Introduction to Commutative Algebra and Algebraic Geometry

INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

The purpose of these notes is to give a quick and relatively painless introduction to the language of modern commutative algebra and algebraic geometry. The point of view which we will emphasize is that commutative algebra and algebraic geometry (at least the local theory) are two diﬀerent languages for talking about the same thing. The algebraic and the geometric aspects of every phenomenon will be introduced and discussed concurrently. The ﬁrst part of the course will consist of standard material; towards the end we will treat some topics of current interest such as valuation theory and singularities. Due to spacetime limitations, we will omit some important topics, notably cohomology, traditionally present in an introductory course. Unless otherwise stated, all rings in these notes will be commutative with 1. N will denote the set of natural numbers, N0 the set of non-negative integers.

1. Varieties and ideals. § Let k be a ﬁeld and x1,...,xn independent variables. The following are some basic examples of rings appearing in algebraic, analytic and formal geometry. The reader should keep them in mind in order to test and illustrate the general theory.

(1) Polynomial rings k[x1,...,xn] and D[x1,...,xn], where D = Z or, more generally, D is the ring of integers in some number ﬁeld, such as Z[√2] or Z[√5i]. (2) Formal power series rings

k[[x ,...,x ]] := c xα1 xα2 ...xαn α =(α ,...,α ) and c k 1 n  α 1 2 n 1 n α ∈  α∈Nn  X0 

and Zp[[x1,...,x n]], where p is a prime number and Zp is the ring of p-adic integers. (3) Convergent power series rings C x1,...,xn and Zp x1,...,xn . { } { } The beauty and power of commutative algebra lie in the fact that, it provides a universal language and methods for studying number theory, local algebraic geometry and local questions in complex analysis, as illustrated by the above examples. Given rings R and S, a homomorphism from R to S is a map R S which preserves the ring operations + and . → · Typeset by AMS-TEX 1 2 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Deﬁnition 1.1. Let R be a ring and I a non-empty subset of R. I is an ideal of R if I is closed under addition and for any a R, x I, we have ax I. ∈ ∈ ∈ For example, 0 is an ideal (usually denoted by (0)); the entire ring R is also an ideal, sometimes{ } called the unit ideal and denoted by (1). Let B = eλ λ∈Λ be a subset of R. Let BR denote the set of all linear combinations of the{ form}

BR = eλi bλi bλi R, eλi B, n N . | ∈ ∈ ∈ (i=1 ) X Exercise 1. If I = BR, prove that I is the smallest ideal of R containing B. If I = BR, we say that I is the ideal generated by B, or that B is a set of generators or a base of I. Of course, a given ideal I may have many different sets of generators. An ideal I is said to be finitely generated if it has a finite set of generators. Definition 1.2. A ring R is Noetherian if every ascending chain I I I ... 1 ⊂ 2 ⊂ 3 ⊂ of ideals of R stabilizes, that is, there exists n0 N such that In = In for all n>n0. ∈ 0 Exercise 2. Prove that a ring R is Noetherian if and only if every ideal of R is finitely generated. Commutative algebra grew out of number theory and geometric invariant theory at the turn of the century. One of the starting points was Hilbert’s basis theorem:

Theorem 1.3. Let R be a Noetherian ring and x1,...,xn independent variables. Then R[x1,...,xn] is also Noetherian for any n. Proof. It is sufficient to consider the case n = 1. We want to establish that R[x] is Noetherian, provided R is Noetherian. Let I be an ideal of R[x]. Let J denote the ideal of R, consisting of all the leading coefficients of elements of I (the reader should check that J is, indeed, an ideal). Since R is Noetherian, J is finitely generated. Hence there exists a finite collection of polynomials f ,...,f R[x], whose leading 1 m ∈ coefficients generate J. Let d = max deg fi. Then for every f I such that 1≤i≤m ∈ m deg f d, there exist h1,...,hm R[x] such that i=1 hifi has the same degree and leading≥ coefficient as f, in other∈ words, P m (1.1) deg f h f < deg f. − i i i=1 ! X

Applying (1.1) repeatedly and using induction on deg f, we can ﬁnd g1,...,gs R[x] such that ∈

m (1.2) deg f g f < d. − i i i=1 ! X INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 3

By (1.2), there exists the smallest l N0 having the following property. There exist ∈ r N, f1,...,fr I such that for any f I there are g1,...,gr R[x] satisfying ∈ ∈ ∈ ∈ r (1.3) deg f g f 0. Let Il denote the set of those elements of I which have degree exactly l 1. Let J denote the ideal − l of R consisting of 0 and all the leading coefficients of elements of Il. Since R is Noetherian, Jl is finitely generated. Hence there exist finitely many elements fr+1,...,fs of Il, such that for every f Il there exist gr+1,...,gs R satisfying s ∈ ∈ deg f g f < l 1. In view of (1.3), this contradicts the minimality of − i i − i=r+1 l. P

Corollary 1.4. Let k be a field and x1,...,xn independent variables. Then the polynomial ring k[x1,...,xn] is Noetherian. Similarly for Z[x1,...,xn]. Let R be a ring and I an ideal of R. The quotient of R by I is the set of R cosets I (viewed as abelian groups with respect to +), endowed with the obvious R ring operations. We have the natural surjective homomorphism R I , whose R → kernel is I. We also say that I is a homomorphic image of R. Note also that every surjective ring homomorphism π : R S is the quotient of R by some ideal (namely, by Ker π). → We are now ready to introduce our basic dictionary between algebra and geometry, at first in the classical setting with algebraic varieties on the geometric side and finitely generated algebras over a field on the algebraic side. Soon we will switch to the more general modern setting with arbitrary schemes on the geometric side and arbitrary rings on the algebraic side. Classically, algebraic geometry studied algebraic subvarieties of kn (i.e. subsets of kn defined by finitely many polynomial equations), where k is a field and n N. ∈ In fact, k was usually taken to be C.

Definition 1.5. Let k be a field and let I be an ideal of k[x1,...,xn]. The algebraic variety defined by I in kn is

V (I) = a kn f(a)=0 for all f I . { ∈ | ∈ }

Since k[x1,...,xn] is Noetherian, we can choose a ﬁnite base (f1,...,fm) for I, so that V (I) = a kn f (a) = = f (a)=0 . { ∈ | 1 ··· m } Associated to every ideal I of k[x1,...,xn] we have the algebraic variety V (I). Conversely, given any set V kn, we may consider the ideal I(V ) deﬁned by I(V ) = f k[x ,...,x ] f⊂(a) = 0 for all a V . A natural quesiton arises: { ∈ 1 n | ∈ } 4 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

what is the relation between I and I(V (I))? To answer this question, we need another definition. Definition 1.6. Let R be a ring and I an ideal of R. The radical of I , denoted by √I, is defined to be

n √I = x R there exists n N0 such that x I . { ∈ | ∈ ∈ } We say that I is radical if I = √I. Exercise 3. Prove that √I is an ideal and that I √I. ⊂ Let k be a field and I an ideal of k[x1,...,xn]. Clearly, V (I) = V (√I). It is also clear that I I(V (I)) for any ideal I k[x ,...,x ]. Hence for any ideal I, ⊂ ⊂ 1 n √I I(V (√I)) = I(V (I)). It turns out that if k is an algebraically closed field (this means⊂ that every non-constant polynomial in k[x] has a root in k), we always have √I = I(V (I)). This is a non-trivial theorem, called Hilbert’s Nullstellensatz. We will state and prove several versions of the Nullstellensatz, all of which are sometimes referred to by this name. Theorem 1.7 (The strong form of Hilbert’s Nullstellensatz). Let k be an algebraically closed field and I an ideal of R = k[x1,...,xn]. Then I(V (I)) = √I. In other words, √I = f R f(a)=0 a V (I) . { ∈ | ∀ ∈ } In other words, there is a one-to-one correspondence between algebraic subvari- n eties of k and radical ideals of k[x1,...,xn]. A proof of Hilbert’s Nullstellensatz will be given shortly. First, we would like to advance a little more in our task of constructing a dictionary between algebra and geometry. To do this, we need to define the notions of maximal and prime ideals. Definition 1.8. Let R be a ring and I $ R an ideal. We say that I is prime if for any x,y R such that xy I, we have either x I or y I. We say that I is maximal if∈I = R and the only∈ ideal of R properly∈ containing∈ I is R itself. The set of all the maximal6 ideals of R will be denoted by Max(R), the set of all the prime ideals of R by Spec R. Associated to the properties of ideals we have encountered so far —radical, prime and maximal — we have the corresponding properties of rings (the precise connection will be explained in a moment). Definition 1.9. Let R be a ring and x an element of R. We say that x is nilpotent if there exists n N such that xn = 0. We say that x is a zero divisor if there exists y R 0 ∈such that xy = 0. The set of all nilpotent elements of R is called ∈ \{ } the nilradical of R, denoted by (0) or nil(R). Remark 1.10. Since (0) is the radicalp of the zero ideal (0), it is itself an ideal. Describing the set of zero divisors of R is more tricky; we will do it somewhat later p in the course. INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 5

Deﬁnition 1.11. Let R be a ring. R is said to be reduced if it has no non- zero nilpotent elements. R is an integral domain (sometimes abbreviated to “domain”) if R has no non-zero zero divisors.

Every ﬁeld is an integral domain. If R is a ring, we will denote by Rred the ring R ; of course, Rred is reduced. √(0)

Exercise 4. Prove that (0) equals the intersection of all the prime ideals of R (Hint: to prove one of the two inclusions, you will need Zorn’s lemma). p The intersection of all the maximal ideals of R is called the Jacobson radical of R, or simply the radical of R), sometimes denoted by J(R). R Exercise 5. Prove the following implications: I is maximal I is a field R ⇐⇒ R = I is prime I is an integral domain = I is radical I is reduced.⇒ As a special⇐⇒ case of this, we have that R is⇒ a field if and only⇐⇒ if the zero ideal is maximal, R is a domain if and only (0) is prime and R is reduced if and only if (0) is radical. By Zorn’s lemma, every ring R (whether Noetherian or not) has at least one maximal ideal, hence, a fortiori, at least one prime ideal. For Noetherian rings the existence of maximal ideals does not require Zorn’s lemma. n Example 1.12. Let k be a field, n N and (a1,...,an) k . Then (x1 a ,...,x a ) is a maximal ideal of k[∈x ,...,x ]. ∈ − 1 n − n 1 n Definition 1.13. The Zariski topology on kn is the topology whose open sets are of the form kn V (I), where I is an ideal of k[x ,...,x ]. \ 1 n Operations on ideals. Next, we define the operations on ideals which correspond to the topological operations of union and intersection. Definition 1.14. Let R be a ring, I a collection of ideals of R. The sum { λ}λ∈Λ λ∈Λ Iλ is defined by

P r

Iλ = xλi r N, λi Λ, xλi Iλi for 1 i r . ( ∈ ∈ ∈ ≤ ≤ ) λ∈Λ i=1 X X

In particular, if Λ = , then λ∈Λ Iλ =0. The product IJ ∅is deﬁned to be the ideal of R generated by the set xy x I, y J . Of course, IJ canP also be thought of as { | ∈ ∈ } n IJ = xiyi n N, xi I, yi J for 1 i r . ( ∈ ∈ ∈ ≤ ≤ ) i=1 X

Remark 1.15. Historically, the word “ideal” ﬁrst appeared in number theory. Kum- mer’s famous mistake in proving Fermat’s Last Theorem consisted in assuming 6 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

that Unique Factorization into prime numbers holds in the ring of integers in any number field. The mistake was soon discovered. For example, in Z[√5i], we have 6=2 3 = (1+ √5i)(1 √5i), and yet 2 is an irreducible element which divides · − neither (1 + √5i) nor (1 √5i). It was then realized that a sensible generalization of the Unique Factorization− Theorem to rings of integers in number fields is not unique factorization for elements but unique factorization for ideals. This is how the notion of “ideal numbers” or ideals was introduced. For instance, in the above example the ideal (6), generated by 6, has a unique factorization into prime ideals: (6) = (2, 1 + √5i)2(3, 1 + √5i)(3, 1 √5i). The following properties are obvious− from definitions and the Nullstellensatz: (1) kn = V ((0)) (2) = V ((1)) (3)∅ If I is a collection of ideals of k[x ,...,x ],then { λ}λ∈Λ 1 n

V (Iλ) = V Iλ . ! λ\∈Λ λX∈Λ (4) V (I) V (J) = V (IJ) = V (I J) ∪ ∩ (5) √J √I = V (I) V (J) and the converse holds if k is algebraically closed.⊂ ⇒ ⊂ In particular, this shows that Zariski topology is, indeed, a topology. The starting point for the theory of schemes is the above Example 1.12, and the partial converse to it, known as the weak form of Hilbert’s Nullstellensatz (the connection between the weak and the strong form will become clear in the course of the proof). Theorem 1.16 (A weak form of Nullstellensatz). Let k be an algebraically closed field. Then (a1,...,an) (x1 a1,...,xn an) is a one-to-one correspon- n ←→ − − dence between k and Max(k[x1,...,xn]). In other words, every maximal ideal of k[x1,...,xn] is of the form described in Example 1.12. Let k be an algebraically closed field. By Theorem 1.16, we may translate all the geometric definitions and statements about points of kn into purely algebraic statements about ideals in the ring k[x1,...,xn]. For example, we can define the Zariski topology on Max(k[x ,...,x ]) as follows: for an ideal I k[x ,...,x ], 1 n ⊂ 1 n V (I) = m Max(k[x1,...,xn]) I m . Thus maximal ideals of k[x1,...,xn] correspond{ to∈ points of kn and radical| ⊂ ideals} correspond to algebraic varieties. Let σ : A B be a homormorphism of rings. Definition 1.17.→ Let I be an ideal of A. The extension IB is the ideal of B generated by σ I). If J B is an ideal of B, the contraction J A is the ideal ( ⊂ ∩ σ−1(J). Remark 1.18. The following are immediate from the definition: (1) If J B, is prime then J A is prime. (2) If J ⊆ B is maximal and σ∩is surjective, then J A is maximal. ⊆ ∩ INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 7

Let x = (x1,...,xn). Let I be an ideal of k[x] and consider the natural homo- k[x] morphism k[x] I . Then the map m m k[x] induces a bijection between k[x] → → ∩ Max( I ) and the set of maximal ideals of k[x], containing I. Hence the above correspondence between kn and Max(k[x]) induces a one-to-one correspondence k[x] k[x] between V (I) and Max( I ). I is nothing but the ring of polynomial functions k[x] defined on the variety V (I). We say that I is the coordinate ring of V (I). For example, k[x] is the coordinate ring of kn. Given two algebraic varieties X kn l ⊂ and Y k , defined by ideals I k[x1,...,xn], J k[y1,...,yl], it makes sense to talk about⊂ polynomial maps from⊂ X to Y . By definition,⊂ they are induced by the maps f : kn kl such that f ∗J I. In that case, f ∗ induces a homomorphism of → ⊂ coordinate rings k[y] k[x] . J → I Next, we show that prime ideals of k[x1,...,xn] correspond to irreducible subvarieties of kn: Definition 1.19. A closed set X in a topological space is said to be irreducible if whenever X = X X , we have either X = X or X = X . 1 ∪ 2 1 2 Let P be a radical ideal of k[x1,...,xn]. We want to show that P is prime if and only if V (P ) is irreducible. We state the result in a form which works for arbitrary rings (so that we can obtain the same fact for schemes in the next section). Proposition 1.20. A radical ideal P in a ring R is prime if and only if whenever P = I J with I and J ideals of R, either P = I or P = J. ∩ Proof. “If”. First, suppose that P cannot be expressed as I J in a non-trivial way. If P were not prime, there would exist x,y k[x ,...,x ]∩ P such that xy P . ∈ 1 n \ ∈ Let I = P +(x), J = P +(y). Then P $ I, P $ J, P I J. On the other hand, ⊂ ∩ IJ P , hence √IJ P . Then I J √I J = √IJ P , so that I J = P , a contradiction.⊂ ⊂ ∩ ⊂ ∩ ⊂ ∩ “Only if” is an immediate consequence of the following lemma.

