S|: cardinal of S; 28, 42 normalization basis; 208, 516 ||S||: ordinal of S; 28, 42-43 normalization; 444, 532 φ|(A,B): restriction of a map φ : S → T to Normalization Theorem and Limitations on (A, B), i.e., to a map A → B where it; 208, 495, 516 A ⊂ S and B ⊂ T with φ(A) ⊂ B; 74, normalize; 549 489, 501 normalizer; 549 φ|A: restriction of a map φ : A → B to A, normalizer of prime power subgroup; 551 i.e., to a map A → φ(A) where A ⊂ S; 74, 489, 501 NOTATION-SYMBOLS ψφ: composition of maps; 2, 501 ∈: element of; 2, 501 τσ: product of permutations; 20 ⊂: subset; 3, 501 αi: power of a homomorphism α : P → P ; $: proper subset; 505 374 ⊃: overset; 3, 501 sσ: scalar product of a homomorphism %: proper overset; 505 σ : P → Q of R-modules by s ∈ R; 374 ∪: union; 3, 501 σ + σ0: sum of homomorphisms σ : P → Q ∩: intersection; 3, 501 and σ0 : P → Q; 374 \: complement; 3, 501 φ−1(U): inverse image; 4, 501 →: map; 2, 501 J−1: inverse module; 258, 520 7→: maps to; 2, 501 R+: underlying additive group; 8, 504 7→: mapping; 404 R×: set of nonzero elements; 8, 504 734 INDEX

× × v : epimorphism K → Gv from α1 ⊕· · ·⊕αn : U1 ⊕· · ·⊕Un → D1 ⊕· · ·⊕Dn: multiplicative group to additive group; direct sum of maps; 171 321 α1 ⊕ · · · ⊕ αn : U → D1 ⊕ · · · ⊕ Dn {x1, . . . , xe}: set of elements; 4, 502 (diagonal direct sum): diagonal direct (γ1, . . . , γm) ∈ S1 × · · · × Sm: with γi ∈ Si sum of maps; 171 Qwo for 1 ≤ i ≤ m; 26, 508 i∈E Wi: well ordered product; 316, 573 S × · · · × S : cartesian product; 26, 508 E Qwo 1 m Wwo = Wi with Wi = W for all m i∈E S = S1 × · · · × Sm: where for 1 ≤ i ≤ m i ∈ E: well ordered power; 316, 573 we have Si = S; 26, 508 Qwo i∈G Wi: well ordered product as a ring; X = X1 × · · · × Xm: direct product of 317 groups; 562 G W = W ((X))G: meromorphic series ring; T wo S : set of all maps or functions T → S; 31 317 T (S )finite: maps of finite support; 31 Qlex i∈E Wi: lexicographic product; 316, 573 (ST ) : maps of well ordered support; wellord W E = W [E] = Qlex W with W = W for 31 lex i∈E i i all i ∈ E: lexicographic power; 317, G[T ]: lexicographic power; 48 573 P V (direct or internal direct sum): i∈I i W d = W [d] = W E if E is the finite set internal direct sum; 169 lex lex {1, . . . , d}: lexicographic power; 34-35, L V (internal): internal direct sum; 169 i∈I i 48, 317, 573 V ⊕ V ⊕ · · · ⊕ V (internal): internal 1 2 n A = (A ): matrix of (i, j)-th entry A ; 51 direct sum; 169 ij ij A∗: adjoint of a matrix; 75 U1 ⊕ U2 ⊕ · · · ⊕ Un: direct sum; 170 0 n A : transpose of a matrix; 75, 406 U = U1 ⊕ U2 ⊕ · · · ⊕ Un with U = Ui for µ∗: adjoint of a homomorphism; 374 1 ≤ i ≤ n: module theoretic direct sum of n copies of U; 170 AX : derivative; 23 AX : partial derivative; 24 W1
···
Wd: lexicographic direct sum; i 317 G+: positive part of ordered abelian group G; 45 (W1 ⊕ W2 ⊕ · · · ⊕ Wn)ν : ν-th homogeneous component of a graded direct sum; 364 G0+,G−,G0−: subsets of ordered abelian qV : backward shift; 364 group G; 45 (q1,... )V : backward shift; 364 G/H: factor group; 4, 502 (q1,... )R → V : canonical homomorphism or Gu: stabilizer of u in G; 553 epimorphism; 365 GU : stabilizer of U in G; 553 −E: negative of a loset; 314, 574 G[U]: elementwise stabilizer of U in G; 553 g ]i∈E ρ(Wi): lexicographic disjoint sum; 574 u : g-image of u ∈ W with g ∈ G under ρ(W1) ]···] ρ(Wd): lexicographic disjoint action G → W ; 553 sum; 574 uH : H-orbit of u ∈ W with H ≤ G under tbi∈F Ti: abstract disjoint union; 573 action G → W ; 553 g −1 ti∈E Si: ordered disjoint union; 573 h : g-conjugate of h, i.e., ghg ; 553 S1 t · · · t Sd: ordered disjoint union; 574 R/I: residue class ring; 6, 503 `: disjoint union; 59, 548 /R: (algebraic, integral, etc.) over R; 10, 65 ` i∈I Wi: partition of a set; 59, 557 x|y: divides or divisible; 14, 505 Q i∈I Ui: cartesian product of sets; 170 [G : H]: index; 4, 502 I Q U = i∈I Ui with U = Ui for all i ∈ I: [L : K]: field degree; 8, 504 set-theoretic I-th power; 170, 513 [G, G]: commutator subgroup; 22 Q i∈I Ui: if the Ui are modules then their [x, y]: commutator of elements; 22 cartesian product is made into a (C : B)V or (C : a)V : module colon; 90 module called the direct product; 170, (C : B)R or (C : a)R: ring colon; 90 513 [U : S]V : bracketed colon; 98 I Q U = i∈I Ui with U = Ui for all i ∈ I: [U :(P1,...,Pn)]V : isolated component (of module theoretic I-th power; 170, 513 a submodule); 98 Q ] i∈I αi: direct product of maps; 170 (J) : relevant portion of an ideal J; L i∈I Ui: direct sum of modules; 170 178-179, 457-458 I L ] U⊕ = i∈I Ui: module theoretic restricted (U) : relevant portion of a submodule U; I-th power; 170, 513 336 E ] W⊕ : restricted power; 316, 573 f : induced isomorphism berween W E : lexicographic restricted power; 573 symmetric groups; 554 L
i∈I αi: direct sum of maps; 170 RS : localization at multiplicative set S; 115 INDEX 735

R → RS : localization map at multiplicative K[[X]]G: generalized power series ring; 35 finite set S; 115 K((X))G : subfield of K((X))G; 572 R ⊂ QR(R): if S ⊂ S (R); 115 S R K((X))newt: subfield of K((X))Q; 36 S ⊂ S (R); called good case; 115 R K[[X]]newt: subring of K[[X]]Q; 36 R : localization at prime ideal P ; 120 P K((X))gnewt: subfield of K((X))Q; 36 R → R : localization map at prime ideal P K[[X]]gnewt: subring of K[[X]]Q; 36 P ; 120 k[U]∗: affine coordinate ring of variety U; RP ⊂ QR(R): if ZR(R) ⊂ P ; 120 122-124 ZR(R) ⊂ P ; called good case; 120 k(U)∗: function field of variety U; 122-124 ] D[P ]: homogeneous localization; 449, 529 k(U) : its alternative function field; AS : generalized homogeneous localization 122-124 at a multiplicative set; 497 k(U)] → k(U)∗: natural isomorphism for (AS )i: i-the component of the generalized function fields; 123-124 ∗ homogeneous localization; 497 Rk(U) : local ring of variety U; 122-124 ∗ ∗ A[P ]: generalized homogeneous localization Rk(U) → k(U) : natural epimorphism for at a prime ideal; 497 local ring to function field; 123-124 VS : localization of a module at k[U]∗: homogeneous coordinate ring of multiplicative set S; 278 variety U; 454-455 V → VS : localization map for a module at k(U)∗: homogeneous function field of multiplicative set S; 278 variety U; 454-455 FS : US → VS : localization map for k(U)]: its alternative homogeneous modules at multiplicative set S; 278 function field; 454-455 VP : localization of a module at prime ideal k(U)] → k(U)∗: natural