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A REVIEW OF COMMUTATIVE UNDERGRADUATE SEMINAR: TORIC VARIETIES

ADRIANO FERNANDES

Contents 1. Basic Definitions and Examples 1 2. Ideals and Quotient Rings 3 3. Properties and Types of Ideals 5 4. C- 7 References 7

1. Basic Definitions and Examples In this first section, I define a ring and give some relevant examples of rings we have encountered before (and might have not thought of as abstract algebraic structures.) I will not cover many of the intermediate structures arising between rings and fields (e.g. integral domains, unique domains, etc.) The interested reader is referred to Dummit and Foote. Definition 1.1 (Rings). The “ring” R is a with two binary opera- tions + and , respectively named and , satisfying · (R, +) is an abelian (i.e. a group with commutative addition), • is associative (i.e. a, b, c R, (a b) c = a (b c)) , • and the distributive8 law holds2 (i.e.· a,· b, c ·R, (·a + b) c = a c + b c, a (b + c)= • a b + a c.) 8 2 · · · · · · Moreover, the ring is commutative if multiplication is commutative. The ring has an , conventionally denoted 1, if there exists an element 1 R s.t. a R, 1 a = a 1=a. 2 8 2 · ·From now on, all rings considered will be commutative rings (after all, this is a review of commutative ...) Since we will be talking substantially about the complex field C, let us recall the definition of such structure. Definition 1.2 (Fields). A in which every nonzero element has a mul- tiplicative inverse is called a field, i.e. if a R there exists b Rs.t.a b = 1. 8 2 2 · Date:02/10/2016. 1 2ADRIANOFERNANDES

It is trivial to see that the form a ring, that integers in modular form a ring, and that functions spaces (with point-wise addition and multiplication) are rings. The rational, real and complex numbers are fields. The concept of subsets of a set gives rise to a natural to a of a ring. Definition 1.3 (Subring). A subring S of a ring R is any subset of R in which the operations of addition and multiplication defined in R make S a ring when restricted to S. Hence, a subring is a ring in itself sitting inside another ring. Transitivity is trivial by the definition. Example 1.4. Z 2Z 4Z 2nZ , for n Z,n>2. This example also shows we can have an infinite chain ··· of ··· inside a ring.2 The rings we will be mostly interested in for this seminar will be Rings. Definition 1.5 (Polynomial Rings). Fix a commutative ring R with identity. Let x be an . The R[x] (read R adjoined x) is the set of all formal n i sums of the form i=0 aix for some n 0 and each ai R. The formal sum is called a polynomial in x with coecients in the ring R. 2 P If an = 0, then the polynomial is said to have degree n. The polynomial is monic if the coe6 cient of the largest power is the multiplicative . Addition and multi- n i plication are defined component-wise in polynomial rings. Let p(x)= i=0 aix and n i n i q(x)= bix be two . Then, p(x) q(x):= (ai+bi)x and p(x)+q(x):= i=0 · i=0 P n [ k (a b )]xk. k=0 Pi=0 i k i P The ring over which the polynomials are formed make a substantial di↵erence, as the followingP P example shows. Example 1.6. Consider p(x)=x+1. If p(x) Z[x], 2p(x)=2x+2 and p(x)2 = x2+2x+1. 2 If p(x) Z/2Z[x], 2p(x) = 0, and p(x)2 = x2 + 1. 2 Before proving some essential facts about polynomial rings, we need some more defini- tions. Definition 1.7 (Integral ). An is any commutative ring R with identity in which there are no zero divisors, i.e. a, b R, a, b =0,ab= 0. 8 2 6 6 Definition 1.8 (Units of a Ring). An element a R is called a unit of the ring if there exists an element b R such that ab = 1. The units2 of a ring form a ring in itself, called 2 the Ring of Units, denoted R⇥ Now we are ready to state an prove some key facts about polynomial rings. Proposition 1.9. Let R be an integral domain, and let p(x),q(x) =0,p(x),q(x) R[x]. Then, 6 2 (1) deg(pq)=deg(p)+deg(q) (2) The units of R and R[x] are the same (3) R[x] is an integral domain. AREVIEWOFCOMMUTATIVERINGTHEORYMATHEMATICSUNDERGRADUATESEMINAR:TORICVARIETIES3