Lemma 1.21. Let P be a prime ideal in a ring R and I1,...,Ir arbitrary ideals of r R. If Ii P , then Ii P for some i, 1 i r. i=1 ⊂ ⊂ ≤ ≤ T Proof. We give a proof by contradiction. Suppose Ii P for any I. For each i r r 6⊂ pick an element xi Ii P . Then xi Ii P , but xi / P for any i. This ∈ \ i=1 ∈ i=1 ⊂ ∈ contradicts the fact that P is prime.Q T

2. Schemes. § In 1 we saw that if k is algebraically closed, points of kn can be identiﬁed with § maximal ideals of k[x1,...,xn]. This suggests generalizing algebraic geometry from kn to Max(R), where R is an arbitrary ring, and where we think of maximal ideals as points in our space. The idea of a scheme, developed in the twentieth century, 8 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

consists, in addition, in enlarging our notion of a point to include all the prime ideals of R. Let R be a ring. Motivated by the above considerations, we define the Zariski topology on Spec R to be the topology whose closed sets are of the form V (I) = P Spec R I P , where I is an ideal of R. { ∈ | ⊂ } Definition 2.1. An affine scheme is apair(R, Spec R) of a ring and a topological space, where Spec R is endowed with its Zariski topology. Remark 2.2. Note that single point sets need not be closed in Spec R. More precisely, we can describe the closure of the set P for P Spec R. Let Q V (P ), so that Q is a prime ideal,containing P . For every{ } ideal I∈such that P V∈(I), we ∈ have I P Q, so that Q V (I). Hence V (P ) P . Conversely, V (P ) is a ⊂ ⊂ ∈ ⊂ { } closed set containing P , hence P V (P ). This proves that P = V (P ). The point P is characterized by being{ } the ⊂ unique point whose closure{ } is V (P ). P is called the generic point of the closed set V (P ). For example, if R is an integral domain, then (0) is a prime ideal. In this case, (0) = Spec R, so that (0) is the generic point of the whole space. More generally,{ the} same conclusion holds for (0) in a ring in which (0) is prime. p Remark 2.2 proves that,p given P Spec R, the one-point set P is closed if and only if P is maximal. In particular,∈ Spec R is not a Hausdorff{ space,} unless every prime ideal of R is maximal. In the category of algebraic varieties, one had the notion of a morphism between two varieties X and Y , which is, by definition, a polynomial map in several variables. We saw that each such morphism f induces a dual homomorphism f ∗ from the coordinate ring of Y to the coordinate ring of X. In scheme theory, the situation is analogous, except, since our basic objects are rings, we start with a homomorphism of rings A B and define the induced morphism Spec B Spec A. → → Let σ : A B be a homomorphism of rings. Define the map σ∗ : Spec B Spec A by σ∗→(P ) = P A σ−1(P ) for P Spec B. Maps Spec B Spec →A which arise in this way are∩ called≡ morphisms∈ of affine schemes. It is→ immediate from definitions that morphisms are continuous with respect to Zariski topology. Two kinds of morphisms are of particular importance in geometry: open and closed embeddings. Open embeddings correspond to localization of rings, which will be the subject of the next section. Ring homomorphisms corresponding to closed embeddings are the surjective homomorphisms, which we will discuss right now. Let I be an ideal of A. Consider A ∗ A the natural homomorphism σ : A I . Then σ : Spec I ֒ Spec A is injective ∗ ∗ → → A and Im(σ ) = V (I). Also, σ is a closed map, for if J is an ideal of I , then ∗ ∗ A σ (V (J)) = V (J A). Thus, σ is a closed embedding from Spec I into Spec A, ∗ ∩ ∗ A i.e., σ is a homeomorphism; σ : Spec I V (I) (with the topology on V (I) induced from Spec A). This shows that the→ closed subset V (I) of Spec A itself has the structure of an affine scheme. We say that V (I) is a closed subscheme INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 9

A of Spec A. I is called the coordinate ring of V (I). As we switch from naive algebraic geometry to this new and more sophisticated view of life, kn is replaced in our minds by Spec k[x1,...,xn] and the algebraic subvariety V (I) by the closed subscheme Spec k[x1,...,xn] Spec k[x ,...,x ]. I ⊂ 1 n Geometrically, prime ideals correspond to irreducible varieties. The point of scheme theory is that we agree to regard irreducible subvarieties themselves as points. One of the advantages of doing so is that it allows to make rigorous all the informal classical arguments of the Italian geometers of the early twentieth century which began with the words “Take a sufficiently general point on the algebraic variety...”. In scheme theory, the canonically defined generic point P takes the place of the intuitive notion of choosing “a generic point” on the irreducible variety V (P ). Apart from rigor, there are other important advantages that the language of schemes has over the classical language of algebraic varieties. The fact that we can do geometry over arbitrary rings, rather than just quotients of k[x1,...,xn] means that we can work over fields of positive characteristic and even over Z itself, which is very important for number theory. In fact, most of the recent advances in number theory would be inconceivable without the use of the geometric language of schemes. For instance, a large part of Faltings’ famous proof of Mordell’s conjecture consists of generalizing the classical theory of algebraic surfaces to arithmetic surfaces, i.e. Z[x,y] surfaces of the form Spec (f(x,y)) . Along with rings over Z, we can consider complex analytic local rings, formal power series rings (which appear naturally in studying local geometry and singularities), rings over the p-adic numbers, etc. Scheme theory provides us with a uniform language for doing geometry in all these different contexts, local and global. Another great advantage of schemes over varieties is the presence of nilpotent structure, which makes the theory much richer. Namely, in classical algebraic geometry, given a non-radical ideal I k[x ,...,x ], one did not distinguish between ⊂ 1 n V (I) with V (√I); indeed, they are identical as topological spaces. In scheme theory, however, they are regarded as different closed subschemes of Spec k[x1,...,xn] by definition. More generally, if R is a ring with nilpotent elements, Spec R and Spec Rred are identical as topological spaces, but are different as schemes. This k[x,y] opens interesting geometric possibilities. For example, Spec (x2) is a double line: it is the y-axis counted twice. We also have a completely new notion of embedded points (which will be discussed in more detail later on, in the section on primary k[x,y] decomposition). For instance, the scheme Spec (x2,xy) is usually thought of as the union of the y-axis with the origin (a meaningless notion in classical algebraic geometry), because (x2,xy)=(x) (x,y)2. In this case, we say that (x,y) is an ∩ k[x,y] embedded point of the scheme Spec (x2,xy) . Even if one is only interested in the usual, reduced algebraic varieties, such non-reduced objects arise in a natural way as limits of families of varieties under deformations. To summarize the nature of the generalization from varieties to schemes, we give 10 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

a definition of finitely generated algebras and morphisms of finite type. Let A be a ring. An A-algebra is a ring B together with a fixed homomorphism σ : A B. The homomorphism σ is sometimes called the structure homomorphism→ for the A-algebra B. An A-algebra homomorphism is a ring homomorphism B C between two A-algebras, which respects the structure homomorphisms. →

Definition 2.3. Let B be an A-algebra and x = xλ λ∈Λ a subset of B. Let A¯ B denote the image of A under the structure homomorphism.{ } The A-subalgebra A⊂¯[x] of B, generated by x, is defined to be the smallest A-subalgebra of B, containing x. In other words, A¯[x] is the set of all elements of B which can be written as multivariable polynomials in the xλ with coefficients in A¯. We say that B is a finitely generated A-algebra, or that B is of finite type over A, if there is a finite subset x B such that B = A¯[x]. A morphism Spec B Spec A is said to be of finite type⊂ if B is finitely generated over A. →

Remark 2.4. B is finitely generated over A if and only if B is isomorphic to A[x1,...,xn] an A-algebra of the form I , where x1,...,xn are independent variables, A[x1,...,xn] denotes the ring of polynomials in n variables with coefficients in A and I is an ideal of A[x1,...,xn]. Coming back to varieties and schemes, we can summarize the situation as follows. Initially, we generalized the naive theory of algebraic varieties in two ways. First, we enlarged our notion of points by agreeing to consider irreducible subvarieties themselves as points. Secondly, we agreed to distinguish between two varieties having different nilpotent structures, even though they may be identical as topological spaces. The effect of these two generalizations is that of replacing the category of algebraic varieties over a field k by the category of affine schemes which are of finite type over k. Rings which are of finite type over some field still form a rather restricted class (for instance, none of the rings listed in the beginning of the first lecture are of this type, except k[x1,...,xn]). Our last generalization consisted in allowing arbitrary rings instead of limiting ourselves to finitely generated algebras over a field.

Deﬁnition 2.5. An aﬃne scheme Spec A is said to be reduced if A is reduced.

Remark 2.6. Just as in the case of algebraic varieties, prime ideals correspond to irreducible closed subschemes. More precisely, let A be a ring and I a radical ideal. A Then by Proposition 1.20, Spec I is irreducible if and only if I is prime. Another way of stating the same fact is that for a reduced ring A, Spec A is an irreducible topological space if and only if (0) is a prime ideal of A if and only if A is an integral domain. In this case, we say that Spec A is an integral scheme. An important object associated with an integral scheme is its ﬁeld of rational functions (which is the analogue of the ﬁeld of meromorphic functions on a complex manifold):

Definition 2.7 Let A be an integral domain. The field of fractions K of A is INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 11 the set of equivalence classes of fractions a K = a,b A, b =0 / b ∈ 6 ∼ n o a c where is the obvious equivalence relation b d ad = bc. K is also called the field∼ of rational functions of the affine∼ scheme⇐⇒ Spec A. If P Spec A, the A ∈ field of fractions of P is called the residue field of P , sometimes denoted by κ(P ). κ(P ) is nothing but the field of fractions of the integral scheme V (P ). We end this section by mentioning the semi-classical precursor of the modern notion of the generic point of an integral subscheme, in order to put things in a historical perspective. Let X be an affine subvariety of kn, where k is a field. Assume X is irreducible, k[x1,...,xn] so that X = V (P ), where P is a prime ideal of k[x1,...,xn]. Let A = P and let K be the field of fractions of A. We have k K, so that k[x1,...,xn] ′ ′ ⊂ ′ n ⊂ K[x1,...,xn]. Let P = PK[x1,...,xn]. Let X = V (P ) K . There exists a ′ ⊂ point ξ X such that for any f k[x1,...,xn], f(ξ) = 0 implies that f P . Such a point∈ ξ is called a k-generic∈ point of X and used to play the role which∈ the modern notion of generic point plays today. One advantage of the modern approach is that the generic point of an irreducible subscheme is canonically defined, while the k-generic point ξ above is not, in general, unique.

3. Localization. § By definition, we have a one-to-one correspondence between morphisms of affine schemes and homomorphisms of rings. We saw in 2 that closed embeddings of affine schemes correspond to surjective homomorphisms§ of rings. In this section, we consider the case of open embeddings, which correspond, appropriately, to the operation of localization in rings. Definition 3.1. Let A be a ring. A subset S A 0 is said to be multiplicative if 1 S and S is closed under multiplication. ⊂ \{ } ∈ Examples: (1) Let f A (0). Then 1,f,f 2,f 3,... is multiplicative. (2) If P ∈Spec\ A, then A P{ is a multiplicative} set. ∈ p \ Definition 3.2 Let A be a ring and S A a multiplicative subset. The localization of A at S is defined to be A =⊆ a a A, s S / , where is S { s | ∈ ∈ } ∼ ∼ the following equivalence relation: a b whenever there exists u S such that s ∼ t ∈ u(at bs) = 0. It is clear that AS is a ring with the operations of addition and − 0 1 multiplication, well known from elementary school. 1 is its zero element and 1 the unit element. Let A be a ring, S A a multiplicative subset. We have the natural homomor- ⊂ a phism σ : A AS, given by a 1 . σ is injective if A is an integral domain, but not in general.→ We have Ker σ 7→= x A sx = 0 for some s S . { ∈ | ∈ } 12 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

An element x in a ring A is said to be invertible,ora unit, if there exists y A ∈ such that xy = 1. Then every element of σ(S) is invertible in AS. Examples: 2 3 (1) Let A be a ring, f A. Let S = 1,f,f ,f ,... . We write Af for AS. Af is the ring consisiting∈ of all the fractions{ whose} denominator is a power of f. (2) Let A be a ring and P Spec A. Let S = A P . By abuse of notation, ∈ \ which has become completely standard, we will write AP for AA\P . Deﬁnition 3.3. A ring A is local if A has only one maximal ideal.

A If A is a local ring with maximal ideal m, the field k = m is called the residue field of A. We will sometimes denote such an A by (A, m) or (A,m,k) to indicate that m is the maximal ideal and k the residue field. Remark 3.4. Let A be a ring and m A and ideal. A is local with the maximal ideal m if and only if every element of A⊂ m is invertible. Indeed, suppose (A, m) is local and let a A m. Suppose a is not\ invertible. Then (a) is a proper ideal of A. By Zorn’s∈ lemma,\ (a) is contained in a maximal ideal, which must be m since m is the only maximal ideal of A. This means that a m, a contradiction. Conversely, suppose every element of A m is invertible, but∈ m is not maximal. \ Then there exists m′ Max(A) such that m $ m′ $ A. There exists a m′ m. By assumption a is invertible.∈ Then (a) = A and so m′ (a) = A, a contradiction.∈ \ ⊃ Remark 3.5. Let A be a ring and P Spec A. Then A is a local ring with the ∈ P maximal ideal PAP . Remark 3.6. Let A be a ring, S A a multiplicative set, and consider the natural homomorphism ⊂ σ : A A . → S Then

(3.1) σ∗(Spec A ) = P Spec A P S = . S { ∈ | ∩ ∅} ∗ −1 Indeed, let P Spec AS. We must show that σ (P ) S = σ (P ) S = . If a σ−1(P ) S∈, then a P and 1 A , hence 1 P and∩ P = A , a contradiction∩ ∅ ∈ ∩ 1 ∈ a ∈ S ∈ S because P is a prime ideal of AS. ∗ Exercise 3.7. Show the opposite inclusion σ (Spec AS) P Spec A P S = in (3.1). Also, show that σ∗ is injective and hence induces⊃{ ∈ a bijection| between∩ Spec∅} A and P Spec A P S = . S { ∈ | ∩ ∅} σ∗ is called the restriction map from Spec A to the subset P Spec A P S = . In the case of algebraic varieties, σ∗ corresponds to restricting{ ∈ a polynomial| ∩ (i.e.∅} an element of A) to the subset where no element of S vanishes. 2 ∗ For example, let f A (0), S = 1,f,f ,... . Consider σ : Spec Af Spec A. Then σ∗ induces∈ a\ bijection between{ Spec A} and Spec A V (f). −→ p f \ INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 13