isomorphism for P ; 278 homogeneous function fields; 454-455 V → V : localization map for a module at ∗ P Rk(U) : homogeneous local ring of variety prime ideal P ; 278 U; 454-455 FP : UP → VP : localization map for ∗ ∗ Rk(U) → k(U) : natural epimorphism for modules at prime ideal P ; 278 homogeneous local ring to WR: ideal generation; 6, 503 homogeneous function field; 454-455 RW : module generation; 7 r(a) = (r1(a), . . . , rn(a)): substitution in a BC: submodule of C; 90 univariate polynomial tuple with a in a aC or {a1, a2 ... }C: submodules of C; 90 field K; 407 R[X]: polynomial ring; 8, 505 r(b) = (r1(b), . . . , rn(b)): substitution in a R[X1,...,Xm]: polynomial ring; 9, 505 univariate polynomial tuple with b in a R[x1, . . . , xm]: ring generation; 9, 505 K-algebra L; 407 A[Y ,...,Yn] : homogeneous polynomials 1 i r(X1,...,Xn−1, a): substitution in a of degree i with coefficients in a subset multivariate polynomial tuple with a A of a ring; 198 in a field K; 407 A[Y1,...,Yn]∞: homogeneous polynomials r(X1,...,Xn−1, b): substitution in a of any degree with coefficients in a multivariate polynomial tuple with b in subset A of a ring; 198 a K-algebra L; 407 W [X1,...,XN ]: polynomial module; 373 (R, S, R0)-transform of J or (J, I): 476-477, α[X1,...,XN ]: polynomial extension of a 530 homomorphism α; 373 R[[X]]: power series ring; 29-30, 506 NOTATION-WORDS R[[X1,...,Xm]]: power series ring; 30, 506 ACC: ascending chain condition; 94, 510 K(X): rational function field; 9, 505 An: alternating group; 5, 21, 503 K(X1,...,Xm): rational function field; 10, annRB: annihilator; 93 505 AppD(F ): approximate root; 49-50 K(x , . . . , x ): field generation; 10, 505 1 m ass V : associator or assassinator; 96, 181 R((X)): meromorphic series ring; 29-30, R Aut(G): automorphisms group; 6-8, 506 502-504 K((X ,...,X )): meromorphic series 1 m avt ( N ): affine variety set; 122 field; 31, 506 k Aκ  1+XYX2  G C: Cohn’s matrix ; 422 K : set of all maps or functions G → K; −Y 2 1−XY 31, 35, 506 CG(H): centralizer of H in G; 549 K((X))G: generalized meromorphic series C(E): core of E; 315, 523, 572 field; 35, 506 cd: common divisor; 15 736 INDEX ch(R): characteristic; 18, 505 DVR: (Real) Discrete Valuation Ring; 298, cm: common multiple; 18 302, 522 cmat(A): characteristic matrix of a matrix (ei = (0,..., 0, 1, 0,..., 0))1≤i≤n: canonical A; 540 basis of (q1,...,qn)R; 365 CM: Cohen-Macaulay; 235, 519 (ei = (0,..., 0, 1, 0,..., 0))1≤i≤n: canonical CMR: Cohen-Macaulay Ring; 298, 302 basis of Rn; 374 cond(R, Re): conductor; 260, 520 (ei = (0,..., 0, 1, 0,..., 0))1≤i≤n: canonical coprin(R)∗: set of all nonmaximal basis of Rn = MT(1 × n, R); 406 0 coprincipal prime ideals in R; 316 (ei)1≤i≤n: canonical basis of MT(n × 1,R); cpol(A): characteristic polynomial of a 406 matrix A; 540 E(R): abbreviation for SE(n, R); 407 u 1
cprojAJ: complementary prospectral E(u): the 2 × 2 matrix −1 0 ; 422 variety; 448 E(2,R): group generated by matrices E(u); crk(A): column rank; 75 422 CR(n): condition characterizing Chinese ED: euclidean domain; 11, 16, 505 Remainder Theorem; 307 emdim(R): embedding dimension; 127, 512 CRT: Chinese Remainder Theorem; 307 estabG(U): elementwise stabilizer; 547 T CT: Classification Theorem; 4-5, 24 (S )finite: maps of finite support; 31 dA(X) fixL∗ (H): fixed field; 13; 505 dX : X-derivative A(X); 9 ∂A(X1,...,Xm) fixW (H) or fix(H): fixed point set; 548 ∂X : partial derivative of the j Gv: value group; 34, 506 function A(X1,...,Xm) relative to the G(R): abbreviation for GL(n, R); 407 variable Xj ; 10 G(R): set of all nonzero fractional R-ideals; ∂(f1,...,fm) : jacobian matrix; 545 ∂(Y1,...,Ym) 306 (D1) and (D2): distributive laws for ideals; Gal(f, K): Galois group; 13, 505 307 Gal(L∗,K): Galois group; 12, 505 D/x: dehomogenization map at x; 449 gcd: greatest common divisor; 15 D/x(f) = F : means F is the GCD: Greatest Common Divisor; 15 dehomogenization of f while f is a gdim(R): global dimension; 262, 521 homogenization of F ; 449 GFI: domain with Group Factorization of ∗ D/x: epimorphism induced by the Ideals; 306, 522 dehomogenization map at x; 450 K((X))gnewt: subfield of K((X))Q; 36 D(n, R): group of all diagonal matrices; 422 K[[X]]gnewt: subring of K[[X]]Q; 36 DE(n, R): group generated by D(n, R) and GE(n, R): general elementary group; 405, SE(n, R); 422 528 DE : closed Dedekind map; 315, 523 GE(n, (K,R)): general elementary group 0 (in a more general situation); 420 DE : open Dedekind map; 315, 523 DF,E : Dedekind map of a pair; 315, 523 GF(p): Galois field; 3, 18, 504 dc: Dedekind completion Edc; 315 GF(q): Galois field; 3, 507 DCC: descending chain condition; 107, 511 GL(n, [R,I]): certain subgroup of DD: Dedekind Domain; 306, 522 GL(n, R); 420 defo: degree form; 78 GL(n, R): general linear group; 51, 528 det(A): determinant of a matrix; 51 GL(n, q): general linear group; 52 det(µ): determinant of a homomorphism; glb: greatest lower bound; 320 374 gnbRV or gnbRI: generating number; 235, DFI: Domain with Factorization of Ideals; 519 306, 522 grad(R,I): associated graded ring; 228, 517 grad(R,I) : n-th graded component; 229 dimK V : dimension of V over K; 8, 504 n n dim(E): dimension of a model; 130, 532 grad(R,I)(h): n-th graded image; 229 n dim(R): dimension of a ring; 105, 511 grad(R,I): n-th graded map; 229 n dim(U): dimension of a variety; 122-124 grad(R,I)(φ): n-th induced homomorphism; DiscY (f): discriminant; 83-86, 510 229 ∗ DiscY (f): modified discriminant; 22, 431, grad(R,I)(φ): induced homomorphism; 229 n n 433, 505, 528 grad(R) or grad(R)n or gradR(h) or gradR n dom(φ): domain of φ where for any map φ or gradR(φ) or gradR(φ): the above we have φ : dom(φ) → ran(φ); 489, 501 six lines when I = M(R) with dptRJ: depth of an ideal; 106, 124, 511 quasilocal R; 230, 518 INDEX 737