n m m Proof. If p(x) and q(x) have leading terms anx and b x respectively, the leading n+m of pq(x)isanbmx .SinceR has no zero divisors, anbm = 0. Hence the (3) and (1) hold. For (2), suppose pq(x)=1 deg(p)+deg(q)=0 deg(6p)=deg(q) = 0, hence both p, q ) ) are elements of R and thus are units in R. ⇤ One can establish -preserving mappings between rings. We now focus on such mappings. Definition 1.10 (Ring ). Let R and S be rings. (1) A is a map : R S satisfying (a + b)=(a)+(b) and (ab)=(a)(b), a, b R 7! (2) The of a ring8 homomorphism2 is the set of elements in R mapping to the addition identity of S,i.e.ker = r R (r)=0S (3) A ring is a bijective ring{ 2 homomorphism.| } Example 1.11. Let n : C[x] Cbedefinedas(p(x)) = p(0). This maps a polynomial to its constant term. Since addition7! and multiplication of the constant terms are clearly preserved under this mapping, this is a ring homomorphism. The kernel is the set of all polynomials with constant term 0 in the complex field. Proposition 1.12. Let : R S be a ring homomorphism. Then, 7! (1) The of is a subring of S; (2) ker is a subring of R and is closed under multiplication by elements of R.

Proof. For (1), if s ,s im s = (r ) ,s = (r ) for some r ,r R.Since is a 1 2 2 ) 1 1 2 2 1 2 2 homomorphism, s1 s2 = (r1 r2) and s1s2 = (r1r2), so s1 s2 ,s1s2 S. For (2), if a, b ker (a)=(b)=0 (ab)=(a b) = 0, so the kernel is2 closed under subtraction2 and) multiplication. Now,) for any r R, (ra)=(r)(a)=(r)0 = 0 = 0(r)=(a)(r)=(ar), so indeed the kernel is2 closed under multiplication by elements of R. ⇤ Kernels are examples of structures appearing more generally in rings, so called ideals. We now focus on these structures.

2. Ideals and Quotient Rings Definition 2.1 (Ideals). Let R be a ring, r R and I R. I is an of R if I is a subring of R that is closed under multiplication2 by the elements⇢ of R,i.e.ifa I,then ra, ar I; one writes rI I r R. 2 2 ⇢ 8 2 It turns out that there is a vital connection between ideals and ring . Ideals are kernels for some ring homomorphism of the ring in which they sit in. The of any element of R,sayr+I, r R, I an ideal in R form a ring in itself if we appropriately define addition and multiplication2 of cosets for the set of cosets. Ideals are in fact what we 4ADRIANOFERNANDES require for those operations as we here present to make the set of cosets into a ring. We define addition and multiplication for cosets of elements of R by an ideal I as follows: (r + I)+(s + I)=r + s + I (r + I)(s + I)=rs + I It is trivial to show that the distributive law holds whence operations are defined as above. Let r, s, t R and I an ideal in R.Then 2 (r + I)[(s + I)+(t + I)] = (r + I)[s + t + I] = r(s + t)+I =(rs + rt)+I =(rs + I)+(rt + I)=(r + I)(s + I)+(r + I)(t + I) So it seems that we can indeed form a ring by considering the cosets of elements of a ring by the ideals of a ring. The discussion above proved the following Proposition 2.2. Let R be a ring and I an ideal. If addition and multiplication are defined as above, the set of cosets of elements of R by the ideal I form a ring. Conversely, any I of R in which the above operations are well-defined form an ideal of R. I skip the proof of the converse direction for succinctness. Definition 2.3 (Quotient Rings). Let R be a ring and I be an ideal of it. The ring R/I with operations defined above form the of R, whose elements of the form r + I for r R. 2 The following theorem summarizes the present discussion: Theorem 2.4 (The First Ring Isomorphism Theorem). Let : R S be a homorphism of rings. Then, ker is an ideal of R,im is a subring of S and R/ker7! is isomorphic as aringto(R). Alternatively, if I is any ideal of R, then ⇡ : R R/I is a surjective ring homomorphism with kernel I, called the natural projection of R7!onto R/I. Hence, every ideal is the kernel of some ring homomorphism and vice-versa. Ideals have set operations rigorously defined: Definition 2.5 (Sum and of Ideals). Let I,J be two ideals of a ring R.Define (1) the sum of I and J as I + J = i + j i I,j J ; (2) the product of I and J as IJ, the{ set| of2 all finite2 } sum of elements of the form ij with i I,j J; 2 2 n (3) the n power of I, denoted I , the set of all finite sums of elements of the form i1i2 in with i I, k =1,...,n ··· k 2 8 We finish this first section on ideals with the idea of a generator for an ideal. Proposition 2.6 (Ideals and Generators). Let R be a ring and let I be an ideal of R.A set J R generates an ideal by setting ⇢ (J)= I J I, I R { | ⇢ ⇢ } Namely, the intersection of arbitrarily\ many ideals is an ideal in itself. AREVIEWOFCOMMUTATIVERINGTHEORYMATHEMATICSUNDERGRADUATESEMINAR:TORICVARIETIES5