Let I A be an ideal. Let fλ λ∈Λ be a set of generators of I. Let P Spec A. Then I ⊆P if and only if f { P }for all λ Λ. ∈ ⊂ λ ∈ ∈ Let U = Spec A V (f ) = Spec A . Then Spec A V (I) = P Spec A I λ \ λ ∼ fλ \ { ∈ | 6⊂ P = P Spec A fλ / P for some λ Λ = (Spec A V (fλ)) = Uλ. } { ∈ | ∈ ∈ } λ∈Λ \ λ∈Λ Thus the open set Spec A V (I) is a union of theS open sets Uλ = SpecS A V (f ) Spec A . Since the\ ideal I was arbitrary, we see that the collection\ λ ∼= fλ Spec A V (f) f A form a basis for the Zariski topology in Spec A. We will { \ | ∈ } sometimes refer to the sets Spec Af as the basic open sets of Spec A. Remark 3.8. If A is Noetherian, we may take Λ above to be finite. Hence in that case any open set is a finite union of basic open sets. We end this section by giving another point of view on localizations in terms of inductive limits. Definition 3.9. Let Λ be a partially ordered set. We say that Λ is a directed set if for any i, j Λ, there exists l Λ such that i l and j l. ∈ ∈ ≤ ≤ Definition 3.10. Let Aλ λ∈Λ be a collection of sets, indexed by a directed set Λ. Suppose that for any{ i, j} Λ with i j we are given a map µ : A A such ∈ ≤ ij i → j that µii = id for all i Λ and µjl µij = µil for all triples i, j, l Λ such that l j i. A is called∈ a direct◦ system or an inductive system∈ . ≥ ≥ { λ}λ∈Λ Let Bλ λ∈Λ be another collection of sets indexed by Λ. Suppose that for any i, j Λ{ with} i j we are given a map φ : B B such that φ = id for all i Λ ∈ ≥ ij i → j ii ∈ and φjl φij = φil for all triples i, j, l Λ such that l j i. Bλ λ∈Λ is called an inverse◦ system or a projective system∈ . ≤ ≤ { }

Deﬁnition 3.11. Let Aλ λ∈Λ be a direct system. The direct limit lim→ Aλ is { } λ

deﬁned to be lim→ Aλ = Aλ/ , where is the following equivalence relation. λ λ∈Λ ∼ ∼ Given xi Ai, xj Aj,` we say that xi xj if there exists l, l i, j such that ∈ ∈ ∼ ≥ µil(xi) = µjl(xj). The inverse, or projective limit lim Bλ is deﬁned to be ←

lim Bλ = xλ λ∈Λ Bλ φij (xi) = xj for all i, j such that i j . ← ({ } ∈ ≥ ) λ∈Λ Y Exercise 3.12. Show that lim Aλ and lim Bλ are characterized by the following → ← λ universal mapping properties. Let A = lim→ Aλ. Then there are natural maps πi : λ ′ Ai A, i Λ, compatible with all the µij. Moreover, given any other set A and a → ∈ ′ ′ collection of maps πi : Ai A compatible with the µij, there exists a unique map A A′, compatible with all→ the π and π′. → i i Similarly, let B = lim← Bλ. Then there are natural maps πi : B Bi, i Λ, λ → ∈ ′ compatible with all the φij . Moreover, given any other set B and a collection of 14 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

′ ′ ′ maps πi : B Bi compatible with the φij, there exists a unique map B B, → ′ → compatible with all the πi and πi.

If the sets Aλ and Bλ are endowed with an additional structure (such as rings, groups, etc.), this structure usually extends in an obvious way to projective and inductive limits. We will take this fact for granted without mentioning it explicitly. Exercise 3.13. Let A be a ring, S A a multiplicative subset. Deﬁne a partial order in S by saying that f g if there⊂ exists h S such that g = fh. Then S is ≤ ∈ a directed set. Show that the localizations Af form a directed system and that the basic open sets Uf = Spec Af form a projective system. Show that

(3.2) lim A = A . → f S f∈S

In particular, if P Spec A, AP = lim Af . ∈ → f/∈P By passing to Specs and reversing all the arrows in (3.2), obtain

lim Spec A = Spec A ← f S f∈S

(as sets).

4. A proof of Hilbert’s Nullstellensatz. § In this section, we give a proof of the various versions of the Nullstellensatz. An injective ring homomorphism is called a ring extension. Let σ : A֒ B be an extension of rings. Let x B. → ∈ Deﬁnition 4.1. We say that x is algebraic over A, if it satisﬁes a polynomial equation of the form

(4.1) a xn + a xn−1 + + a x + a =0, n n−1 ··· 1 0

with ai A. x is integral, or finite over A if, in addition, we may take an = 1 in (4.1).∈ If x is not algebraic, it is said to be transcendental. We say that σ is algebraic, or that B is algebraic over A, if every x B is algebraic. σ is integral if every x B is integral over A. Extensions which∈ are not algebraic are called transcendental∈ . Note that if A is a field then “integral over A” and “algebraic over A” are the same thing. Proposition 4.2 (an intermediate version of the Nullstellensatz). Let k be a field and A a finitely generated k-algebra. Then A is a domain, algebraic over k, if and only if A is a field. INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 15

First, let us assume Proposition 4.2 and show that it implies the two other versions of Nullstellensatz which were stated previously. n Proof of Theorem 1.16. It is obvious that for every a = (a1,...,an) k , (x1 ∈ ′ −n a1,...,xn an) is a maximal ideal of k[x1,...,xn], and that distinct points a,a k give rise to− distinct maximal ideals. It remains to show that every maximal ideal∈ is n of the form (x1 a1,...,xn an), a k . Take any m Max(k[x1,...,xn]). Then k[x1,...,xn] − − ∈ ∈ m is a field, which is a finitely generated k-algebra (since it is generated k[x1,...,xn] by the images of the xi). By Proposition 4.2, m is algebraic over k, hence equals k (k being algebraically closed). Then for each i, 1 i n, there exists ≤ ≤ ai k such that xi ai mod m. Then m (x1 a1,...,xn an), hence m = (x ∈ a ,...,x a ),≡ as desired. ⊃ − − 1 − 1 n − n To prove Theorem 1.7, we state and prove the following more general result, which holds without the assumption that k is algebraically closed. Proposition 4.3. Let k be any field and R a finitely generated k-algebra. For an ideal I R, let V˜ (I) = m Max(R) I m . For a subset V Max(R), let ⊂ { ∈ | ⊂ } ⊂ I(V ) = f R f m for all m V . Then for any ideal I, I(V˜ (I)) = √I. { ∈ | ∈ ∈ } By Theorem 1.16, Proposition 4.3 implies Theorem 1.7. Proof of Proposition 4.3 (assuming Proposition 4.2). Clearly, √I I(V˜ (I)). We must prove the opposite inclusion. ⊂ √ R R 1 Take f / I. We may consider B = ( I )f = I [ f ] (where by abuse of notation ∈ R we identify f with its natural image in I ). We have a natural map σ : R B. Take any m Max(B). Then I m R and f / m R. If we can show→ that ∈ ⊂ ∩ ∈ ∩ m R is maximal, we will have m R V˜ (I) and hence f / I(V˜ (I)), as desired. ∩ ∩ ∈ ∈ B R To see that m R is maximal, apply Proposition 4.2 to m and m∩R . B is a ∩ 1 finitely generated k-algebra (it is generated by the generators of R and f ). By Proposition 4.2, we have m maximal = B is a field = B is algebraic over ⇒ m ⇒ m k = R isa domain, algebraic over k (since R m is prime and R B ) ⇒ R∩m ∩ R∩m ⊂ m = R is a field = R m is a maximal ideal of R. ⇒ R∩m ⇒ ∩ To complete the proof of Proposition 4.3, and hence of Theorem 1.7, it remains to prove Proposition 4.2. Proof of Proposition 4.2. First, suppose A is a domain, algebraic over k. Take any x A. We must show that x is invertible in A. Since A is algebraic, x satisfies an ∈ n equation of the form anx + + a0 = 0, with ai k. Since A is a domain, we ··· 1 n−1 ∈i may assume that a0 = 0. Then x( ) ai+1x = 1, which proves that x is 6 − a0 i=0 invertible. Conversely, suppose A is a field, whichP is not algebraic over k. Since A is finitely generated, we may write A = k[x ,...,x ] for some x ,...,x A. We can 1 n 1 n ∈ renumber the xi such that x1,...,xr are algebraically independent over k and A is algebraic over k[x1,...,xr]. Replacing k by the field k(x1,...,xr−1) does not change 16 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

the problem, so that we may assume r = 1. For 2 i n, let b k[x ] be the ≤ ≤ i ∈ 1 leading coeﬃcient of a polynomial equation over k[x1], satisﬁed by xi. Then all the

xi are integral over k[x1]b2...bn . At this point, we need the following basic fact about integral extensions: Lemma 4.4. Let A B be a ﬁnitely generated ring extension. If each of the generators is integral over⊂ A, then B is integral over A.

First, we ﬁnish the proof, assuming Lemma 4.4. Since each of the xi is integral

over k[x1]b2...bn , by Lemma 4.4 A is integral over k[x1]b2...bn . Since A is a field, it contains k(x1), so that k(x1) is integral over k[x1]b2...bn . Definition 4.5. Let A B be a ring extension. The integral closure of A in B is the A-subalgebra A¯ of→B consisting of all the elements of B, integral over A (the fact that A¯ is a subalgebra follows from Lemma 4.4). A is said to be integrally closed in B, if A = A¯. If A is a domain, we will say that it is integrally closed or normal if A is integrally closed in its field of fractions. Definition 4.6. A domain A is said to be a unique factorization domain, or UFD, if every element of A can be written uniquely (up to multiplication by units of A) as a product of irreducible elements.

Let b = x1b2 ...bn + 1, so that deg b 1 and b is relatively prime to b2 ...bn. 1 ≥ We have k[x1]b ...bn $ k(x1), because k(x1) k[x1]b ...bn . At the same time, 2 b ∈ \ 2 k(x1), which is the ﬁeld of fractions of k[x1]b2...bn , is integral over k[x1]b2...bn . To

get a contradiction, it is suﬃcient to show that k[x1]b2...bn is normal. Now, it is well known that k[x1] is a UFD (this is proved by using Euclidean algorithm, in other

words, division with remainder). Thus the normality of k[x1]b2...bn follows from two facts, which we leave to the reader as easy exercises:

Exercise 4.7. If B is a UFD and S a multiplicative subset of B, then BS is a UFD. Exercise 4.8. Any UFD is normal. To complete the proofs of all the Nullstellensatzes, it remains to prove Lemma 4.4. To do this, we introduce the concept of a module over a ring. Deﬁnition 4.9. Let A be a ring. An A-module is an abelian group M together with an operation A M M (multiplication by elements of A), such that for all a,b A, x,y M, we× have:→ ∈ ∈ (1) a(bx)=(ab)x (2) (a + b)x = ax + bx (3) a(x + y) = ax + ay (4) 1x = x.

Any ideal of A is an A-module. Any A-algebra is an A-module. INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 17

Definition 4.10. Let A be a ring and M an A-module. An A-submodule of M is an abelian subgroup of M, which is an A-module under the operations inherited from M. For example, the A-submodules of A are precisely the ideals of A. Definition 4.11. Let M be an A-module and x = x a subset of M. The { λ}λ∈Λ A-submodule B, generated by x (sometimes denoted by λ∈Λ Axλ), is defined to be the smallest A-submodule of M, containing x. In other words, λ∈Λ Axλ is the set of all elements of M which can be written as A-linearP combinations of the P xλ. We say that M is a finitely generated or a finite A-module if there is a finite n subset x1,...,xn M such that M = i=1 Axi. An A-algebra B is said to be finite over{ A if B}is ⊂ a finite A-module (of course, this is stronger than saying that B is a finitely generated A-algebra). P Exercise 4.12. Let A B C be ring extensions. If B is finite over A and C finite over B, then C is⊂ finite⊂ over A. Now we are in the position to prove Lemma 4.4. We prove Lemma 4.4 in the following stronger form. Lemma 4.13. Let A֒ B be an extension of rings. The following conditions are equivalent. → (1) B is finite over A. (2) B is integral and finitely generated over A (as an algebra). (3) B is finitely generated over A as an algebra, and each of the generators is integral over A.

Proof. (2) = (3) is trivial. ⇒ (1) = (2). Assume that B is finite over A. Clearly, B is finitely generated as ⇒ an A-algebra. Let x1,...,xn be a set of generators of B as an A-module. Take any y B; we want to show that y is integral over A. There exist aij A, 1 i, j n, such∈ that yx = n a x . Let T denote the n n matrix with∈ entries≤a , I≤the i j=1 ij j × ij n n identity matrix. Consider the n n matrix yI T with entries in B. Let x denote× the columnPn-vector with entries×x . By construction,− (yI T )x = 0. Hence, i − by linear algebra, det(yI T )xi =0 for 1 i n. Since the xi generate B as an A-module, det(yI T )x −= 0 for every x ≤B,≤ so that det(yI T ) = 0. We have constructed a monic− polynomial equation∈ satisfied by y over A−, which proves that y is integral over A. (3) = (1) Suppose B is finitely generated, with generators integral over A. Since a composition⇒ of finite extensions is finite (Exercise 4.12), we may assume that, as an algebra, B is generated over A by a single element x, integral over A. This means that as an A-module, B is generated by 1,x,x2,...,xn,... . By assumption, n n−1 n x satisfies a monic equation x + an−1x + + a0 = 0 over A. Hence x belongs to the A-submodule of B generated by 1,x2,...,x··· n−1. Then by induction on l, any xl with l n belongs to the A-submodule of B generated by 1,x2,...,xn−1. Hence ≥ 18 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

B is generated by 1,x2,...,xn−1 as an A-module, which proves that B is ﬁnite over A. This completes the proof of Lemma 4.13, Propositions 4.2 and 4.3 and Theorem 1.7.