Qlex grad(R,I,J): generalized associated graded i∈E Wi: lexicographic product; 316 ring; 490 E [E] Qlex Wlex = W = i∈E Wi with Wi = W for grad(R,I,J)n: n-th graded component; 490 all i ∈ E: lexicographic power; 317 gradn (h): n-th graded image; 490 d [d] E (R,I,J) Wlex = W = Wlex if E is the finite set n grad(R,I,J): n-th graded map; 490 {1, . . . , d}: lexicographic power; 34-35, n 48, 317 grad(R,I,J)(φ) and grad(R,I,J)(φ): n-th induced homomorphism and induced lo: linear order; 27 homomorphism respectively; 490 loset: linearly ordered set; 27, 506 GST: Gorenstein; 252, 520 lub: lowest upper bound; 320 H∞ = HN+1: hyperplane at infinity; 460 M(R): maximal ideal of a quasilocal ring; H(R): set of all nonzero ideals in R; 306 71, 508 H(µ): matrix of a homomorphism; 374 max(R): the maximum or largest or hdimRV : homogeneous dimension; 365, 527 greatest element of a poset R; 320 HL(n, k): homothety group; 52 mcprojAJ = mproj(A) \ mvprojAJ: hnbRV or hnbRI: homogeneous generating maximal complementary prospectral number; 364 variety; 448 htRJ: height of an ideal; 106, 124, 511 MDT: monoidal transformation; 467 IR(α, β): invariance-set; 374 min(R): the minimum or smallest or least N iavtk(Aκ ): irreducible affine variety set; element of a poset R; 28, 320 122 mspec(R): maximal spectrum; 95, 123, 510 id(R): ideal set; 313, 510 mproj(R): maximal prospectrum; 448 ijsvt(R): set of all irreducible members of mpsvt(A): maximal prospectral variety set; jsvt(R); 219 448 im(φ): image of φ; 4, 501 msvt(A): maximal spectral variety set; 123, impsvt(A): set of all irreducible member of 510 mpsvt(A); 448 mvspecRJ: maximal spectral variety; 95, imsvt(A): set of all irreducible member of 123 msvt(A); 123 mvprojAJ: maximal prospectral variety; inf: infimum; 320 448 info: initial form; 77 MT(m × n, R): set of matrices over R; 51 int: intersection multiplicity; 55 mult: multiplicity; 54 iprojAU: prospectral ideal; 448 MNC: minimal condition; 107, 511 ipsvt(A): set of all irreducible members of MXC: maximal condition; 94, 510 psvt(A); 448 NG(H): Normalizer of H in G; 549 N ipvtk(Pκ ): irreducible projective variety NL/K (z): norm of an element z relative to set; 454 a field extension L/K; 539 ispecRW : spectral ideal; 96, 123 NR(R/I): nilpotents modulo ideal I: 181 isvt(A): set of all irreducible members of nassRV : minimal associator or (the svt(A); 123 colorful) minimal assassinator; 96, 181 J(f1,...,fm) : jacobian; 546 ncprojAJ = nproj(A) \ nvproj J: minimal J(Y1,...,Ym) A complementary prospectral variety; JL/K (r): invariant ideal; 407 jspec(R): Jacobson spectrum; 219, 517 448 jsvt(R): Jacobson spectral variety set; 219 NC: normal crossing; 59 jrad(R): Jacobson radical of a ring R; K((X))newt: subfield of K((X))Q; 36 183-184, 219, 514 K[[X]]newt: subring of K[[X]]Q; 36 jvspecRJ: Jacobson spectral variety; 219 nid(R): nonunit ideal set; 313, 510 ker(φ): kernel of φ; 6, 502 NNC: noetherian condition; 94, 510 krdimAK: kroneckerian dimension of K nrad(R): nilradical; 219, 517 over A; 444 nspec(R): minimal spectrum; 193, 223, 510 lcm: least common multiple; 18 nvspecRJ: minimal spectral variety; 96, LCM: Least Common Multiple; 18 181, 510 LX : left regular representation; 554 nvprojAJ: minimal prospectral variety; 448 ledi(R,I)(J): leading ideal; 229, 517 orbG(u): orbit; 547 lediR(J) or lefoR(h): either the above line orbsetH (W ): orbit set; 548 or the following line when I = M(R) ord: order of meromorphic series; 29-36, with quasilocal R; 230, 518 506 lefo(R,I)(h): leading form; 229, 517 ord: order of power series; 29-31, 54, 506 738 INDEX

∗ ∗ ordRh: order relative to a quasilocal ring; Rk(U) → k(U) : natural epimorphism for 71 homogeneous local ring to ord(R,I)h: order relative to an ideal; 78, homogeneous function field; 454-455 509 Rn and Sn: Serre conditions; 298, 522 pa: arithmetic genus; 334 Rv: valuation ring; 34, 67, 157-160, 506 p-group: group of p-power order; 550 radRI: radical of ideal I; 90 p-power: power of prime p; 550 radV U: radical of submodule U; 90 p-subgroup: subgroup of p-power order; 550 ran(φ): range of φ where for any map φ we P (G): set of all nonzero principal isolated have φ : dom(φ) → ran(φ); 489, 501 subgroups of G; 316, 572 rd(R): radical ideal set; 96, 123; 510 reg V :(R,J)-regularity of module V , P (R): set of all nonzero prime ideals in R; (R,J) 306 with (R,J) replaced by by R when R is local; 235, 519 pdim V : projective dimension of a R reg(R): regularity of local ring R; 235, 519 module; 262, 521 Res (f, g): resultant; 83-85, 510 PFI: domain with the property Prime Y Resmat (f, g): resultant matrix; 83-85, 510 Factorization of Ideals; 307 Y restrdeg R0: residual transcendence PGL(n, k): projective general linear group; R degree; 210, 464 52 rk(A): rank; 75 PGL(n, q): projective general linear group; RNR: Reduced Normal Ring; 298, 304 52 rrk(A): row rank; 75 PIC: primary ideal conditions; 98 Sn: symmetric group; 2, 21, 502 PID: principal ideal domain; 11, 16, 505 Sn and Rn: Serre conditions; 298, 522 PIR: principal ideal rings; 293-297, 521 0 00 000 0 00 0 00 (Sn), (Sn), (Sn ), (Sen), (Sen), (Sn), (Sn): PMC: primary module conditions; 98 alternative Serre conditions together po: partial order; 27 with their variations; 303, 493 ∗ poset: partially ordered set; 27, 506 (Sn): Serre quasicondition; 303, 494 prd(A): homogeneous radical ideal set; 448 SR(V ): nonzerodivisor set; 99 prodim(A): projective dimension of a S(G): set of all nonzero isolated subgroups subintegrally graded ring; 448 of G; 313, 572 prodim(U): projective dimension of a S(G): set of all isolated subgroups of G; variety; 449, 454 313 proj(A): projective spectrum; 447, 529 Sb(G): set of all segments of G; 313 0 PSL(n, k): projective special linear group; S (G): set of all isolated positive upper 52 segments of G; 313 PSL(n, q): projective special linear group; Sb0(G): set of all positive upper segments of 51-52 G; 313 psvt(A): prospectral variety set; 448 sc: segment completion Esc; 315, 523 N SE(n, [R,I]): certain subgroup of SE(n, R); pvtk(Pκ ): projective variety set; 453 Q(R): abbreviation for QSE(n, R); 407 420 Q(R): set of all nonzero maximal ideals in SE(n, R): special elementary group; 405, R; 306 528 SF(f, K): splitting field; 12, 505, 534 QDT: quadratic transformation; 465 sgn(σ): signature or sign; 22, 51, 503 QF(E): quotient field; 31, 506 SL([2, n],R): image of SL(2,R) in SL(n, R) QR(R): total quotient ring; 114-115 by canonical monomorphism; 406 QSE(n, R): quasielementary group; 406 SL([m, n],R): image of SL(m, R) in QSE([m, n],R): quasielementary group (of SL(n, R) by canonical monomorphism; general dimensions); 420 420 QSEP(n, R): quasielementary pregroup; SL(n, R): special linear group; 51, 528 406 SL(n, q): special linear group; 52 r(G): rational rank; 47 soc(R): socle; 252, 520 RX : right regular representation; 554 socz(R): socle-size; 252, 520 ∗ Rk(U) : local ring of variety U; 122-124 spec(R): spectrum; 95, 123, 510 ∗ ∗ ∗ Rk(U) → k(U) : natural epimorphism for spec(R) : nonmaximal spectrum; 313 local ring to function field; 123-124 SPIR: special principal ideal rings; 293-297, ∗ Rk(U) : homogeneous local ring of variety 521 U; 454-455 stabG(U): stabilizer; 547 INDEX 739