Proof. We need only to show the last claim. So let Ik be arbitrarily many ideals of a ring R, and consider I = I . Let i I i I k. Hence ri I , k,thusri I. k 2 ) 2 k 8 2 k 8 2 ⇤ To better understandT the concept of a set generator for an ideal, notice that (J)isthe smallest ideal containing J as a set, and its elements are finite R-linear combinations of n elements of J,i.e. i=1 riji for ri R, ji J i. The following example would be one of interest going forward in this seminar.2 Naming-wise,2 8 an ideal generated by a single element is called a PrincipalP Ideal, whereas an ideal generated by a finite set is called a Finitely Generated Ideal. Example 2.7. Consider the two-indeterminate polynomial ring over the complex field C[x, y]. Consider further the polynomials without constant terms. Clearly, any polynomial in C[x, y] can be made into a “zero” constant polynomial by multiplying the entire poly- nomial by one of the indeterminates. Hence, we can form an ideal I of C[x, y] given by the set of all zero constant polynomials. A representative element of such ideal takes the form xP (x, y)+yQ(x, y), for P, Q C[x, y]. Thus, (x, y)=I. 2 3. Properties and Types of Ideals Definition 3.1 (Maximal Ideals). An ideal M in a ring R is called a if M = R and no ideal I exists such that M I R. Equivalently, if M I R M = I. 6 ⇢ ⇢ ⇢ ⇢ ) Definition 3.2 (Prime Ideals). An ideal P in a ring R is called a if (1) P = R; (2) if ab6 R for a, b R,thena P or b P The following2 propositions2 (which2 we state2 without proof) make it easy to identify maximal and prime ideals. Proposition 3.3. Assume a commutative ring R and an ideal I of R. (1) I is maximal in R if the quotient ring R/I is a field; (2) I is prime in R if the quotient ring R/I is an integral domain. Example 3.4. The quotient ring Z/2Z is a field, so the ideal generated by (2,x) is maximal in Z[x]. The ideal (x) is not a maximal ideal in Z[x], since Z is not a field. However, (x) is a prime ideal, since Z is an integral domain. Definition 3.5 (Radical Ideals). An ideal I of R is called a Radical Ideal if I = pI = r R rn I for some n>0 . { 2 | 2 } The first ring isomorphism theorem preserves the types of ideals. That is, if I is a prime, maximal or radical ideal in the domain ring, its image in the codomain ring is also a prime, maximal, or radical ideal, respectively. We can now solve some exercises from Smith et al.

Exercise 2.1.1. Proof. (1) We want to show first that every maximal ideal is prime. From the proposition, let R be our ring and M be a maximal ideal of R, so M/R is a field, hence an integral domain, therefore M is also prime. 6ADRIANOFERNANDES

(2) Now, we show that every prime ideal is radical. So consider M as a prime ideal. I first show that if r R and n N,n>0, then rn M r M. Proceed by induction. If n = 1 (base2 case), there2 is nothing to show.2 So let) n 2 1 and rn M r M. Then, rn+1 = rnr M r Morrn M, in which caser M regardless.2 ) Now,2 let 2 ) 2 2 2 r pM,sorn M for some n>0,n N. By the above, r M,sopI I.The converse2 inclusion2 is trivial. 2 2 ⇢ (3) The final claim is that a radical ideal is an ideal. So let I be an ideal. That I pI is ⇢ trivial, for a = a1 I. Let a, b pI an I and bm I for some n, m > 0,n,m 2 2 ) 2 2 2 N. A binomial expansion shows that m+n m+n m + n k m+n k (a + b) = a b k Xk=0 ✓ ◆ k m+n k If k n,thena I.Ifk