5. Primary decomposition. § Definition 5.1. A topological space X is said to be Noetherian if any descending chain of closed sets in X stabilizes. Exercise 5.2. Show that in a Noetherian topological space every closed set can be written uniquely as a finite union of irreducible closed sets. Clearly, if R is a Noetherian ring, then the affine scheme Spec R is Noetherian. Let I be a radical ideal of R. We saw earlier (Proposition 1.20) that the reduced closed subscheme V (I) Spec R is irreducible if and only if I is prime. Hence, Exercise 5.2 applied to Spec⊂ R is equivalent to saying that in a Noetherian ring every radical ideal can be written uniquely as an intersection of primes (the uniqueness follows easily from Lemma 1.21). Since in scheme theory we are interested in including the case of non-reduced schemes, we would like to extend this result to ideals of R which are not necessarily radical. The corresponding result in the non-radical case is called primary decomposition. First, we note that the elementary argument of Exercise 5.2 applies verbatim to show that every ideal can be written as an intersection (not necessarily unique) of irreducible ideals: Definition 5.3. Let I be an ideal of A. I is irreducible if it cannot be written in the form I = I I , where I and I properly contain I. 1 ∩ 2 1 2 Lemma 5.4. Let A be a Noetherian ring. Then every ideal of A can be written as a finite intersection of irreducible ideals in A. Proof. Let denote the collection of all the proper ideals of A which cannot be expressed as an intersection of finitely many irreducible ideals of A. We want to show that ΣP = . Suppose not. Since A is Noetherian, Σ contains a maximal element I. Since∅I Σ, I is not irreducible. Then I can be written as I = I I ∈ 1 ∩ 2 for some ideals I1, I2 of A such that I $ I1 and I $ I2. By the maximality of I, we have I1,I2 / . Hence both I1 and I2 can be expressed as finite intersections of irreducible∈ ideals of A and hence so can I. Then I / Σ, a contradiction. Therefore = . P ∈ ∅ P The point of primary decomposition is to give a more explicit description of irreducible ideals and to relate the irreducible ideals appearing in a decomposition A of I to the set of zero divisors in I . Definition 5.5. Let A be a ring and Q $ A an ideal. Q is primary if for any x,y A such that xy Q, either x Q or y √Q. ∈ ∈ ∈ ∈ INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 19

Note that in the above definition we think of x,y as an ordered pair. The definition is not symmetric in x and y. Remark 5.6. Let Q be a primary ideal of A. Then P = √Q is prime. Indeed, xy P = (xy)n Q for some n N = xn Q or (yn)m Q for some ∈ ⇒ ∈ ∈ ⇒ ∈ ∈ m N. Thus xn Q or ynm Q, so that either x P or y P . This proves that P is∈ prime. We say∈ that Q is∈ a P -primary ideal or∈ that it is∈ primary to P . Let Q be an ideal such that √Q = P is prime. Then Q is primary if and only if Q = QA A. In particular, the intersection of two P -primary ideals is P -primary. P ∩ Next, we show that every irreducible ideal is primary, so that every ideal in a Noetherian ring is a finite intersection of primary ideals. In the rest of this section, we will discuss to what extent this primary decomposition is unique. Exercise 5.7. Let m Max(A). Show that an ideal Q is m-primary if and only if √Q = m. ∈ Lemma 5.8. Let A be a Noetherian ring, I A an irreducilbe ideal. Then I is primary. ⊂ Proof. Suppose ab I and b / I. Let I : a = x A ax I . I : a is an ideal. I : a I : a2 (∈I : a3) ...∈ is an ascending{ sequence∈ | of ideals∈ } in A. Since ⊂ ⊂ ⊂ A is Noetherian, there exists n N such that I : an = I : an+i for all i 1. ∈ ≥ Now, I (I +(an)) (I +(b)). Conversely, let r (I +(an)) (I +(b)). Then r = g + can⊂= h + db where∩ g, h I and c, d A. So∈ra = ga + ca∩n+1 = ha + dab, so that g,h,ab I. Then c (I∈: an+1)=(∈I : an), so that r = g + can I. We obtain I =(I +(∈an)) (I +(∈b)). ∈ ∩ Now, I is irreducible and I $ I +(b) because b / I. Hence I = I +(an), so ∈ an I. We have proved that a √I. Therefore I is primary, as desired. ∈ ∈ Corollary 5.9. Let A be a Noetherian ring, I A an ideal. There exists a decomposition ⊂

(5.1) I = Qi, i=1 \ where each Qi is primary. Example 5.10. Primary decomposition is not unique. Let A = k[x,y]. We have (x2,xy)=(x) (y,x2)=(x) (y,x)2, which are two diﬀerent primary decomposition of the same ideal.∩ ∩ We will now further analyze primary decomposition of a given ideal. Deﬁnition 5.11. Let A be a ring, I A an ideal. Consider a decomposition n ⊂ I = Qi where each Qi is primary. Let Pi = √Qi; by Remark 5.6, the Pi are i=1 prime.T The Pi are called associated primes of I. The minimal ones among the Pi are called the minimal primes of I. Associated primes which are not minimal are called the embedded components or embedded points of I. Sometimes the 20 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

A Pi are also referred to as the associated (resp. minimal) primes of the A-module I . In particular, the associated (resp. minimal) primes of (0) are called the associated (resp. minimal) primes of A. This terminology seems ambiguous, but usually it causes no confusion. Note 5.12. By Lemma 1.21, the minimal primes of I are precisely the minimal elements of V (I) under inclusion, so that the name is appropriate. For instance, in Example 5.10, (x) and (x,y) are the associated primes, (x) is the only minimal prime and (x,y) is an embedded point. In this example, we see that the collection of associated primes depends only on the ideal I and not on the choice of the primary decomposition. We will now show that this is true in general by describing the Pi in terms of zero divisors mod I. Lemma 5.13. Let Q A be a primary ideal, P = √Q, a an element of A. ⊆ (1) If a Q, then (Q : a) = A. ∈ (2) If a P Q, then Q $ (Q : a) and Q : a is P -primary. (3) If a∈ / P ,\ then (Q : a) = Q. ∈ Proof. (1) is obvious. We always have Q (Q : a) because Q is an ideal. ⊂ (3) Let b (Q : a); then ab Q. Since a / P = √Q, b Q. This proves that (Q : a) = Q.∈ ∈ ∈ ∈ (2) First, we show that (Q : a) P . Indeed, take b (Q : a); then ab Q. Since a / Q and Q is primary, we have ⊂b P . This proves that∈ (Q : a) P . ∈ ∈ ∈ ⊂ Now, P = √Q (Q : a) √P = P (the last equality holds since P is prime), ⊆ ⊆ therefore (Q : a) = P . p Now, take b, c A such that bc (Q : a) and b / (Q : a) = P . Then abc Q. p ∈ ∈ ∈ ∈ Since b / P = √Q and Q is primary, ac Q. Hence c (Q : a). This proves that p (Q : a) is∈ primary. ∈ ∈ It remains to show that Q : a = Q. Indeed, since a P Q, there exists a unique 6 ∈ \ n N such that an / Q but an+1 Q. Then an (Q : a) Q. This completes the proof∈ of Lemma 5.13.∈ ∈ ∈ \ We will now use Lemma 5.13 to interpret primary decomposition in terms of the set of zero divisors mod I. Theorem 5.14. Let I be an ideal in a Noetherian ring A. Consider a primary decomposition (5.1). Let P = √Q , 1 i n. Then: i i ≤ ≤ (1) P1,...,Pn are precisely all the prime ideals which are of the form I : a for some a A. (2) Let P¯ =∈ P A , 1 i n. Then the set of zero divisors in the ring A is i i I ≤ ≤ I n P¯ . ∪i=1 i In particular, the set of zero divisors in A is precisely the union of the associated primes of A (recall that nil(A) is the intersection of the associated primes of A). INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 21

Corollary 5.15. The collection Pi 1≤i≤n depends only on I and not on the choice of primary decomposition. { } If A is a ring and a A, the annihilator of a is the ideal Ann a = (0) : a. Thus another way of stating∈ Theorem 5.14 is to say that the associated primes of A are precisely all the prime ideals of the form Ann a for a A. ∈ Proof of Theorem 5.14. (1) We may assume that for all i = 1, 2,...,n, Qi n n 6⊃ Qj. Take any i, 1 i n and any a Qj Qi. We have j=1 ≤ ≤ ∈ j=1 \ j=6 i j=6 i T T

n n (5.2) (I : a) = Q : a = (Q : a) = Q : a P ,  j j i ⊂ i j\=1 j\=1   where the last two statements hold by Lemma 5.13. Let P˜i denote a maximal n element among all the ideals of the form I : a, where a Qj Qi. By (5.2) and ∈ j=1 \ j=6 i Lemma 5.13 (2),(3), T

(5.3) Pi = P˜i. q We want to show that P˜i = Pi, that is, all the Pi are of the form (I : a) for some n a A. Suppose not. Let a Qj Qi be such that P˜i = I : a and take b Pi P˜i. ∈ ∈ j=1 \ ∈ \ j=6 i T l ˜ l+1 ˜ l ˜ By (5.3), there exists l N such that b / Pi but b Pi. Since b / Pi = I : a, n ∈ ∈ ∈ ∈ l ab / I. Since a Qj, ∈ ∈ j=1 j=6 i T n l (5.4) ab Qj. ∈ j=1 j\=6 i

n l l l+1 Combined with ab / I, (5.4) gives ab Qj Qi. On the other hand, b a I ∈ ∈ j=1 \ ∈ j=6 i ˜ Tl by the choice of l. Then Pi = I : a $ I :(ab ), where the inclusion is strict because b I : (abl) P˜ . This contradicts the maximality of P˜ . We have proved that all ∈ \ i i the Pi, 1 i n are of the form I : a for some a A. Conversely,≤ ≤ let P Spec A be of the form P ∈= I : a for some a A. Then n ∈ ∈ P = (Qj : a). Since prime ideals are irreducible (Proposition 1.20), we must j=1 have PT= Q : a for some i, 1 i n. Hence Q : a = A, so that, by Lemma 5.13, i ≤ ≤ i 6 22 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

(Qi : a) = Pi. We obtain P = √P = (Qi : a) = Pi. This completes the proof of (1). p p A ¯ (2) We may replace A by I , I by (0) and Pi by Pi. In particular, we take the n Qi to be primary components of the zero ideal: (0) = i=1Qi. By (1), each of the ∩ n Pi is the annihilator ideal of some a A, hence every element in i=1Pi is a zero divisor. ∈ ∪ n Conversely, take any a A i=1Pi. Then by Lemma 5.13 (3), Qi : a = Qi ∈ \ ∪ n n for 1 i n. We have Ann a (0) : a = i=1(Qi : a) = i=1Qi = (0). Thus ≤ n≤ ≡ ∩ ∩ a A i=1Pi implies that a is not a zero divisor, as desired. This completes the proof∈ of\ ∪ Theorem 5.14.

6. Dimension Theory. § Continuing with our dictionary between algebra and geometry, we come to the concept of dimension of an algebraic variety or scheme. Intuitively, we all know what is meant by dimension in geometry; for example, everybody knows that the n-space kn has dimension n. Therefore, it is natural to look for the corresponding definition on the algebraic side: that of the dimension of a ring. There are several alternative ways of thinking about dimension; the purpose of this section is to describe their algebraic interpretation and to prove that all give the same answer. The first one is the maximal length of a descending chain of irreducible subvarieties. For example, a maximal chain of irreducible subvarieties in kn is kn kn−1 k2 k 0 ; it has length n (not counting kn itself). We model our⊃ definition⊃···⊃ of the⊃ dimension⊃{ } of a ring on this geometric idea. The second approach is to consider the number of elements in a local coordinate system on X. More precisely, since we want to include singular varieties, we should consider the smallest number of equations, whose set of common zeroes in X consists of isolated points. This is the idea behind the integer δ(A), associated to a local ring A later in this section. A third way of defining dimension, of great importance both in theory and in computations, is the degree of the Hilbert-Samuel polynomial. Finally, in the case of an algebraic variety X over a field k, the dimension equals the largest transcendence degree of the field of rational functions on an irreducible component of X. We now give precise definitions and prove the equality of all four numbers. Definition 6.1. Let A be a ring. The Krull dimension of A, denoted dim A, is the maximal length of an ascending chain of prime ideals in A:

(6.1) P0 $ P1 $ $ Pd ··· not counting the minimal prime P0. In other words, if (6.1) is the longest, we say that dim A = d. If there are arbitrarily long chains of primes in A, we say that dim A = . For P Spec A, the height of P is deﬁned by ht P = dim AP (which is the∞ maximal∈ length of a chain of prime ideals contained in P ). INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 23

Exercise 6.2. Let A be a ring. Prove that dim A = max dim A m Max(A) . { m | ∈ } (Hint: first, show that for any P Spec A, the natural map Spec AP Spec A induces a bijection between Spec A∈ and Q Spec A Q P .) → P { ∈ | ⊂ } Soon we will show that dim A < if A is a Noetherian local ring. ∞ Problem 6.3. Give an example of an infinite dimensional Noetherian ring. Exercise 6.2 says that dimension can be computed locally. In other words, in order to study dimension in arbitrary rings, it is sufficient to understand the case of local rings. Definition 6.4. Let (A,m,k) be a Noetherien local ring. Let δ(A) denote the smallest number of generators of an m-primary ideal in A. Definition 6.5. Let A be a ring and M an A-module. The length of M is defined to be the maximal length of a chain of submodules (0) $ M1 $ M2 $ $ ··· Mn = M, not counting (0). If there exist arbitrarily long such chains, we say that length(M) = . ∞ For example, if V is a vector space over a field k, length(V ) as a k-module is nothing but dimk V . A homomorphism of A-modules is a homomorphism of abelian groups, which respects multiplication by elements of A. M Given A-modules N M, the quotient N of M by N is defined to be the quotient of abelian groups,⊂ endowed in the obvious way with the operation of mul- I tiplication by elements of A. In particular, we may talk about the quotient J of I A two ideals of A, J I. We may view J as an ideal in the ring J . Exercise 6.6. Show⊂ that length is an additive function, that is, if 0 M ′ M M ′′ 0 is an exact sequence, then length(M) = length(M ′) + length→ (M ′′→). → → Definition 6.7. Let (A,m,k) be a Noetherian local ring. The Hilbert Samuel A function of A is the function HA,m : N N, defined by HA,m(n) = length mn+1 (considered as an A-module). By additivity→ of length, A n mi (6.2) length = dim , mn+1 k mi+1 i=0 X mi where the mi+1 are k-modules, that is, k-vector spaces. Note that since A is Noetherian, each of mi is finitely generated, so that all the quantities in (6.2) are finite. Theorem 6.8. H (n) is a polynomial for n 0. In other words, there exists a A,m ≫ polynomial P (n) with rational coefficients, such that P (n) = HA,m(n) for n 0. P (n) is sometimes called the Hilbert polynomial of A. ≫ A proof of this Theorem 6.8 will be given shortly. First, we finish our list of the different characterizations of dim A. 24 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Notation 6.9. Let d(A) denote the degree of the Hilbert polynomial of A.

Finally, let A be a k-algebra. Let x1,...,xn be elements of A. n α Notation. Given x =(x1,...,xn) and α =(α1,...,αn) N0 , x will denote the α α1 α2 αn ∈ monomial x = x1 x2 ...xn .

x1,...,xn are said to be algebraically independent over k if they satisfy no α non-trivial polynomial relations of the form aαx = 0, aα k. The transcendence degree of A over k, denoted tr. deg(A/k), is defined∈ to be the maximal number of elements of A, algebraically independentP over k. If A is a finitely generated k-algebra, then tr. deg(A/k) < . ∞ Example 6.10. Let k be a field, A = k[x1,...,xd] the polynomial ring in d variables. (0) $ (x1) $ (x1,x2) $ $ (x1,...,xd) is a chain of prime ideals of length d. ··· d Now let m =(x1,...,xd) be the maximal ideal corresponding to the origin in k . Consider the localization Am. m is generated by d elements. The Hilbert-Samuel A n+d function of Am is HAm,m(n) = length( mn+1 ) = d , which is a polynomial in n of degree d. In this case, H (d) is a polynomial for all n, not merely for n Am,m sufficiently large. Thus, in this example, we have

(6.3) tr. deg(A/k) = d(A ) = d dim A dim A δ(A). m ≥ ≥ m ≥ We will soon show that all the inequalities in (6.3) are, in fact, equalities. We now turn to the main purpose of this section, which is to prove that for any Noetherian local ring A, dim A = d(A) = δ(A). If A is a finitely generated algebra over a field k, we will prove that dim A = tr. deg(A/k). The first step is to prove that d(A) is well defined, that is, HA,m(n) is, indeed, a polynomial for n 0. To this end, we introduce the notion of graded algebras and graded modules. ≫

Definition 6.11. Let G0 be a ring. A graded G0-algebra is a G0-algebra of the ∞ form G = Gn, such that for all n,l N0, we have GnGl Gn+l (in particular, n=0 ∈ ⊂ each Gn isL a G0-module under the ring operations in G). If G is a graded G0-algebra, a graded G-module is a G-module of the form ∞ M = Mn where each Mn is a G0-module and for all n,l N0, we have GnMl n=0 ∈ ⊂ Mn+l.L For n N0, elements of Mn are called homogeneous of degree n. ∈ Example 6.12. Let A be a ring and I A an ideal. Associated to A and I is ∞ ⊂ A In the graded I -algebra grI A = In+1 . If I is a finitely generated ideal, then grI A n=0 A A is a finitely generated I -algebraL (its generators as an I -algebra are precisely the INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 25 generators of I as an ideal, viewed as elements of grI A of degree 1). By the Hilbert basis theorem, if A is Noetherian then grI A is Noetherian (cf. Exercise 6.13 below). In particular, if (A,m,k) is a local Noetherian ring, grmA is a finitely generated mn graded k-algebra. Moreover, for any n N0, dimk mn+1 < . In other words, the homogeneous elements of degree n form∈ a finite dimensional∞ vector space. For the moment, this last example of a graded algebra is our main interest for applications. Later in the course, other important examples of graded algebras will appear.