0 sup: supremum; 320 trdegRR : transcendence degree of the 0 Subp(G): set of all p-power subgroups; 550 quotient field of R over the quotient svt(R): spectral variety set; 96, 123 field of R; 209 0 0 supp(y): support of an element in a direct trdegR/P R /P : transcendence degree of sum; 175 the quotient field of R0/P 0 over the supp(φ): support of a function or map in quotient field of R/P ; 209 general; 140, 170, 316, 513 two-transitive: t-transitive with t = 2; 556, suppRV : support of a module; 165, 281 570 Supp(A): support of a meromorphic series U(R): group of units in R; 11, 33, 51, 505 or power series or polynomial (or of a Un(R): set of all unimodular n-tuples (or function or map leading to these); row vectors); 406 0 29-36; 506 Un(R) : set of all unimodular column Sylp(G): set of all p-Sylow subgroups; 551 vectors; 406 Sym(S): symmetric group; 2, 502 UFD: unique factorization domain; 4, 11, n 16, 67-69, 505 syzRV : n-th R-syzygies of V ; 364, 527 n UFI: domain with Unique Factorization of syz[R]V : n-th homogeneous R-syzygies of V ; 366 Ideals; 306, 522 vproj J: prospectral variety; 448 NL/K (z): norm of an element z relative to A a field extension L/K; 539 vspecRJ: spectral variety; 95, 123 t-transitive: permutation group which WDT: Weierstrass Division Theorem; sends any t points to any t points; 556, 70-74, 508 T 570 (S )wellord: maps of well ordered sup.; 31 t-antitransitive: permutation group in wideg(f): Weierstrass degree; 70, 508 which only the identity fixes t points; wo: well order; 28 Qwo 556, 570 i∈E Wi: well ordered product; 316 E Qwo (t, τ) group: permutation group which is Wwo = i∈E Wi with Wi = W for all t-transitive and τ-antitransitive; 556, i ∈ E: well ordered power; 316 Qwo 570 i∈G Wi: well ordered product as a ring; (T ), (T 0), (T 00), (T 000), (T ∗): reduced normal 317 G ring conditions; 304 Wwo = W ((X))G: meromorphic series ring; (T ∗): reduced normality quasicondition; 317 494 woset: well ordered set; 28, 506 T (n, σ) = (T (n, σ)ij ): permutation matrix; WPT: Weierstrass Preparation Theorem; 430, 528 70, 72, 508 Tij (n, λ): transvection matrix; 404 Z(G): center of group G; 549 Tij (n, R): set of all transvection matrices; Zr: cyclic group of order r; 4, 502 404 ZR(V ): zerodivisor set; 99 Tii(n, λ): dilatation matrix; 404 ZR(R/I): zerodivisors modulo ideal I: 181 Tii(n, R): set of all dilatation matrices; 404 ZR(V/U): zerodivisors modulo submodule ∗ Tii(n, R): set of all restricted dilatation U: 181 matrices; 404 N Ak : affine space; 56-57, 122 T (R): set of all invertible members of N N Abκ = Aκ,N+1: affine portion; 460 G(R); 306 N σ (A ) : spectral affine space; 126 T ( N ),T ( N ),T 0( N ): temporary k b Aκ Pκ Pκ ( N )µσ: maximal spectral affine space; 126 notation for certain sets of varieties; Ak ( N )ρσ: rational spectral affine space; 126 460 Ak N δ 0 ( ) : modelic affine space; 127 Tb(Bb),T (B),T (B): temporary notation for Ak N µδ certain sets of ideals; 461 (Ak ) : minimal modelic affine space; 127 0 ( N )ρδ: rational modelic affine space; 127 T (A),T (AS ),TS (A), Tb(A[S]): again Ak temporary notation for certain sets of C: complex numbers; 3, 44-50, 502 ideals; 497 Ik(U): ideal of U; 122 tassRV : tight associator or (the colorful) Jk(U): homogeneous ideal of U; 453 tight assassinator; 98, 181 N+: positive integers; 5, 502 transform: (R, S, R0)-transform of an ideal N: nonnegative integers; 5, 502 P or a pair of ideals (J, I): 476-477, 530 N⊕ (with special member φp): free additive trdegK L: transcendence degree; 11, 124, monoid on P ; 307 N 505 Pk : projective space; 56-57 740 INDEX

N N+1 N N+1 N Pk = ∪i=1 (P \ Hi) = ∪i=1 Ak,i: αP,n: ring homomorphism of MT(n × n, R) decomposition of projective space into to MT(n × n, RP ) induced by the ring affine spaces as complements of homomorphism ψP ; 420 hyperplanes out of which H∞ = HN+1 β: map induced by intersection map is called the hyperplane at infinity and J 7→ J ∩ A[S]; 498 N N its complement Abκ = Aκ,N+1 is called βP,n: homomorphism of SL(n, R)/SE(n, R) the affine portion; 57, 132, 451, 460 to SL(n, RP )/SE(n, RP ) induced by N Pκ = set of all [u] with u = (u1, . . . , uN+1) the ring homomorphism αP,n; 420 N+1 varying over κ \{(0,..., 0)} where βv: order isomorphism Γv → Gv; 321 [u] is the equivalence class (under γ: dehomogenization map in its operational proportionality) containing u: incarnation; 459 projective space as proportional tuples; γ∗: map induced by above 453, 460 dehomogenization map γ; 498 N σ (Pk ) : spectral projective space; 458-459 γR: divisibility valuation induced by N µσ valuation ring R; 157, 321, 523 (Pk ) : maximal spectral projective space; 458-459 γv: divisibility valuation induced by N ρσ valuation ring R ; 321, 523 (Pk ) : rational spectral projective space; v 458-459 Γ(R): divisibility group of R; 157, 321, 523 N δ (Pk ) : modelic projective space; 132, Γv: divisibility group of Rv; 321, 523 452-453, 458-459 Γi: dehomogenization map; 452 N µδ (Pk ) : minimal modelic projective space; Γ: dehomogenization map; 459 132, 452-453, 458-459 Γ:b monomorphism related to the above N ρδ (Pk ) : rational modelic projective space; dehomogenization map; 459 132, 452-453, 458-459 δ: operational homogenization or minimal Q: rational numbers; 3, 502 homogenization map which is sort of Q+: positive rational numbers; 44 inverse of the operational Q0+, Q−, Q0−: subsets of Q; 44 dehomogenization map γ given above; Qn and Qn,p: subsets of Q; 35-36 461 [d] ∗ Q : lexicographically ordered tuples; δ : map induced by above homogenization 34-35 map δ; 498 R: real numbers; 3, 44-50, 502 δij : Kronecker’s delta; 51 R+: positive real numbers; 45 ∆v: residue field Rv/M(Rv); 321 R0+, R−, R0−: subsets of R; 45 (U): see ι given below; 460 [d] x R : lexicographically ordered tuples; 34-35 Θ(R,I) : k[(Xl))l∈L] → grad(R,I) with ring Vκ(J): affine variety defined by a set of k = R/I: graded ring homomorphism polynomials J; 122 induced by (R,I) and a family of Vκ(f, g, . . . ): affine variety defined by generators of I; 229-230 x polynomials f, g, . . . ; 122 ΘR : k[(Xl))l∈L] → grad(R) with field Wκ(J): projective variety defined by a set k = R/M(R): graded ring of polynomials J; 453 