Proof.(1) First we show M is a maximal ideal i↵ R/M is a field. To show ( ), suppose R/M is a field, and J is an ideal of R that properly contains M,i.e.j + M(is a nonzero element of R/M. Hence, there exists an element k+M such that (k+M)(j+M)=M +1. Since j J, kj J,so1 kj M J.Then1=(1 kj)+kj J, so J=R, for an ideal generated2 by2 unity is the2 ring itself.⇢ For (), suppose M is maximal,2 and consider j R M. Let J = rj + k r R, k M , which is an ideal properly containing M. For2 Mis maximal, R{= M,hence1+| 2 2M =} jr + M =(j + M)(r + M), so units can indeed be found for arbitrary elements. (2) Now we show that if P is a prime ideal of R,thenR/P is an integral domain. For suciency, consider x + P, y + P such that (x + P )(y + P )=xy + P = P (since P is the zero in R/P ). Since P is prime, either x P or y P . WLOG, suppose x P .Then x + P = P , as we wanted to show. For necessity,2 let2 R/P be an integral domain,2 and let a, b R such that ab P .Then,P = ab + P =(a + P )(b + P ). Since R/P is an integral2 domain, a + P = 2P or b + P = P . WLOG, let a + P = P ,hencea P ,soP is 2 prime. ⇤ Exercise 2.1.3.

Proof. First, we want to show that if I S is an ideal, any ring map : R S induces ⇢1 7! 1 an injective homomorphism of rings R (I) S/I. A representative element of R (I) 1 7! 1 is r + (I), so by the structure preserving map, (r + (I)) = (r)+I.Soif(r + 1 1 (I)) = (s + (I)) r = s, showing the map is injective. From previous discussion, we have argued that the) pre-image of kernels are also kernels under ring homomorphisms, which completes the proof ⇤

Exercise 2.1.4. AREVIEWOFCOMMUTATIVERINGTHEORYMATHEMATICSUNDERGRADUATESEMINAR:TORICVARIETIES7

Proof. For suciency, suppose the zero ideal is radical, i.e. 0 = (0) = r R rn (0) for some n>0 . Hence, r R, rn =0i↵ r = 0, proving that{ R} is reduced.{ For2 necessity,| 2 suppose the ring} is reduced,8 2 so rn =0i↵ r = 0. Thus, the zero ideal is also its radical, trivially. ⇤ Exercise 2.1.5. Proof. For necessity, suppose R/I is reduced, so (r + I)n = rn + I = I i↵ r + I = I,i.e. rn I i↵ r I, so the ideal I is clearly radical. For suciency, let I be a radical ideal, and let 2i I = sqrtI2 , so that in = 0 for some n>0 in R. Thus, in + I = I in R/I if i + I I. Since2I is the zero of R/I, R/I is reduced. Conversely, let i I, and since the ideal2 is radical, in I,soinR/I, i + I = I = in + I. 2 2 ⇤ 4. C-algebras Definition 4.1 (C-). AringR is a C-algebra if C R as a subring. ⇢ Every C-algebra is a vector over R, where vector addition is defined as the addition in R, and multiplication for a given z C and a vector r R is given by multipli- 2 2 cation in R. We extend ideas from previous section to the specific case of C-algebras. Definition 4.2 (Generators for a C-algebra). The C- generated by a subset J of the C-algebra S is A J A, A S, S a C-algebra . { | ⇢ ⇢ } This subalgebra isT the smallest C-subalgebra containing the subset J. Its elements are all elements of S that can be expressed as polynomials with elements in S and coecients in C. Definition 4.3 (Finitely generated ). An algebra R is finitely generated if there exists a finite subset J R generating R. ⇢ Example 4.4. In the polynomial ring C[x, y], we see a C-algebra, for C C[x, y]bysetting ⇢ x = y = 0. It is finitely generated since (x, y) generate C[x, y]. Definition 4.5 (C-). If R, S are two C-algebras, a map : R S 7! is a C-algebra Homomorphism if it is a ring map that is linear over C,i.e.(zr)=z(r), z C,r R. 8 2 2 Example 4.6. Consider the C-algebra C[x, y] C[z] x2 + y3 7! . The images of the generators must be preserved under the projection homomorphism, so a candidate for is (x)=z3 and (y)= z2, for these satisfy the original relationships between ideal generators in the kernel.

References David S. Dummit, Richard M. Foote, , Third Edition, Wiley.

[0] [1] Smith et al., Algebraic , Springer, 2000