Exercise 6.13. Let G be a graded G0-algebra. Prove that G is Noetherian if and only if G0 is Noetherian and G is finitely generated over G0. Now, suppose that this is the case and let M be a finitely generated graded G-module. Show that each Mn is a finite G0-module. The Hilbert-Samuel function of a Noetherian local ring (A,m,k) is

n mi H (n) = length . A,m mi+1 i=0 X We start our proof of Theorem 6.8 by the following simple observation. Let P (n) (1) n be a real valued function of n. Let P (n) = P (n) and let d N0. Then P i=0 ∈ is a polynomial in n of degree d if and only if P (1) is a polynomial of degree d + 1. P Hence to prove Theorem 6.8 it is sufficient to prove, for a local Noetherian ring mn (A,m,k), that for n 0 length mn+1 is a polynomial in n. We can generalize this statement as follows.≫ ∞ Let G be a graded algebra, M = Mn a finite graded G-module, as in the n=0 Example 6.12 (in particular, each Mn Lis a finite G0-module). Let λ be an additive function from the category of finite G0-modules to N0. For n N0, we may consider ∈ λ(Mn) as a function of n. To prove Theorem 6.8, it is enough to show that λ(Mn) is a polynomial for n 0. To this end, we consider the generating function of ∞ ≫ n M: P (M,t) = λ(Mn)t , where t is an independent variable. n=0 Theorem 6.14P (Hilbert, Serre). Let x1,...,xd be a set of homogeneous gener- f(t) ators of M, ki = deg xi. Then P (M,t) = d , where f is a polynomial in t (1−tki ) i=1 with integer coefficients. Q Proof. Induction on d. First, let d = 0. Then A = A0, hence Mn = 0 for n 0, so that λ(M ) = 0 for n 0 and the result is true in this case. ≫ n ≫ Next, suppose the result is known for d 1. For each n N0, multiplication by − ∈ xd is a G0-module homomorphism from Mn to Mn+kd . Let Kn and Ln+kd denote, respectively, the kernel and the cokernel of this homomorphism. For each n N0, we obtain an exact sequence ∈

(6.4) 0 K M M L 0. → n → n → n+kd → n+kd → 26 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

By additivity of λ, (6.4) implies that

(6.5) λ(K ) λ(M ) + λ(M ) λ(L )=0. n − n n+kd − n+kd ∞ ∞ n+kd Let K = Kn, L = Ln. Multiply (6.5) by t and sum over n. We obtain n=0 n=kd L P (6.6) tkd P (K,t) tkd P (M,t) + P (M,t) P (L,t) + g(t)=0, − −

where g(t) is a polynomial in t. Since both K and L are annihilated by xd, both K and L are graded G0[x1,..., xd−1]-modules. Hence by the induction assumption, d−1 d−1 (1 tki )P (K,t) and (1 tki )P (L,t) are polynomials in t. By (6.6), we obtain i=1 − i=1 − f(t) (1Q tkd )P (M,t) = tkdQP (K,t) + P (L,t) g(t). Hence P (M,t) = for − − − d−1 (1−tki ) i=1 some polynomial f(t) and the Theorem is proved. Q We have the following Corollary, which obviously implies Theorem 6.8. Corollary 6.15. Let G = Gn, M = Mn, λ, xi be as in Theorem 6.14. Suppose that ki deg xi =1, 1 i d. Then, for n 0, λ(Mn) is a polynomial in n of degree at≡ most d 1. L ≤ ≤ L ≫ − f(t) Proof. By Theorem 6.14, P (M,t) = (1−t)d , where f is a polynomial in t. Write N ∞ k 1 n+d−1 n f(t) = akt . Since (1−t)d = d−1 t , we have, for n 0, λ(Mn) = k=0 n=0 ≫ N n−Pk+d−1 P ak d−1 , which is a polynomial in n of degree at most d 1. k=0 − P Remark 6.16. Notice in the above proof that the degree of λ(Mn) for large n is f(t) precisely the order of the pole of (1−t)d at t = 1 minus 1. We now state the main result of this section. Theorem 6.17. Let (A,m,k) be a Noetherian local ring. Then dim A = d(A) = δ(A). First, we study the case dim A = 0. Definition 6.18. A ring A is said to be Artinian if every descending chain of ideals in A stabilizes. Despite the apparent similarity of this definition with the Noetherian property, we will now show that the Artinian property is much stronger. Namely, we will show that A is Artinian if and only if A is Noetherian and dim A = 0. Proposition 6.19. Let A be an Artinian ring. Then every prime ideal of A is maximal (in other words, dim A = 0). Proof. Let P A be a prime ideal. Then A is an Artinian domain. We want ⊂ P to show it is a field. Take any x A . It suffices to show that x is invertible. ∈ P INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 27

Consider (x) (x2) (x3) ... . This chain of ideals stabilizes, so that n n+1 ⊃ ⊃ ⊃ A n+1 n (x )=(x ) for some n. Hence there exists b P such that bx = x . Since A ∈ A P is a domain, bx = 1, so that x is invertible. This proves that P is a field, i.e. P is maximal. Corollary 6.20. Let A be an Artinian ring. Then nil(A) = J(A) (recall that J(A) stands for the Jacobson radical of A). Proof. This follows from Exercise 4 of 1. § Exercise 6.21. Let A be any ring, m1, m2 two distinct maximal ideals of A. Prove that m m = m m . 1 2 1 ∩ 2 Theorem 6.22. Let A be an Artinian ring. Then A has finitely many maximal ideals. Proof. Suppose not. Then there is an infinite set m1, m2, . . . ,mn,... of distinct maximal ideals of A. Consider the descending chain of ideals m1 m1 m2 m m m ... . Since A is Artinian, we must have m m ⊃ ∩ m =⊃ 1 ∩ 2 ∩ 3 ⊃ 1 ∩ 2 ∩···∩ n m1 mn mn+1 for some n. By Lemma 1.21, mn+1 = mi for some i n. This∩···∩ is a contradiction∩ and the Theorem is proved. ≤ Exercise 6.23. Let A be a ring and I a finitely generated ideal, contained in nil(A). Show that there exists n N such that In = (0). More generally, show that if I is ∈ any finitely generated ideal, there exists n N such that (√I)n I. ∈ ⊂ Theorem 6.24. Let A be an Artinian ring. Let Max(A) = m1,...,ms . Then n { } (m1m2 ...ms) = 0 for some n N. ∈ Proof. Let I = m1m2 ...ms. By Exercise 6.21, I = J(A). By Corollary 6.20, I = (0). Now, we cannot apply Exercise 6.23, because we have not yet proved that I is finitely generated. p Consider the descending chain of ideals I I2 I3 ... In ... . Then In = In+1 for some n. We claim that In = (0).⊃ Suppose⊃ ⊃ not.⊃ Let Σ⊃ denote the collection of all the ideals H˜ A, for which HI˜ n =0. Σ = since I Σ. Since A is Artinian, Σ contains a minimal⊂ element H (in6 the sense6 ∅ of inclusion).∈ Since HIn = 0, there exists x H such that xIn = 0. We have (x) H, hence (x) = H by the6 minimality of H.∈ Now, (xIn)In = xI6 2n = xIn = (0) and⊂ xIn (x), hence xIn = (x) by the minimality of H. Then there exists6 y In such that⊂ x = xy. Then ∈

(6.7) x = xy = xy2 = = xyl = .... ··· Since y In I = (0), (6.7) implies that x = 0, which is a contradiction. This proves Theorem∈ ⊂ 6.24. p Theorem 6.25. Let A be a ring. Then A is Artinian if and only if length(A) < (where we view A as an A-module). ∞ Proof. is obvious. ⇐ 28 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Let Max(A) = m , . . . ,m and let I = m ...m = J(A). By the previous ⇒ { 1 s} 1 s theorem, In = 0 for some n N. Then there exists r s and maximal ideals ∈ ≥ m1,...,mr, not necessarily all distinct, such that m1m2 ...mr = 0. Consider the chain A m m m ... m m ...m (0) of submodules of A. By the ⊃ 1 ⊃ 1 2 ⊃ ⊃ 1 2 r−1 ⊃ additivity of length, it is sufficient to show that length m1...mi < . Now, m1m2...mimi+1 ∞ m1...mi is annihilated by m , hence it is an A -module, that is, an A - m1m2...mimi+1 i+1 mi+1 mi+1 vector space. In a vector space, a submodule is the same thing as a vector subspace, so that a vector space has finite length if and only if it has finite dimension if and only if every descending chain of submodules stabilizes. This last condition holds m1...mi by the Artinian property of A, hence dim A m m ...m m < . This proves that mi+1 1 2 i i+1 ∞ length(A) is finite, as desired. Theorem 6.26. A ring A is Artinian if and only if A is Noetherian and dim A = 0. Proof. Suppose A is Artinian. By the previous theorem, length(A) < , hence A is Noetherian. Since every prime ideal in A is maximal, dim A = 0. ∞ s Conversely, suppose A is Noetherian and dim A = 0. Let (0) = Qi be a i=1 primary decomposition of (0). Let Pi = √Qi be the associated primes.T Since s dim A = 0, each of the Pi is maximal. Moreover, (0) = i=1Pi, hence by Exercise 6.23 there exists n such that (P ...P )n = 0. ∩ 1 s p Now we can use the same reasoning as in the last theorem. We have A P P P ... P P ...P (0), where r>s and P ,...,P is a sequence⊃ 1 ⊃ 1 2 ⊃ ⊃ 1 2 r−1 ⊃ 1 r consisting of P1,...,Ps with some repetitions. Since A is Noetherian, P1 ...Pi is a finitely generated ideal, hence P1...Pi is a finite A-module, hence a finite A - P1...Pi+1 Pi+1 module, i.e. a finite dimensional A -vector space. Now the reasoning in the proof Pi+1 of the last theorem applies, and we conclude that length(A) < , so that A is Artinian. ∞ Now, we shall prove a special case of Theorem 6.17. Theorem 6.27. Let (A,m,k) be a Noetherian local ring. Then dim A =0 if and only if δ(A)=0 if and only if d(A)=0. Proof. dim A = 0 m is the only prime ideal m = (0) (0) ⇐⇒ ⇐⇒ A ⇐⇒ is m-primary δ(A)=0 n such that A = n n n 0 N m p 0 H (n) = const.⇐⇒for n 0 ⇐⇒d(A ∃)=0.∈ ∀ ≥ ⇐⇒ A,m ≫ ⇐⇒ This takes care of the zero dimensional local rings. Let (A,m,k) be any Noetherian local ring and Q A an m-primary ideal. Next, we show that d(A) δ(A). ⊂ ≤ Proposition 6.28. A (1) length Qn < . A ∞ (2) length n is a polynomial H (n) for n 0 of degree at most s where s is Q Q ≫ the minimal number of generators of Q. INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 29

(3) deg H is independent of Q (hence deg H = d(A) δ(A)). Q Q ≤ A Proof. 1) is clear, since Qn is Artinian. ∞ A Qn A 2) Apply Theorem 6.14 to the Q -algebra grQA = Qn+1 , generated over Q by n=0 Qn s elements. By Theorem 6.14, length Qn+1 is a polynomialL in n of degree at most n A Qi s 1. Since HQ(n) = length Qn+1 = length Qi+1 , we have deg H s, as − i=0 ≤ desired. P 3) It is enough to prove that deg HQ = deg Hm. Since Q is m-primary, there exists r N such that mr Q m. Then mrn Qn mn for any n N, so that ∈A A⊂ ⊂ A ⊂ ⊂ ∈ length rn length n length n . m ≥ Q ≥ m Since Hm(rn) HQ(n) Hm(n) for all n N and since both Hm and HQ are ≥ ≥ ∈ polynomials in n, we have deg HQ = deg Hm. QED To complete the proof of Theorem 6.17, it remains to show that d(A) dim A δ(A). We now state a crucial lemma, whose proof will be the subject≥ of the next≥ two sections.

Lemma 6.29. Let x A. Assume that x is not a zero divisor. Then d A ∈ (x) ≤ d(A) 1. − The rest of this section is devoted to proving Theorem 6.17, assuming Lemma 6.29. Proposition 6.30. d(A) dim A. ≥ Proof. Induction on d(A). If d(A) = 0, the Proposition follows from Theorem 6.27. Suppose d(A) > 0. Let P0 $ P1 $ $ Pr be a chain of prime ideals. We want to show r d(A). ··· ≤ Let A¯ = A . Take a non-zero element x P1 . Since A¯ is a domain, x is not P0 ∈ P0 a zero divisor in A¯. By Lemma 6.29, d A¯ d(A¯) 1 d(A) 1. On the (x) ≤ − ≤ − ¯ ¯ ¯ ¯ other hand, P1A $ P2A $ Pr A is a chain of distinct prime ideals of A . Hence (x) (x) ··· (x) (x) dim A¯ r 1. Using the induction hypothesis, we obtain r 1 dim A¯ (x) ≥ − − ≤ (x) ≤ d A¯ d(A) 1 therefore r d(A), as desired. Q.E.D. (x) ≤ − ≤ Corollary 6.31. If A is a Noetherian local ring, then dim A < . In particular, if P Spec A, then ht P < . ∞ ∈ ∞ It remains to show that dim A δ(A). ≥ Given any ideal I A, deﬁne ht I = min ht P I P,P Spec A . By Corollary 6.31, ht I <⊂ for any ideal I in a Noetherian{ | ring⊂ A. ∈ } ∞ 30 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Lemma 6.32. dim A δ(A). ≥ Proof . Let d = dim A. We will construct, recursively in i, elements x ,...,x 1 i ∈ m such that ht(x1,...xi) i. Assume that i d and that x1,...,xi−1 with ht(x ,...,x ) i 1 are already≥ constructed. If ht(≤ x ,...x ) d, then m is the 1 i−1 ≥ − 1 i−1 ≥ only minimal prime of (x1,...xi−1), hence (x1,...xi−1) is m-primary, hence δ(A) i 1 d and the Lemma is proved. Thus we may assume that ht(x ,...x )

Lemma 6.33. Let I,P1,...,Pr be ideals in a ring A, where all except possibly two r of the Pj are prime. If I Pj, then I Pj for some j. ⊂ j=1 ⊂ S We ﬁnish the proof of Lemma 6.32 assuming Lemma 6.33. By Lemma 6.33, m = r r 6 Pj. Take an element xi m Pj. We want to show that ht(x1,...,xi) i. j=1 ∈ \ j=1 ≥ Indeed,S take any minimal prime P Sof (x ,...,x ). Since (x ,...,x ) P , P P 1 i 1 i−1 ⊂ ⊃ j for some j, 1 j r. Since xi P Pj, Pj $ P . Since ht Pj ht(x1,...,xi−1) i 1, we have≤ ht≤P ht P +∈ 1 =\i. We have constructed the≥ desired elements≥ − ≥ j (x1,...,xi). For some i d, we obtain ht(x1,...,xi) d, hence (x1,...,xi) is m-primary, hence d = ht m≤ δ(A). It remains to prove Lemma≥ 6.33. ≥