homomorphism induced by quasilocal Wκ(f, g, . . . ): projective variety defined by ring R and a family of generators of homogeneous polynomials f, g, . . . ; 453 M(R); 229-230 N N Z: integers; 5, 502 ι: natural bijection Abκ → Aκ with image [d] N Z : lexicographically ordered tuples; 34-35 (U) = ι(U ∩ Abκ ) giving intersection P Z⊕: free additive additive abelian group on π(Ub); 460 P ; 307 µ: certain map used for discussing relations `R(V ): length of a module; 103, 511 between ideals and homogeneous U(G): set of all elements in monoid G ideals; 498 having an inverse; 306 µ(a, b): Mennike symbol; 419 P(T ): power set; 28, 37, 506 ∗ ∗ b µ , µe : certain maps used for discussing Pb×(T ): restricted power set; 28, 37, 506 relations between ideals and α, αe: maps induced by localization map homogeneous ideals; 498 ∗ φ : A → AS ; 497, 498 ν, ν : certain maps used for discussing αa,n: ring homomorphisms of relations between ideals and MT(n × n, R) to MT(n × n, Ra) homogeneous ideals; 498 induced by the ring homomorphism π: natural homomorphism of grad(R,I) ψa; 420 into grad(R,I,J); 490 INDEX 741

π(Ub): see ι given above; 460 V(A): modelic spec; 127, 530 πv: residue class epimorphism Rv → ∆v; W(A;(xl)j∈Λ): modelic proj; 131, 530 321 W(A; x1, . . . , xn): modelic proj; 131, 530 × × πv : epimorphism U(Rv) → ∆v ; 321 W(A, P ): modelic blowup; 133, 530 ∆ πv,w: surjective map Rv → πw(Rv); 321 W(A, P ) : dominating modelic blowup as × πv,w: surjective map U(Rv) → πw(U(Rv)); a subset of W(A, P ); 474, 530 321 W(D): modelic proj of a semihomogeneous ρ(G): real rank; 47-48 domain; 449, 530 ρ(Rv): real rank; 314 ρ(v): real rank; 314, 523 Nullstellensatz (including its Spectral and ρ0(G): principal rank; 576 Projective versions); 126, 218, 222-223, 0 ρ (Rv): principal rank; 576 456, 516 ρ0(v): principal rank; 576 null ring; 5, 503 τf (g): Tschirnhausen relative to; 49-50 φP : K → KP : canonical (local) Obvious Lemma; 556 homomorphism; 420 odd permutation; 5, 21, 502 ψP : R = K[X] → RP = KP [X]: canonical one dimensional CM local rings; 259 homomorphism induced by φP ; 420 one dimensional special GST rings; 260 φa : K → KS(a): canonical (local) One Dimensional GST Ring homomorphism; 420 Characterization Theorem; 258, 520 ψa : R = K[X] → Ra = KS(a)[X]: operational dehomogenization map; 460 canonical homomorphism induced by operational homogenization or minimal φP ; 420 homogenization map; 461 ω: first infinite ordinal; 28 orbit; 547 Ω(V ): positive portion of a graded module; orbit set (= orbset); 548 336, 526 Orbit Counting Lemma; 548 Ω(S): positive portion of a graded ring; 178 Orbit-Stabilizer Lemma; 547 E(R,J): equimultiple locus; 475, 530 Ord Valuation Lemma; 234; 518 Ei(R,J): i-dimensional members of Ord Valuation Lemma, Another aspect; E(R,J); 475, 530 251, 518 g: Hilbert degree; 333, 524 Ord Valuation Theorem; 128 gi: Hilbert subdegree; 333, 524 Order Lemma; 557 gµ: modulized Hilbert degree; 333, 524 order isomorphism; 47 µ gi : modulized Hilbert subdegree; 333, 524 order of a group; 4, 502 bh: Hilbert function; 332, 524 order of a meromorphic series; 29-36, 506 h: Hilbert polynomial; 333, 524 order of a minor; 75 bhµ: modulized Hilbert function; 332, 524 order of a power series; 29-36, 506 K(D): homogeneous quotient field of D; order of a square submatrix; 75 449, 529 order relative to a quasilocal ring; 71, 128 N(R, K): normalization of a quasilocal order relative to an ideal; 78, 509 subring; 444, 532 ordered abelian group; 34, 506 N(E, K): normalization of a set of ordered completion; 46 quasilocal subrings; 444, 532 ordered disjoint union; 573 hµ: modulized Hilbert polynomial; 333, 524 ordered domain; 45 R(K): Riemann-Zariski space; 129, 531 ordered field; 45 R(K/A): Riemann-Zariski space (for a ordered (additive abelian) monoid; 174 subring); 129, 531 ordered set; 27, 506 R0(K): quasitotal Riemann-Zariski space; order-type; 47 129, 531 orderwise complete; 46 R0(K/A): quasitotal Riemann-Zariski orderwise completion; 46 space (for a subring); 129, 531 ordinal; 28, 42-43, 47 R00(K): total Riemann-Zariski space; 129, orthogonal group; 52 531 orthogonal idempotents; 288 R00(K/A): total Riemann-Zariski space (for orthogonal to; 288 a subring); 129, 531 overgroup; 4 t: Hilbert transcendence; 333, 524 overfield, overring, etc.; 5, 503 tµ: modulized Hilbert transcendence; 333, overnormal domain; 66, 508 524 overposet; 320 742 INDEX overset; 3, 501 poset; 27, 506 positive portion of a graded module; 336, parabola; 53 526 paraboloid; 53 positive portion of a graded ring; 178 parameter; 54 positive upper segment; 313 parameter ideal; 252, 520 Possible Real Ranks; 576 parameters for a local ring; 252, 520 power of a homomorphism; 374 parametrically; 54 power series; 29-33, 506 parametrize; 125 power series ring; 29-33, 506 parenthetical colon; 98 power set; 28, 37, 506 Parenthetical Colon Lemma; 323, 524 predecessor (immediate); 315, 523, 572 parity; 5, 22, 503 prefree resolution; 262 Parshall; 82 preprojective resolution; 262, 364, 527 partial order; 27 premodel; 130, 531 partially ordered set; 27, 506 primary decomposition; 86, 89, 95 partition; 4, 31, 59, 501 primary decomposition summary; 111-114 Pdim Lemma (Supplemental); 327 primary decomposition uniqueness; 98-99 perfect field; 431, 529 primary ideal colons; 91 Perfect Field Theorem (Basic); 435 primary ideal for a prime conditions; 91 perfect group; 571 primary ideal or submodule; 90, 510 permutation; 2, 20, 502 primary ideals intersections; 91, 187 permutation isomorphism; 554 primary nonirreducible ideal; 95 permutation matrix; 430, 528, 570 primary not power of prime; 91 perpendicular; 54 primary submodule colons; 93-94 PID Lemma; 402 primary submodule conditions; 93-94, 142, PID Theorem; 402 163-164 PID Unimodularity Lemma; 407 primary submodule intersections; 93-94 PIDs (modules over); 402-403 Primary Ideal Blowup Theorem; 463 plane curve; 50, 53, 60, 84, 124 Prime Avoidance Corollary; 285 plane curves with their tangents counted in Prime Avoidance Lemma; 285, 521 terms of tangential multiplicities; 466 prime ideal; 6, 14, 503 plane curves with their total transforms prime ideal conditions; 182 consisting of proper transforms and prime ideal generalities; 106-107 powers of the exceptional line; 466 prime ideal (nonmaximal coprincipal); 316 Pl¨ucker; 82 prime ideals avoidance; 107, 285 po; 27 prime ideals intersect in rad zero; 106 points at finite distance; 57, 460 prime power group; 550 points at infinity; 55-57, 460 prime power not primary; 91 point-set is zero dimensional; 124 prime power orbit; 551 polar; 52-58 prime power subgroup; 550 pole; 52 prime power subgroup (existence of); 553 polynomial; 8-11, 140, 505, 543 Prime Ideal Blowup Theorem; 469 polynomial automorphism 76-78, 82 primitive element; 125, 431, 528 polynomial extension of a homomorphism; primitive (permutation) group; 558 357, 373 primitive root of one (or unity); 434 polynomial in ω; 28 Primitive Element Theorem; 125 polynomial module; 373 Primitive Element Theorem (Basic polynomial ring; 8-11, 505 version); 436 polynomial ring in a (not necessarily finite) Primitive Element Theorem (Projective family of indeterminates; 140, 543 version); 455 polynomial rings (ideals or maximal ideals Primitive Element Theorem (Supplemented in and regularity of the localization version); 443 of); 355-360 Primitive Root of Unity (or One); 434 Polynomials in a Family (Not Primitivity Lemma; 558 NECESSARILY FINITE) of Princeton Book; 167 Indeterminates; 140, 543 principal component; 101 portion at finite distance (or affine principal ideal; 6, 503 portion); 57, 460 principal ideal domain; 11, 505 INDEX 743 principal ideal rings; 293-297, 521 properly dominates; 130, 531 PID; 11, 505 properties of PIDs, PIRs, SPIRs, and PIR; 293-297, 521 UFDs; 293-297, 521 Principal Ideal Theorem; 193, 485 prospectral ideal; 448 Principal Ideal Theorem with its Corollary, prospectral variety; 448 194 prospectral variety set; 448 principal isolated subgroup; 316, 572 pseudonormal crossing; 474-476, 532 principal rank; 576 purely inseparable element; 431, 529 principle of idealization of Nagata; 188-191, purely inseparable extension; 431, 529 514 Purely Inseparable Extensions (Derivations processes on roots; 63 of); 545 products of ideals or modules; 89-90 product spaces (diagonals and intersections quadric; 52-53 in); 362-363 quadratic and monoidal transformations Product Theorem (Codimension); 350, 527 (for resolving singularities); 476-483 projection; 84 quartic discriminant; 86 projective decomposition of ideals and quadratic equation, 1 varieties; 457 quadratic transform; 477, 531 projective dimension (pdim) of a module; quadratic transformation; 465, 476 262, 521 quasielementary group; 407 projective dimension (prodim) of a quasielementary group (of more general subintegrally graded ring; 448 dimensions); 420 projective dimension of a variety; 449, 454 Quasielementary Group Lemma; 408 projective general linear group; 52 quasielementary pregroup; 407 projective model; 131-132, 531 quasilocal ring; 71, 508, 518 projective module; 262, 371, 521 quasinormal crossing; 474-476, 532 projective modules over polynomial rings quasiordered abelian monoid; 499 (or over PIDs); 371-430, 528 quasiprimary decomposition; 99-103 projective normalization; 443, 529 quasiprimary decomposition (summarized); Projective Dimension Theorem; 455 113-114 Projective Inclusion Relations Theorem; quasiprimary ideal or module; 99, 511 455 quasisemilocal ring; 307 Projective Normalization Theorem; 446, quasispecial ED; 11, 505 529 quasispecial subset; 11, 505 Projective Nullstellensatz; 456 Quillen; 371 Projective Primitive Element Theorem; 455 Quillen-Suslin Theorem; 371 projective resolution; 262, 521 quotient field; 31 Projective Resolution Lemma; 263 quotient rule; 9-10 projective space; 55-58 quotients of ideals or modules; 89-90 projective space as proportional tuples; 57, 131, 453 radical; 89-90, 510 projective space is decomposed into affine radical description; 205, 486 spaces as complements of hyperplanes; radical ideal; 96, 123, 510 57, 132, 451 radical ideal set; 96, 123, 510 projective special linear group; 51 Radical Description Lemma; 205, 486 projective spectrum; 447, 529 radicals of ideals or modules; 89-90, 510 Projective Theorems; 497 ramified covering; 490 projective varieties; 453-456 range of a map (see map); 489, 501 projective variety set; 453 rank; 75, 509 proper containment; 253, 505 rational completion; 46 proper domination; 130 rational function; 9-10, 505 proper normal subgroup; 4, 502 rational function field; 9-10, 505 proper overideal; 253 rational function ring; 358 proper subideal; 253 rational modelic affine space; 127 proper subset; 505 rational numbers; 3, 502 proper subgroup; 502 rational points; 127, 459 proper subvariety; 122, 123, 448, 454 rational rank; 47 proper transform; 133-134, 466-468 rational spectral affine space; 126 744 INDEX rational spectral projective space; 458-459 relative algebraic closure; 19 rationalization of surds; 23 relative independence characterization in a real completion; 46 Lemma dealing with conditions sharp, real discrete valuation; 298, 522 dagger-prime, and double-dagger; 199 real numbers; 3, 44-50, 502 relative independence of elements over a real valuation; 298 ring by itself or over a ring with an real valuations and their characterization; assigned ideal; 198-202, 515 298, 302, 312-314, 523 relative independence of parameters; Real Valuation Characterization; 298 198-202, 252, 485, 515, 520 real rank (of ordered abelian group); 47-48 Relative Independence Theorem; 201, 515 real rank (of valuation); 312-320, 523 relative independence via a Lemma about real rank (of valuation ring); 312-320, 523 univariate polynomials and a Blowup Real Ranks (Characterization Theorem Lemma; 199-201 for); 319, 523 relatively prime; 16 Real Ranks (Possible or Impossible); 576 relevant ideal; 178 reciprocate the roots; 63 relevant submodule; 336, 526 reduced normality quasicondition; 494 relevant portion of an ideal; 178-179, reduced (rings); 286, 521 457-458 Reduced Normal Rings (Conditions for); relevant portion of a submodule; 336-337 304 residual dimension and gnb over quasilocal reducible variety; 86-87, 122-123, 448, 454 rings; 275-276 refinement of a normal series; 103-105 residual properties and coefficient sets; 464 reflexive relation; 31 residual transcendence degree; 209, 464, regular local ring; 87, 128, 512 516 regular local rings and their basic residually algebraic; 464 properties; 483, 498 residually algebraically dependent; 464 regular local rings and their properties; 87, residually algebraically independent; 464 128, 231-234, 244-248, 251, 252, 258, residually finite algebraic; 464 276-277, 286, 299, 323-329, 483, 498, residually finite