Proof of Lemma 6.33. We may assume that P3,...,Pr are prime. The proof is by induction on r. First, suppose r = 2. Suppose that I P and I P . 6⊂ 1 6⊂ 2 Pick elements x1 I P1 and x2 I P2. Then x1 P2 and x2 P1, so that x + x I (P ∈P ),\ a contradiction.∈ \ ∈ ∈ 1 2 ∈ \ 1 ∪ 2 Next, suppose the result is known for r 1, with r 3. Suppose I Pj for 1 j r. By the induction assumption, we− may assume≥ that I is not contained6⊂ in ≤ ≤ r the union of any smaller subcollection of the Pj. For each j, pick xj I i=1Pi. ∈ \ ∪i6=j r−1 r−1 Now consider the element x = xr + xj. Clearly, x I Pi. On the other j=1 ∈ \ j=1 hand, since xj / Pr for j

Let (A,m,k) be a Noetherian local ring of dimension d. Let x ,...,x m 1 d ∈ be such that (x1,...,xd) = m. The elements x1,...,xd are called a system of parameters of A. p To complete the proof of Theorem 6.17, it remains to prove Lemma 6.29. To do so, we need to study ﬁltrations, ideal-adic topologies and the Artin-Rees lemma. This is the content of the next section. INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 31

7. Filtrations, ideal-adic topologies and the Artin-Rees lemma. § Let A be a ring, I an ideal of A, M an A-module. The notion of multiplying M by an ideal is defined in the obvious way. Namely IM is the submodule of M given by IM = ax M a I,x M . Definition{ 7.1∈ An|I-filtration∈ ∈ of}M is a chain of submodules M = M M 0 ⊃ 1 ⊃ ... Mn such that IMn Mn+1 for all n N0. ⊃ ⊂ ∈ A filtration is stable if IMn = Mn+1 for n 0. The I-adic topology on M is the topology whose basic open sets are all≫ the sets of the form x + InM, x M,n N. ∈ ∈ Example 7.2. Let A = M = Z and let I = (p), where p is a prime number. We have the p-adic filtration and the p-adic topology on Z. Clearly this filtration is stable. ′ ′ Remark 7.3. Let Mn and Mn be two stable filtrations. Then Mn and Mn { } { } { ′ } { } have bounded difference, that is, there exists n0 N such that Mn+n0 Mn ′ ∈ ⊂ ⊂ Mn−n0 for all n 0 (this is so because there exists n0 such that, for all n N, ′ n ′ ≫ n ∈ M = I M I M Mn and similarly in the other direction). n+n0 n0 ⊂ ⊂ Lemma 7.4 (the Artin-Rees Lemma). Let A be a Noetherian ring, M ′ M ⊂ finitely generated A-modules. Let Q be an ideal. Let Mn be a stable Q-filtration ′ ′ ′ { } ′ of M. Let Mn = Mn M . Then Mn is a stable Q-filtration of M . ∩ { } ∞ Proof. Consider the graded A-algebra G = Qn and the graded G-module n=0 ∞ ∗ L M = Mn. n=0 ∗ LemmaL 7.5. Mn is stable if and only if M is a finite G-module. { } ∗ ∗ Proof. For each n, let Mn denote the graded G-module Mn = M M1 Mn QM Q2M ... . We get a chain of submodules ⊕ ⊕···⊕ ⊕ n ⊕ n ⊕ (7.1) M ∗ M ∗ ... M ∗ ... 1 ⊂ 2 ⊂ ⊂ n ⊂ of M ∗. ∗ By Exercise 6.13, each Mn is a finite A-module. Hence since G is Noetherian, M is finitely generated as a G-module it is generated by its homogeneous elements ⇔ of degree less than some fixed number (7.1) stabilizes there exists n0 N such n ⇔ ⇔ ∈ that Mn+n = Q Mn for all n N the filtration is stable. 0 0 ∈ ⇔ We return to the proof of the Artin–Rees Lemma. We need the following characterization of finite modules over a Noetherian ring, which we leave to the reader as an easy exercise. Exercise 7.6. Let A be a Noetherian ring, M an A-module. Then M is finite over A if and only if it is Noetherian, that is, any ascending chain of submodules N N ... N ... stabilizes. In particular, if A is Noetherian, M a finite 1 ⊂ 2 ⊂ ⊂ k ⊂ 32 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

A-module and M ′ M a submodule, then M ′ is also a finite A-module. (Hint: if M is a finite A-module⊂ then M can be written as a quotient of a direct sum of finitely many copies of A.) It is easily seen that M M ′ is a Q-filtration. { n ∩ } ∞ ′∗ ′ ′∗ ∗ ∗ Let M = Mn. We have M M . Since M is a finite G-module, by n=0 ⊂ Exercise 7.6 soL is M ′∗. By Lemma 7.5, this implies the Artin-Rees lemma. Corollary 7.7. Let A be a Noetherian ring, Q A an ideal and M a finite ′ ⊂ A-module. Let M M be a submodule. Then there exists n0 N such that n+n0 ′ ⊂n n0 ′ ∈ (Q M) M = Q (Q M M ) for all n N0. ∩ ∩ ∈ ′ ′ Corollary 7.8. Keep the above notation. Let Mn be a stable Q-filtration of M . n+n0 {′ } ′ n−n0 ′ There exists n0 N such that (Q M) M Mn (Q M) M for all n n . ∈ ∩ ⊂ ⊂ ∩ ≥ 0 8. Dimension theory, continued. § We are now in a position to prove Lemma 6.29. First, we prove the following generalization of Proposition 6.28. Proposition 8.1. Let (A, m) be a Noetherian local ring, Q A an m-primary ⊂ ideal. Let M be a finite A-module, Mn a stable Q-filtration of M. Let s be the minimal number of generators of Q.{ Then:} M (1) length < for all n N0. Mn ∞ ∈ M (2) length is a polynomial HM,Q(n) for n 0, of degree at most s. Mn ≫ (3) deg HM,Q depends only on M, not on Q and Mn . The highest coefficient of H depends only on M and Q, not on the{ filtration} M . M,Q { n}

Mn A Proof . (1) Since Mn is a Q-filtration, is an -module. Since Mn is { } Mn+1 Q a finite A-module, Mn is a finite A -module. Hence it is a quotient of the direct Mn+1 Q sum of finitely many copies of A . Now the fact that length A < implies that Q Q ∞ length Mn < and hence length M < . Mn+1 ∞ Mn+1 ∞ (2) Follows immediately from Corollary 6.15 of the Hilbert-Serre Theorem, ap- ∞ n ∞ A Q Mn plied to the graded -algebra G = n+1 and the graded G-module . Q Q Mn+1 n=0 n=0 ′ (3) Take any other stable Q-filtrationL M . There exists n0 N suchL that { n} ∈ ′ (8.1) Mn−n M Mn+n for all n N. 0 ⊂ n ⊂ 0 ∈ ′ ′ Let HM,Q denote the Hilbert polynomial associated to Mn . By (8.1) HM,Q(n ′ { } ′ − n0) H (n) HM,Q(n n0) for all n N, hence HM,Q and H have the same ≥ M,Q ≥ − ∈ M,Q degree and highest coefficient. To show that, in addition, deg HM,Q is independent INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 33 of Q, compare deg HM,Q with deg HM,m. Since the degree is independent of the choice of the filtration, we may take the filtrations to be QnM and mnM, respectively. Since Q is m-primary, there exists r N such that mr Q m rn n n ∈ ⊂ ⊂ (Exercise 6.23). Then m Q m for any n N. Hence HM,m(rn) ⊂ ⊂ ∈ ≥ HM,Q(n) HM,m(n) for all n N. This proves that deg HM,Q = deg HM,m, as desired. QED≥ ∈

Notation. The common degree of the polynomials HM,Q will be denoted by d(M). We now prove Lemma 6.29 in the following stronger form. Lemma 8.2. Let (A, m), M be as above. Let x m. Assume that x is not a zero ∈ divisor for M, that is, xa = 0 for any a M 0 . Then d M d(M) 1. 6 ∈ \{ } xM ≤ − Proof . Let N = xM, M ′ = M . Then N = M since x is not a zero divisor. Let N ∼ N = mnM N. We have an exact sequence n ∩ 0 N M M ′ 0, → → → → which induces an exact sequence

N M M ′ (8.2) 0 n n ′ 0. → Nn → m M → m M → for each n N. By additivity of length, (8.2) gives ∈

(8.3) H (n) H (n) + H ′ (n)=0. N,m − M,m M ,m

By the Artin–Rees Lemma , Nn is stable. By Proposition 8.1, HN,m and HM,m have the same degree and highest{ } coeﬃcient, hence, by (8.3),

deg HM ′,m < deg HM,m.

This proves that d(M ′) d(M) 1. This completes the proof≤ of the− Dimension Theorem 6.17. ∞ Exercise 8.3. Let G = Gn be a Noetherian graded G0-algebra, with G0 Ar- n=0 ∞ L tinian. Let M = Mn be a ﬁnite graded G-module and let d(M) denote the n=0 degree of the HilbertL polynomial

n H(n) = length(Mi) Xi=0 for n 0. Let x Gs be a homogeneous element of degree s, which is not a zero divisor≫ in M. Show∈ that d( M ) d(M) 1. (x)M ≤ − Now we can relax and derive several important corollaries from Theorem 6.17. 34 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Corollary 8.4. Let (A,m,k) be a Noetherian local ring. Then dim A dim m . ≤ k m2 m m Proof. Let x1,...,xs be elements of m whose images in m2 generate m2 . Then

2 (8.4) m =(x1,...,xs) + m .

We now invoke one of the most famous lemmas in all of commutative algebra—the Lemma of Nakayama—to conclude that (8.4) implies m = (x1,...,xs). This will show that dim A = δ(A) s, as desired. ≤ It remains to state and prove Nakayama’s Lemma. Lemma 8.5 (Nakayama’s Lemma). Let A be a ring, I an ideal of A, M an A- module, N M a submodule. Assume either that M is ﬁnite or that I is nilpotent ⊂ (that is, In = (0) for some n N). Suppose that M = IM + N. Then there exists x I such that (1 + x)M = N∈. If, in addition, I J(A), then M = N. ∈ ⊂ Applied to (8.4) with I = M = m,(x1,...,xs) = N, Nakayama’s Lemma implies that m =(x1,...,xs). Proof of Nakayama’s Lemma. If I is nilpotent, we have In = (0) for some n N. Substituting IM for M recursively in the expressions IiM + N, we obtain M∈= IM + N = I2M + N = = InM + N = N, and the Lemma is true in this case. ··· Next, suppose M is ﬁnite over A. Let w1,...,wn be the generators of M. Replac- M ing M by N , we may assume that N = (0). Then M = IM, hence w1,...,wn IM. Then there exist a I, 1 i, j n, such that ∈ ij ∈ ≤ ≤ w = a w + a w + + a w 1 11 1 12 2 ··· 1n n w = a w + a w + + a w 2 21 1 22 2 ··· 2n n (8.5) . . w = a w + a w + + a w n n1 1 n2 2 ··· nn n Let a11 a12 ... a1n a21 a22 ... a1n T =  .  . .    an1 an2 ... ann    By (8.5),

w a w ... a w 0 1 − 11 1 − − 1n n w2 a21w1 ... a2nwn 0 (8.6)  . − − −  =  .  . . .      wn an1w1 ... annwn   0   − − −    INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 35

w1 0 . . Let 1n denote the n n identity matrix. (8.6) says (1n T ) . = . . Since × −  .    wn 0 det(1n T )1n = adj(1n T )(1n T ), we have det(1n T )wi = 0, i =1 , 2,...,n. Since− all the a I, det(1− −T )=1+ x for some x− I. We have ij ∈ n − ∈ (8.7) (1+ x)w =0, 1 i n. i ≤ ≤ n Since M = i=1 Awi, (8.7) implies (1+ x)M = (0). If, in addition, x J(A), then 1 + x is not in any maximal ideal of A, hence 1 + x is invertible. Then∈M = (0), as desired. P m m dimk m2 is called the embedding dimension, m2 is called the Zariski cotan- m gent space of A. Its dual Homk m2 ,k is called the Zariski tangent space. We continue with Corollaries of Theorem 6.17. Corollary 8.6. Let A be a Noetherian ring and let x1,...,xr be elements of A, such that (x ,...,x ) = A (in other words, there exists m Max(A), containing 1 r 6 ∈ all the xi). Then for any minimal prime P of (x1,...,xr), we have ht P r (in particular, ht(x ,...,x ) r). ≤ 1 r ≤ Proof. Let P be any minimal prime of (x1,...,xr). Exercise 8.7. Let I be an ideal in a Noetherian ring A, P a minimal prime of I. Show that √IAP = PAP . By Exercise 8.7, (x ,...,x )A = PA , hence ht P dim A = δ(A ) r. 1 r P P ≡ P P ≤ Theorem 8.8 (Krull’sp Principal Ideal Theorem). Let A be a Noetherian ring, x an element of A, which is neither a unit nor a zero divisor. Then every minimal prime ideal of (x) has height 1. Proof. Let P be a minimal prime of (x). By Corollary 8.6, ht P 1. It remains to show that ht P = 0. Indeed, suppose ht P = 0. Then P is a minimal≤ prime of A. By Theorem 5.14,6 every element of P is a zero divisor, which contradicts the fact that x P . ∈ Exercise 8.9. Let A be a ring, S A a multiplicative set and P a prime ideal such that P S = . Show that ht P⊂= ht(PA ). ∩ ∅ S Exercise 8.10. For any n N, give an example of a Noetherian local ring A and x A which is not a zero divisor,∈ such that (x) has an associated prime of height n.∈ Let (A, m) be a Noetherian local ring, x an element of m which is not a zero divisor. From Lemma 6.29 and the Dimension Theorem we know that A (8.8) dim dim A 1. (x) ≤ −

The next Corollary says that the inequality is, in fact, an equality. 36 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Corollary 8.11. In this situation, we have dim A = dim A 1. (x) − A A ′ ′ Proof. Let d = dim (x) = δ (x) . Let x1,...,xd be elements of A whose A m ′ images in (x) generate an (x) -primary ideal. Then x1,...,xd ,x generate an m- primary ideal in A. Then dim A = δ(A) d′ + 1 = dim A + 1, which proves the ≤ (x) opposite inequality of (8.8). Hence the Corollary is proved.

9. Dimension and transcendence degree in §finitely generated algebras over a field. Notation. If P Spec A for a ring A, let κ(P ) = AP denote the residue field of ∈ PAP the local ring AP . Let k be a field and A a finitely generated k-algebra. If x1,...,xn are the generators of A over k, A can be written as the quotient of the polynomial ring k[x ,...,x ] by an ideal I k[x ,...,x ]; that is, A = k[x1,...,xn] . 1 n ⊂ 1 n I Theorem 9.1. (1) dim A = tr. deg(A/k). (2) If P Spec A and K is the field of rational functions of some irreducible ∈ component of Spec A containing P (in other words, K = κ(P0) for some minimal prime P of A, contained in P ), we have dim AP +tr. deg(κ(P )/k) = 0 P0AP tr. deg(K/k).