purely inseparable; 464 512 residually finite separably algebraic; 464 regular noetherian ring; 252, 520 residually purely inseparable; 464 regular parameters; 252, 520 residually rational; 464 regular permutation group; 554, 556 residually separably algebraic; 464 regular polynomial; 78, 208 regular sequence; 235, 519 residue class; 6, 503 Regular Sequence Lemma; 237 residue class epimorphism (it is the obvious regular sequence, maximal or in an ideal; epimorphism from a group it a factor 235, 519 group, or ring to a residue class ring); regularity and inverse of the maximal ideal; 74 492 residue class map (same as above); 91 regularity (of a module or local ring); 235, residue class ring; 6, 503 519 Resolution Book; 167, 474, 476 regularity of localization of polynomial resolution (fame); 143 ring; 358 resolved (ideal in a ring); 475, 532 regularity of localization of power series restricted domains and projective ring (and its history); 286, 306 normalization; 444-447, 529 regularity of localizations of regular rings; restricted domain; 444, 529 261, 277 restricted dilatation matrix; 404 regularity of power series rings; 196 restricted power; 170, 316, 513, 573 Regularity versus Independence Theorem restricted power set; 28, 37, 506 together with its Corollary; 240-244 restriction of a map to a pair (whose first regular representation (left and right); 554 member is a subset of the domain and relation between affine and projective the second member is a subset of range varieties; 459-462 containing the image of the first relation between polynomial operations and member); 74, 489, 501 matrix operations; 149 restriction of a map to a singleton (same as relations preserving permutations; 13, 505, above when the second member 538 coincides with the image of the first INDEX 745

member which is the singleton); 74, Separable Generation Theorem; 437 489, 501 Separable Algebraic Extensions (Extending resultant; 83-85, 138, 146-157, 510 Derivations through and Criterion for); resultant matrix; 83, 138, 510 542-545 resultant properties (itemized in the thirty Separably Generated Extensions lines under (R6) of Detailed Content, (Extending Derivations throgh); 542 Lecture L4, §12 Remarks); 148-157 separating normalization basis; 431, 528 Riemann-Zariski space of a field; 129, 531 Separating Normalization Basis Theorem Riemann-Zariski space over a ring; 129, 531 (in connection with affine domains Riemann-Zariski space (quasitotal); 129, over an infinite field); 441 531 separating transcendence basis (of a field Riemann-Zariski space (total); 129, 531 extension); 431-432, 528, 529 right regular representation; 554 separating transcendental; 432, 529 ring; 5, 503 Serre; 277, 286, 298, 371 ring-isomorphic to a direct sum of rings or Serre conditions; 298-306, 522 finite direct sum of rings; 287, 521 Serre conditions in their alternative ring-theoretic compositum; 348, 527 versions and variations; 493 ring-theoretic direct sum; 286, 251 Serre conjecture; 371 Roots of Unity Theorem (Basic); 433 Serre quasicondition; 494 root field; 12, 505 Serre Criterion; 304, 522 row; 51 sesqui-transitive permutation group; 556 row rank; 75 set; 2, 501 set of all maps or functions from one set to Salmon; 82 another; 31, 35, 506 saturated chain of prime ideals; 209, 516 set-theoretic map; 4 scalar matrix; 52 set-theoretic power; 170, 513 scalar product of a homomorphism; 374 Severi; 82 sum of homomorphisms; 374 sharp-Frobenius group; 556 Schroeder-Bernstein Theorem; 28, 42 sharply transitive permutation group; 556, secant line; 53 570 segment; 47, 312-313, 523 Shiva; 1 segment complete; 315, 523 short exact sequence; 262 segment completion; 312-319, 523 short exact sequence (graded); 365 Segment-Completions (and short exact sequence splits; 262 Characterization Theorem for them); Shreedharacharya; 1 317 Shreedharacharya’s proof of Newton’s segment cut; 314, 523 Theorem; 64 segment-full; 315, 523 sign; 51 segment-full (conditions for a segment-full signature; 22, 503 loset); 318 similar triangles; 52 segment (positive upper); 313 simple group; 4, 502 semigenerating set; 307 simple point; 55-58, 128, 474 semigroup; 5, 503 simple point (in the sense of local rings); semihomogeneous ring; 178, 513 474-476, 530 semilocal ring; 307 simple module; 103 semimodel; 130 simple ring extension; 225, 517 semi-Frobenius group; 556 Simple Ring Extension Lemma; 225, 517 semi-regular permutation group; 554, 556 Simple Center Blowup Theorem; 469 semi-transitive permutation group; 556 Simplicity Criterion; 569 separable (algebraic); 432 Simplicity of the Alternating Groups; 5, 21, separable algebraic extension; 431 503, 564 separable element; 431, 529 Simplicity of the Projective Special Linear separable extension; 431, 529, 533 Groups; 569 separable extensions and primitive simplifying singularities by blowups (theory elements; 431-443 of quadratic and monoidal separable polynomial; 12, 341, 431, 505, 529 transformations); 462-483, 529-532 separable polynomials and multiple roots; singular; 53-58 341, 505 singular point; 53-58, 128, 474 746 INDEX size of a set; 2, 502 strongly independent; 491 skew-field, skew-ring, etc.; 5-6, 504 strongly restricted domain; 444, 529 smallest (element of a set of subsets); 253 subdimension formula; 492 smallest (or least or minimum) element in a subfield, subring, etc.; 5, 503 poset; 27, 320 subgroup; 4, 502 smallest proper overideal; 253 subgroup generated by; 404 Snake Lemma; 263 subintegrally graded ring; 175 Snake Sublemma; 326 subscript notation; 553 socle; 252, 520 subset; 3, 501 socle-size; 252, 520, 571 subset injection (it is the obvious injective Socle Size Lemma; 256, 520, 571 map of a subset into the set); 322 solid; 124-125 subset injection; 322 solid ball; 124 submatrix; 75 Solvability Theorem; 13 submonic (polynomial); 15, 407 solvable group; 4, 502 substitution homomorphism; 32 Something Is Twice Something Theorem on substitution map; 9, 32 Gorenstein Rings; 261, 520 substitution map for polynomial tuples; 407 special ED; 11, 505 subvariety; 87 special elementary group; 405, 528 subvariety (also with qualification of proper Special Jordan H¨olderTheorem; 103-105 or reducible or irreducible); 122, 123, special linear group; 51, 528 448, 454 special linear groups over polynomial rings; sum of homomorphisms; 374 419-430 sums of ideals or modules; 89-90 special principal ideal rings; 293-297, 521 Supplemental Pdim Lemma; 327 Special PIR; 293-297, 521 Supplementary (additional) Modelic SPIR; 293-297, 521 Blowup Theorem; 474, 498 specializations of valuations; 320-323, 523 Supplementary (First and Second) spectral affine space; 126 Dimension Theorem; 223-224 spectral ideal; 96, 123, 510 Supplemented Primitive Element Theorem; Spectral Nullstellensatz; 222-223, 489 443 spectral projective space; 458-459 support of an element in a direct sum Spectral Relations Theorem; 220-222 (denoted by supp); 175 spectral variety; 95, 123, 510 support of a function or map in general spectral variety set; 96, 123, 510 (denoted by supp); 140, 170, 316, 513 spectrum; 95, 123, 510 support of a meromorphic series or power sphere; 124 series or polynomial (or of a function sphere with handles; 334 or map leading to these, denoted by split homomorphism; 373 Supp); 29-36; 506 split monomorphism; 373 support of a module (denoted by supp); split monomorphisms (a criterion); 495 165, 281 split short exact sequence; 262 support and annihilator of a module; 165 splits (a homomorphism); 373 support and quasiprimary decomposition of splits (a monomorphism); 373 a submodule; 165 splits (a short exact sequence); 262 supremum; 320 splits homogeneously (a graded short exact surds; 23 sequence); 365 surface; 25-26, 50, 53-55, 86, 124-125 splitting field; 12, 19, 505 surjection; 2, 501 spread notation; 149 surjective; 2, 489, 501 spur; 540 Suslin; 371 square matrix; 75 Suslin’s Localization; 577 square submatrix; 75 Suslin’s Localization Lemma; 421 stabilizer; 547 Suslin’s Theorem; 371, 417, 419 stabilizes; 547 Sylow subgroup; 551 stably free module; 417 Sylow transitivity; 551 Stably Free Module Theorem; 417 Sylow’s Theorem; 552 Steinitz; 39 Sylvester; 82, 83, 138, 146, 364, 527 strict normal crossing; 475, 532 symmetric function; 536 strong relative independence; 491 symmetric function (elementary); 535, 536 INDEX 747 symmetric group; 2, 21, 502 UFD Lemma; 324 symmetric rational function; 536 UFD Theorem; 329, 524 symmetric relation; 31 Ujjain; 499 symplectic group; 52 underlying additive abelian group; 8 system of imprimitivity; 557 underlying homomorphism (of an algebra); syzygies; 364, 527 286, 521 syzygies (homogeneous); 366 unequicharacteristic = syzygy; 364, 527 nonequicharacteristic (quasilocal ring); Syzygy Theorem (Hilbert); 370 465 uniformization (thesis); 143 tacnode; 60 unimodular column; 406 tight assassinator = colorful (somewhat like unimodular row; 406 the word annihilator) long form of unimodular tuple; 406 tass; 98, 140-142, 181, 511 unique factorization domain; 4, 11, 16 tight associator; 98, 140-142, 181, 511 unique factorization in polynomial rings; 68 tangent; 52-58 unique factorization in power series rings; tangential multiplicity; 466 68, 330 topologically equivalent; 334 unique factorization in regular local rings; torsion subgroup; 45 323-329, 524 total quotient ring; 114-115 union; 3, 501 total quotient rings of reduced noetherian unit ideal; 6, 503 rings and their normalizations; 286-293 Unit Ideals in Polynomial Rings; 40, 565 total transform; 133-134, 466-468 unitary group; 52 trace; 539 units in a matrix ring; 51 trace (behaviour under finite algebraic field units in a power series ring; 33 extension); 540 units in a ring; 11, 505 trace (properties of); 541 univariate; 12 transform; 133-134, 466-468, 476-477, 530 univariate ideal extensions; 197, 517 transforms of one or two ideals relative to univariate ideal extensions lemma together local ring triples; 476-477, 530 with its sharper versions called Simple transcendental element; 10, 505 and Multiple Ring Extension Lemmas; transcendence basis; 11, 41, 505 197, 225-226, 517 transcendence degree; 11, 124, 209, 505, 516 univariate polynomial and power series transitive action; 551, 570 rings (heights of ideals); 251 transitive group; 551, 570 universally catenarian; 225, 517 transitive relation; 31 University Algebra; 500 translation of coordinates; 58 unmixed (ideal); 236, 298, 519 transportation; 554 unmixedness theorem holds; 236, 519 transpose; 75, 509 unresolved (ideal in a ring); 475, 532 transposition; 5, 21, 503 Unsolvability Theorem; 14 transversally; 474 upper bound in a poset; 27-28 transvection matrix; 404, 570 upper segment (positive); 313 Transvection Theorem; 405 triple point; 55 valuation; 33-34, 157-160, 506 trivariate; 25 Valuation Characterization Theorem; 130 trivial block; 557 Valuation Existence Theorem; 129 trivial valuation; 34, 298, 522 Valuation Extension Theorem; 129 triviality of associated graded ring; 491 valuation function; 230, 518, 572 Tschirnhausen; 49-50 valuation functions (Theorem about them); tuple notation and functional notation; 171 231, 518, 572 two-transitive action (of special linear and Valuation Maximality Theorem; 129 projective specual linear group) on valuation ring; 34, 129, 506 projective space; 570 valuation ring of a field; 129, 321, 523 type of a graded ring is an additive abelian valuation ring of a field over a ring; 129 monoid; 172 valuation theoretic composition; 577 valuation, trivial or real or real discrete; UFD = unique factorization domain; 4, 11, 298, 522 16, 67-69, 505 Valuations and L’Hospital’s Rule; 167-168 748 INDEX value group; 34, 506 Vandermonde Determinant Theorem; 431, 432 variable; 11, 30 varieties in affine or spectral space; 122-126 varieties in modelic space; 127-132 varieties in projective space; 447-462 variety; 26, 86-89, 122-132, 447-462 Veblen; 82 vector; 8, 504 vector space; 8, 504 vector space basis; 41 weak Zorn property; 37 Wedderburn; 6 Weierstrass; 70 Weierstrass degree; 70, 508 Weierstrass Division Theorem; 70-74, 508 Weierstrass Preparation Theorem; 70-74, 82, 508 well ordered power; 316, 573 well ordered product; 316, 573 well ordered set; 28, 506 well ordering; 27-28, 37-44, 506-507 Weyl; 52 wo; 27 word problem; 1 woset; 27, 506

Young; 82

Zariski; 82, 500 Zariski ring; 183-184, 514 Zariski rings characterization; 191 Zariski-Samuel’s book (cited in Limitations on Normalization Theorem); 495-496 Zassenhaus; 6 Zero Dimensional GST (Gorenstein) Ring Characterization Lemma; 254 zerodivisor; 90, 99, 181 zerodivisor set; 99, 181, 511 Zerodivisor Theorem; 181, 514 zerodivisors modulo a submodule: 181 zerodivisors modulo an ideal: 181 zero ideal; 6, 503 zero matrix; 152 Zorn property; 28, 37 Zorn’s lemma; 27-28, 37-44, 506-507