Proof. By the following exercise, we may assume that A is a domain in (1):

Exercise 9.2. Let A be a ﬁnitely generated k-algebra and P1,...,Pr the minimal primes of A. Show that (1) dim A = max dim A 1≤i≤r{ Pi } (2) tr. deg(A/k)= max tr. deg( A /k) . 1≤i≤r{ Pi }

In (2) of Theorem 9.1, we may replace A by A . From now on, we will assume P0 that A is a domain. Let d = dim A and let (0) $ P1 $ $ Pd be a maximal chain of prime ··· ideals. Then dim APd = d. Moreover, Pd is maximal, hence by Proposition 4.2, tr. deg( A /k) = 0. Since tr. deg(A/k) is obviously unaffected by localization (here Pd we are using the fact that A is a domain), we have (2) = (1) in Theorem 9.1. ⇒ It remains to prove (2). Let P Spec A. Let t = tr. deg AP /k . Renumber- ∈ PAP A ing the xi, we may assume that the images of x1,...,xt in P form a transcendence base of AP over k. Let L = k(x ,...,x ); we have the obvious inclusion PAP 1 t L ֒ AP (this is where we are using the fact that A is a domain: the natural map→A A is injective). We have a natural map φ : L[x ,...,x ] A ; → P t+1 n → P by definition, AP = L[xt+1,...,xn]φ−1(P ). Let K be the field of fractions of A. Since transcendence degree is additive in field extensions, we have tr. deg(A/k) = INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 37 tr. deg(AP /k) = tr. deg(K/k) = t + tr. deg(K/L). Therefore we may replace k by L and A by L[xt+1,...,xn]. In other words, we may assume that P is maximal and t = 0. It remains to prove that under these assumptions, dim AP = tr. deg(A/k). Lemma 9.3. Let (A,m,k) be a Noetherian local ring of dimension d. Let x1,...,xd be a system of parameters of A and let Q =(x1,...,xd). Let f A[t1,...,td] be a homogeneous polynomial of degree s in d variables t ,...,t . If f(x∈ ,...,x ) Qs+1, 1 d 1 d ∈ then f mA[t1,...,td]. ∈ ¯ A Proof . Let f denote the natural image of f in Q [t1,...,td]. Consider the map

A ∞ Qi (9.1) α : [t ,...,t ] , Q 1 d → Qi+1 Mi=0

i deﬁned by α(t ) = x , 1 i d. By Proposition 6.28, d Q = dim A = d. i i ≤ ≤ Qi+1 i (t1,...,td) A d+i−1 A Also length i+1 = length , so that d L[t1,...,td] = d. (t1,...,td) Q · d−1 Q By assumption, f ker α. Since both algebras in (9.1) have the same degree of ∈ ¯ A Hilbert polynomial, by Exercise 8.3 f is a zero divisor in Q [t1,...,tn]. We leave it as an Exercise to ﬁnish the proof: A Exercise 9.4. Show that (0) is a primary ideal in Q [t1,...,td] and that

A A m [t ,...,t ] = nil [t ,...,t ] Q 1 d Q 1 d A is the only associated prime of Q [t1,...,td]. In particular, every zero divisor in A A Q [t1,...,td] belongs to m Q [t1,...,td]. Q.E.D. Corollary 9.5. Let (A, m) be a Noetherian local ring of dimension d, x1,...,xd a system of parameters of A. Suppose A contains a ﬁeld k0. Then x1,...,xd are algebraically independent over k0. Proof . Let f(x1,...,xd) = 0 be an algebraic relation among the xi over k0. Write f = fs + fs+1 + + ft where each fi is homogeneous of degree i. Then f (x ,...,x ) Qs+1, hence··· by Lemma 9.3 f mA[t ,...,t ], which implies f 0 s 1 d ∈ ∈ 1 d ≡ since f is a polynomial with coeﬃcients in k0.

Corollary 9.5 proves that dim AP tr. deg(A/k). For the opposite inequality, we need the Noether Normalization Lemma.≤ Lemma 9.6 (Noether Normalization Lemma). Let k be a ﬁeld and A = k[x1,...,xn] a ﬁnitely generated k-algebra. Then there exist y ,y ,...,y A, alge- I 1 2 d ∈ braically independent over k, such that A is integral over k[y1,...,yd].

Proof. We may assume that x1,...,xd are algebraically independent over k and xd+1,...,xn are algebraic over k[x1,...,xd]. We will prove, by induction on i, 38 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

that there exists a choice of generators xj such that xd+1,...,xi are integral over k[x1,...,xd]. Suppose this is known for i, d i

λj (9.2) xj = xj + xi+1, 1 j d. By assumptions, there is an algebraic dependence relation ≤ ≤ e (9.3) F (x ) = a xl + a xl−1 + + a =0, i+1 0 i+1 1 i+1 ··· l

aj k[x1,...,xd]. It is enough to ﬁnd λj N, 1 j d, such that after the ∈ ∈ ≤ ≤ substitution (9.2), (9.3) becomes an integral equation for xi+1 over k[˜x1,...,x˜d]. Then xi+1 will be integral over k[˜x1,...,x˜d] and xd+1,...,xi over k[˜x1,...,x˜d,xi+1], hence also over k[˜x1,...,x˜d]. α α1 αd Letx ¯ =(x1,...,xd),x ¯ = x1 ...xd . Write

α j (9.4) F (xi+1) = aαx¯ xi+1. Xαj d d ′ ′ Choose λ1 λ2 λd 2 such that s=1 αsλs + j = s=1 αsλs + j for ≫ ≫···≫ ≫ ′ ′ 6 any two monomialsx ¯αxj =x ¯α xj appearing in (9.4) with non-zero coefficients i+1 6 i+1 P P (the existence of such λj is easy to see by induction on d). Substituting (9.2) into (9.3), the coefficient of the highest order term in xi+1 is a non-zero element of k by the choice of λj. Hence after the substitution (9.2), (9.3) becomes an integral dependence relation of xi+1 over k[˜x1,...,x˜d], as desired. k[x1,...,xn] By induction in i, we find a choice of generators xi such that I is integral over k[x1,...,xd]. QED Remark 9.7. If k is infinite, we can achieve the same result by a transformation of the form xj = xj + cjxi+1, 1 j d, where cj are sufficiently general constants in k. ≤ ≤ It is also worthe mentioning that the analogue of Noether normalization for formal or convergent power series ring is stronger. Namely, we have Theorem 9.8. Let (A,m,k) be a formal (resp. convergent) power series ring over k (resp. C). Let y1,...,yd be a system of parameters of A. Then A is finitely generated as a module over k[[y1,...,yd]] (resp. C y1,...,yd ) (it is easy to see { } that there are no formal or convergent relations among the yi, that is, no formal or convergent power series in the yi is equal to 0 in A).

Thus in the formal and convergent cases it is much easier to choose y1,...,yd: any system of parameters will do. Theorem 9.8 can be proved rather easily using the Weierstrass preparation theorem, but we will not do it here (see [4, Chapter V, (31.6) and Chapter VII, (45.3) and (45.5)]). INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 39

Let A, y1,...,yd be as in Noether Normalization Lemma. Clearly,

d = tr. deg(A/k).

To prove Theorem 9.1, it remains to prove that dim A = d for every m Max(A). m ∈ It is a little easier to prove that dim Am = d for some m Max(A); this is already enough for Theorem 9.1 (1). We prove this weaker result∈ ﬁrst in order to warm up. The plan is to show that dim k[y1,...,yd] = d and then analyze the behaviour of dimension under integral extensions and show that it is preserved by such extensions. Lemma 9.9. Let m Max(k[y ,...,y ]). Then ht m = d. In particular, ∈ 1 d

dim k[y1,...,yd] = d.

′ k[y1,...,yd] Proof. Take any m Max(k[y1,...,yd]) and let k = κ(m) = m . By Propo- ′ ∈ sition 4.2, k is a finite algebraic extension of k. Lety ¯i = yi mod m be the natural image of y in k′; then k′ is generated byy ¯ ,...,y¯ as a k-algebra. For 1 i d, i 1 d ≤ ≤ let f¯i denote the minimal polynomial ofy ¯i over the field k[¯y1,...,y¯i−1]. For each i, pick and fix a polynomial f k[y ,...,y ] whose natural image in k[¯y ,...,y¯ , y¯ ] i ∈ 1 i 1 i−1 i is f¯. Then, by induction on i, k′ = k[y1,...,yd] . Hence m = (f ,...,f ); in par- i (f1,...,fd) 1 d ticular, dim Am = δ(m) d. On the other hand, let Pi k[y1,...,yi] denote the kernel of the natural map≤ k[y ,...,y ] k[¯y ,...,y¯ ].⊂ It is easy to see that 1 i → 1 i (0) $ P1k[y1,...,yd] $ $ Pd = m is a chain of d distinct prime ideals, so that ··· dim Am = d. To complete the proof of Theorem 9.1, it remains to analyze the behaviour of dimension in integral ring extensions. Proposition 9.10. Let A B be an integral ring extension. Then dim A = dim B. ⊂ Lemma 9.11. Let A B be an integral ring extension and P Spec A, Q Spec B such that Q A⊂= P . Then P is maximal if and only if Q ∈is maximal. ∈ Proof. A ֒ B is∩ an integral extension of integral domains. Thus we can further P → Q reduce Lemma 9.11 to Lemma 9.12. Let A B be an integral extension of integral domains. Then A is a field if and only if B⊂is a field. Proof. = is proved by the same argument as = in Proposition 4.2. ⇒ ⇒1 ( =) Assume that B is a field. Let x A. Then x B. Since B is a domain, ⇐ 1 ∈ ∈ −m −m+1 integral over A, x satisfies an integral dependence relation x + b1x + + 1 m−1 ··· bm = 0, where bi A. Then x = b1 b2x bmx A, which proves that A is a field. QED∈ − − −···− ∈ Corollary 9.13. Let A B be an integral ring extension, P Spec A. Then ⊂ ∈ (1) There exists Q Spec B such that Q A = P (2) There are no inclusion∈ relations between∩ primes Q B, satisfying (1). ⊂ 40 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Proof.(1) Localize both A and B by the multiplicative set S = A P . Then \ AP ֒ BS, is an integral ring extension. Let Q be any maximal ideal of BS. Then Q →A must be maximal by Lemma 9.11, hence Q A = PA . ∩ P ∩ P P Then Q B lies over P : (Q B) A = Q A =(Q AP ) A = PAP A = P . /par ∩ ∩ ∩ ∩ ∩ ∩ ∩ 2) Suppose Q Q are two primes of B such that Q A = Q A = P . Then 1 ⊂ 2 1 ∩ 2 ∩ Q1BS and Q2BS are both maximal in BS which contradicts the fact that Q1 $ Q2. Corollary 9.14 (The going up theorem). Let A B be an integral extension of integral domains. Let P P ′ A be prime ideals.⊂ Let Q B be a prime ideal such that Q A = P . Then⊂ there⊂ exists a prime Q′ of B, such⊂ that Q Q′ and P ′ = Q′ A.∩ ⊆ ∩ A B P ′ . Proof. Apply Corollary 9.13 (1) to the extension P ֒ Q and the ideal P ′ → There exists Q′′ Spec B such that Q′′ A = P . Put Q′ = Q′′ B. ∈ Q ∩ P P ∩ Proof of Proposition 9.10. Let A B be an integral ring extension. Let ⊂ Q0 $ $ Qn be a chain of prime ideals of B. By Corollary 9.13 (2), Q0 A $ $ Q A···is a chain of distinct prime ideals of A. This proves that dim B∩ dim···A. n ∩ ≤ Conversely, let P0 $ $ Pn be a chain of prime ideals of A. By Corollary 9.13 (1), there exists Q ···Spec B lying over P . By the going up theorem and induction 0 ∈ 0 on i, we can successively construct a chain of prime ideals Q0 $ $ Qn of B such that Q A = P . This proves that dim A dim B, so that dim···B = dim A. i ∩ i ≤ We come back to the proof of Theorem 9.1. We have an integral domain A of finite type over k. By Noether Normalization Lemma, there exist y1,...,yd, algebraically independent over k, such that A is finite over k[y1,..., yd]. By Lemma 9.9 and Proposition 9.10, dim A = dim k[y1,...,yd] = d. In other words, the largest height of a maximal ideal in A is d. This proves (1) of Theorem 9.1. To prove (2) of Theorem 9.1, we need a stronger result: every maximal ideal in A has height exactly d. The existence of a maximal ideal of height d was proved by starting with a maximal chain of primes in k[y1,...,yd] and constructing a chain of primes in A lying above it recursively, using the going up theorem at each step. Now, we would like to start with a fixed maximal ideal m of A. Since m k[y ,...,y ] is maximal by Lemma ∩ 1 d 9.11, we can consider a chain of d primes (0) $ P1 $ $ Pd = m k[y1,...,yd] in ··· ∩ k[y1,...,yd]. To imitate the procedure in the proof of Proposition 9.10 and construct a chain of primes in A lying over the Pi, we now need a going down theorem. The fact that the going down theorem holds in our context is a somewhat more delicate result; it depends not only on the fact that A is integral over k[y1,...,yd] but also on the fact that k[y1,...,yd] is normal. Thus to complete the proof of Theorem 9.1, it remains to prove two things: Proposition 9.15 (The going down theorem). Let φ : A ֒ B be an integral extension of integral domains, such that A is normal. Let P,P→′ Spec A, Q′ Spec B be such that P P ′ and P ′ = Q′ A. Then there exists Q∈ Spec B such∈ that Q Q′ and P = Q⊂ A. ∩ ∈ ⊂ ∩ INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 41

Lemma 9.16. k[y1,...,yd] is normal.

Once the going down theorem and the normality of k[y1,...,yd] are established, Theorem 9.2 (2) will follow from the following result, proved by imitating the proof of Proposition 9.10. Exercise 9.17. Let A B be a ring extension. Let Q Spec A, P = Q A. Show that: ⊂ ∈ ∩ (1) If A B has the going down property, then ht Q ht P . (2) If for⊂ any prime P ′ P , there are no inclusion relations≥ among the primes of B contracting to ⊂P ′, then ht Q ht P . ≤ Proof of the going down theorem. Let K and L denote, respectively, the fields of fractions of A and B. Replacing B by its integral closure in L and Q′ by a prime ideal of the integral closure lying over Q′ does not change the problem. Furthermore, we may replace L by a normal extension of K and B by the integral closure of A in L. Thus we will assume that L is a normal extension of K and B is integrally closed in L (in particular, any automorphism of L over K maps B to itself). By Corollary 9.13 (1), there exists Q˜ Spec B such that Q˜ A = P . By the ∈ ∩ going up theorem there exists a prime Q˜′ Q˜ in B such that Q˜′ A = P ′. Hence to prove the going down theorem, it is sufficient⊃ to prove the following∩ lemma. Lemma 9.18. Let A, B, K and L be as in Proposition 9.15. Assume that L is a normal extension of K and that B is integrally closed in L. Let P Spec A, Q ,Q Spec B be such that Q A = Q A = P . Then there∈ exists an 1 2 ∈ 1 ∩ 1 ∩ automorphism σ of L over K such that σ(Q1) = Q2. By Lemma 9.18 there exists an automorphism σ such that σ(Q˜′) = Q′. Then Q = σ(Q˜) is the desired prime ideal. It remains to prove Lemma 9.18 (note that for Theorem 9.1 we only need it in the case when L is finite over K). Proof of Lemma 9.18. First, assume that L is finite over K, so that the group G of automorphisms is finite: G = σ1,...,σs . Suppose Q2 = σi(Q1) for any i. Since s{ } 6 s Q1 = σi(Q2) for any i, Q1 σi(Q2) by Lemma 6.33. Take x Q1 σi(Q2). 6 6⊂ i=1 ∈ \ i=1 Then σi(x) / Q2 for any i, henceS S ∈ s (9.5) y = σ (x) / Q . i ∈ 2 i=1 Y But y is invariant under all the automorphisms of L over K, hence there is r N such r r ∈ that y K. Since y is integral over A, we have y A. Then y A Q1 = P Q2, which contradicts∈ (9.5). This proves Lemma 9.18∈ in case L is finite∈ ∩ over K. ⊂ Next, suppose L is not finite over K. Let K′ denote the maximal purely insep- arable extension of K contained in L (K′ is nothing but the invariant field of the group G of automorphisms). Then L is Galois over K′. Let A′ denote the integral closure of A in K′. 42 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Exercise 9.19. Show that the natural map Spec A′ Spec A is a bijection. → By Exercise 9.19, we may replace A by A′ and K by K′; that is, we may assume that L is Galois over K. Let L′ be a finite Galois extension of K contained in L, and let F (L′) = σ G σ(Q L′) = Q L′ . { ∈ | 1 ∩ 2 ∩ } Since K L′ is a finite field extension, Lemma 9.18 holds with B replaced by the integral closure⊂ of A in L′. Hence F (L′) is not empty. The Krull topology on G is defined to be the topology whose basis consists of all the cosets of the form σH, where σ G and H is a normal subgroup of G of finite index. ∈ Exercise 9.20. Show that F (L′) is closed in the Krull topology. Now, the set of finite groups which are homomorphic images of G form a projective system, whose maps are the natural surjective homomorphisms: whenever H H are two normal subgroups of finite index, we have a natural surjective map i ⊂ j G G . Since an automorphism σ G is completely determined by the collection Hi → Hj ∈ of all of its restrictions to finite subextensions K L′ of K contained in L, we have G = lim G . Moreover, the Krull topology on G⊂is, by definition, nothing but the ← H G inverse limit of the discrete topologies on H (that is, it is the topology whose basis G is formed by all the inverse limits of all the sets in H or, equivalently, the coarsest G topology which makes all the natural maps G H of Exercise 3.12 continuous). → G Since we are considering only normal subgroups of finite index, H is finite, so that G the discrete topology on H is compact and Hausdorff. Now, the inverse limit of compact Hausdorff spaces Xi is compact. Indeed, by definition lim Xi Xi. Xi is compact by Tychonoff’s theorem and one checks ← ⊂ i i easily that lim Xi isQ closedQ in Xi using the Hausdorff property of the Xi. ← i Hence G is compact in theQ Krull topology. Now, any finite collection of finite Galois subextensions of L is contained in another finite Galois subextension. Hence any finite collection the closed sets F (L′) has a non-empty intersection. Since G is compact, this implies that F (L′) = . Any element σ F (L′) satisfies the L′ 6 ∅ ∈ L′ conclusion of Lemma 9.18. T T

This completes the proof of the going down theorem. To finish proving Theorem 9.1, it remains to prove the normality of k[y1,...,yd] (Lemma 9.15). By Exercise 4.8, it is sufficient to prove that k[y1,...,yd] is a UFD. This follows by induction on d from the Gauss lemma: Exercise 9.21 (the Gauss lemma). Let A be a UFD and y an independent variable. Then A[y] is a UFD. (Hint: let K be the field of fractions of A. Since K[y] INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 43 is a UFD, the question is reduced to the following: given polynomials f,g,h A[y] and an irreducible element a A such that a fg, either a f or a g.) ∈ ∈ | | | This completes the proof of Theorem 9.1. We end this section by giving an alternative proof of Lemma 9.9 using 9.15–9.17.

A second proof of Lemma 9.9. . Take any m Max(k[y1,...,yd]). We want to ′ k[y1,...,yd] ∈ ′ prove that ht m = d. Let k = m . Then k is a ﬁnite extension of k, so ′ that k[y1,...,yd] k [y1,...,yd] is a ﬁnite ring extension. Let a1,...,ad denote the ⊂ ′ natural images of y1,...,yd in k . Then (y1 a1,...,yd ad) is a maximal ideal of ′ − − k [y1,...,yd] of height d by Example 6.10. Since (y1 a1,...,yd ad) k[y1,...,yd] = m, we have ht m = d by Exercise 9.17. − − ∩

10. Regular Rings. § Regular rings are those rings which correspond to complex manifolds in classical algebraic geometry. In other words, they correspond to algebraic varieties which are locally diffeomorphic to Cn at every point. Yet another way of saying this is that regular varieties do not have self-intersections, corners or cusps (these latter kinds of points are called singularities). Algebraically, we express this intuitive idea by saying that locally at every point there is a local coordinate system with d coordinates, where d is equal to the dimension. In other words, a point ξ on a d- dimensional variety or scheme is regular if there exist d equations locally at ξ whose common set of zeroes is ξ with its reduced structure. Now for precise definitions. Let (A,m,k) be a Noetherian local ring of dimension d. Definition 10.1. A is regular if m can be generated by exactly d elements ∞ ⇐⇒ m mn dimk m2 = d mn+1 = k[t1,...,td] . ⇐⇒ n=0 ∼ Definition 10.2. LIf A is regular of dimension d, a set x1,...,xd of d generators of m is called a regular system of parameters for A {(this is equivalent} to saying 2 that x1,...,xd are linearly independent mod m ). Definition 10.3. Let A be any Noetherian ring. We say that A is regular if Am is regular for all m Max(A). ∈ Properties of regular local rings: 1) A regular local ring is a domain. This follows from the graded algebra description appearing in Definition 10.1 and the following two exercises: Exercise 10.4. Let A be a ring, I A an ideal such that ⊂ ∞ (10.1) In =0. n=1 \

If grI A is a domain then A is a domain. 44 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Exercise 10.5. Let A be a Noetherian ring, m $ A an ideal. Assume either that ∞ m J(A) or that A is a domain. Then mn = (0). ⊂ n=1 T Problem 10.6. Prove that a regular local ring is normal. (Hint: prove that if grI A is normal, then A is normal, provided thatA is Noetherian and I J(A) (in particular, I satisﬁes (10.1)).) ⊂

More generally, it is known that a regular local ring is a UFD (this is a non-trivial theorem of Auslander and Buchsbaum).

Example 10.7. Let k be a ﬁeld. Then the polynomial ring k[x1,...,xd] is regular. Indeed, let m k[x1,...,xd] be a maximal ideal. In the ﬁrst proof of Lemma 9.9, we explicitly constructed⊂ a set of d generators for m. More generally, it is easy to see that if A is a regular ring, then so are A[x1,...,xn] and A [x1,...,xn] , where x ,...,x are independent variables. 1 n Proposition 10.8. Let (A,m,k) be a regular local ring of dimension d. Let l d. ≤ Let x1,...,xl be elements of m, none of which belongs to the ideal generated by the others. The following are equivalent: 2 (1) x1,...,xl are linearly independent mod m . (2) x1,...,xl can be extended to a regular system of parameters of A. (3) A is a regular local ring. (x1,...,xl)

If (1)–(3) hold, we have ht(x1,...,xl) = l. Proof. 1) 2). By Nakayama’s Lemma, x1,...,xd is a regular system of ⇐⇒ 2 m parameters if and only if xi mod m form a basis for m2 . Hence 2) follows from the fact that a collection of linearly independent elements in a vector space can be extended to a basis. 2) = 3). Extend x ,...,x to a system of parameters x ,...,x . Let A′ = ⇒ 1 l 1 d A , m′ = m . Then dim A′ d l, for if an m′-primary ideal was (x1,...,xl) (x1,...,xl) ≥ − generated by fewer than d l elements, together with x1,...,xl we would obtain a set of fewer than d generators− for an m-primary ideal in A. On the other hand, m′ is generated by x ,...,x . Hence dim A′ = d l and A′ is regular. l+1 d − 3) = 1). Let x ,...,x be the largest subset of x ...x which are linearly ⇒ 1 r 1 l independent mod m2. Suppose r

Definition 10.9. Let k be a field. k is perfect if either char k = 0 or char k = p > 0 and kp = k. -Definition 10.10. Let σ : k ֒ k′ be a field extension. We say that k′ is sep arably generated over k if σ→can be decomposed into a purely transcendental extension and a separable algebraic extension. If k(x1,...,xt) is a purely transcen- ′ ′ dental subextension of k over which k is separable algebraic, x1,...,xt is called a separating transcendence base. Exercise 10.11. (1) Any finite field is perfect. (2) If k is perfect, then any algebraic extension of k is separable. (3) More generally, if k is perfect, any finitely generated field extension of k is separably generated.

-Exercise 10.12. Let k ֒ k′ be a finitely generated, separably generated exten → ′ sion of transcendence degree t. Let x1,...,xn be a set of generators of k over k (as a field). Show that there exists a subset of x1,...,xn which is a separating transcendence base. Theorem 10.13. Let k be a field. Let A = k[x1,...,xn] , P Spec A. Let Q be the (f1,...,fl) ∈ inverse image of P in k[x1,...,xn]. Then:

∂fi (1) ht(f1,...,fl)k[x1,...,xn]Q rk mod P . ≥ ∂xj (2) If equality holds in (1), then APis regular. (3) If κ(P ) is separably generated over k and AP is regular, then equality holds in (1). Note that by Exercise 10.11 κ(P ) is always separably generated over k in the case when k is perfect. ′ A ′ Proof. First, assume P is maximal. Let k = P . Then k k is a finite field ′ → extension and k[x1,...,xn] k [x1,...,xn] a finite ring extension. Let ai be the ′ → ′ natural image of xi in k and consider the ideal Q = (x1 a1,...,xn an) ′ ′ − − ⊂ k [x1,...,xn]. Then Q k[x1,...,xn] = Q. The homomorphism k[x1,...,xn] ∩ ′ → k′[x ,...,x ] induces a map ϕ : Q Q between two k′-vector spaces of dimen- 1 n Q2 → Q′2 sion n. Lemma 10.14. ϕ is an isomorphism if and only if k′ is separable over k. Proof. Letg ¯ denote the minimal polynomial of a over k(a ,...,a ) k′, 1 i i 1 i−1 ⊂ ≤ i n and let gi an arbitrary inverse image ofg ¯i in k[x1,...,xi]. Then, as in the ≤ Q proof of Lemma 9.9, g1(x1),...,gn(x1,...,xn) generate Q. Therefore φ( Q2 ) is the vector space generated by the linear forms of g1,...,gn. Hence φ is an isomorphism if and only if det ∂gi / Q. Now, the extension k k′ is separable each of the ∂xj ∈ ⊂ ⇐⇒ n g is separable ∂g i / Q for all i det ∂gi = ∂gi / Q, as desired. i ∂xi ∂xj ∂xi ⇐⇒ ∈ ⇐⇒ i=1 ∈

46 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Now, let r = rk ∂fi . Say, f ,...,f are such that rk ∂fi = r. ∂xj 1 r ∂xj Q Q 1≤i,j≤r ′ This means that the natural images of f ,...,f in Q are linearly independent, 1 r Q′2 Q k[x1,...,xn]Q hence they are linearly independent in 2 is a regular local ring by Q ⇒ (f1,...,fr ) Proposition 10.8 (in particular, it is a domain). Then r = ht(f ,...,f )k[x ,...,x ] ht(f ,...,f ), which proves (1). Now, 1 r 1 n Q ≤ 1 l suppose r = ht(f1,...,fl)k[x1,...,xn]Q. Since (f1,...,fr)k[x1,...,xn]Q is prime, this equality of heights implies that

(f1,...,fr)k[x1,...,xn]Q =(f1,...,fl)k[x1,...,xn]Q,

which shows that A = k[x1,...,xn]Q is regular. P (f1,...,fl) ′ Next, assume that k is separable over k and AP is regular. Then ϕ is an isomorphism. Since AP is regular, (f1,...,fl)=(f1,...,fr) by Proposition 10.8. ′ ∂fi ′ Q rk ∂x is equal to the dimension of the k -vector subspace of ′2 , spanned j Q Q i≤i≤r by the images of f1,...,fr. Since φ is an isomorphism and since f1,...,fr are linearly

2 ∂fi independent mod Q , we have rk ∂x = r, as desired. j Q i≤i≤r Finally, suppose P is not maximal. Let r = rk ∂fi ) mod P . Say, ∂xj ∂f rk i = r. ∂xj Q!1≤i,j≤r

Letx ¯i denote the natural image of xi in κ(P ). Renumbering the xi, we may assume thatx ¯r+1,...,x¯r+t are algebraically independent over k andx ¯r+t+1,...,x¯n are algebraic over k(¯xr+1,...,x¯r+t). Let

k[xr+1,...,xn] k′ = P ∩k[xr+1,...,xn] = k(¯x ,...,x¯ ), P k[x ,...,x ] r+1 n ∩ r+1 n ′′ ′ ′′ ′′ k = k(¯xr+1,...,x¯r+t). By the choice of t, k is algebraic over k and k AP . ′′ ⊂ Since k [x1,...,xr,xt+r+1,...,xn] is obtained from k[x1,...,xn] by localizing with respect to the multiplicative system k[xr+1,...,xr+t], disjoint from P , replacing k by k′′ in the Theorem does not change the problem. In other words, we may assume that t = 0 and k′ is algebraic over k. Then k[x ,...,x ] k′[x ,...,x ] is 1 n → 1 n a finite ring extension. Let P0 =(xr+1 x¯r+1,...,xn x¯n) k[x1,...,xn]. By the going up theorem, there exists an ideal−Q′ k′[x ,...,x− ] lying∩ over Q containing ⊂ 1 n (xr+1 x¯r+1,...,xn x¯n). Since height of ideals is preserved by finite extensions, none of− the statements− in Theorem 10.13 are affected by replacing k by k′, A by ′ k [x1,...,xn] , Q by Q′, etc. For (3) in Theorem 10.13, note that if κ(P ) is separably (f1,...,fl) INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 47 generated over k then by Exercise 10.12 we may choose k′ so that κ(P ) is separably generated over k′. To complete the proof of Theorem 10.13, it is sufficient to show that, under the assumption t = 0, Q is a maximal ideal of k[x1,...,xn], since the theorem has already been proved in this case. By Proposition 4.2, this is equivalent to proving that κ(P ) is algebraic over k. The assumption t = 0 means thatx ¯r+1,...,x¯n are algebraic over k. Thus it remains to prove thatx ¯1,...,x¯r are algebraic over k(¯x ,...,x¯ ). Since k′ is a quotient of k(¯xr+1,...,x¯n) , everything is reduced r+1 n (f1,...,fl)k(¯xr+1,...,x¯n) to the following lemma. Lemma 10.15. Consider elements f ,...,f in k[x ,...,x ] and a prime ideal Q 1 r 1 r ∈ ∂fi V (f1,...,fr), such that det / Q. Then Q Max(k[x1,...,xr]). ∂xj ∈ ∈

Note: When we apply the Lemma to ﬁnish the proof of Theorem 10.13, the ′ ﬁeld k of the Lemma plays the role of k = k(¯xr+1,...,x¯n) in the Theorem; the ideal Q of the Lemma corresponds to the kernel of the natural homomorphism ′ ψ : k [x1,...,xr] κ(P ). Thus the Lemma will imply that κ(P ) is the quotient of ′ → ′ k [x1,...,xr] by a maximal ideal, hence is algebraic over k by Proposition 4.2, as desired. First proof of Lemma 10.15. By Nullstellensatz, there exists m V (Q) such that ∈ ∂fi det / m. By Theorem 10.13, applied to the maximal ideal m k[x1,...,xr], ∂xj ∈ ⊂ ht(f ,...,f )k[x ,...,x ] = r. Since k[x ,...,x ] =(k[x ,...,x ] ) and height 1 r 1 r m 1 r Q 1 r m Q is preserved under localization (Exercise 8.9), ht(f ,...,f )k[x ,...,x ] = r 1 r 1 r Q ≥ dim k[x1,...,xr]. Hence ht(f1,...,fr)k[x1,...,xr]Q = r, which proves that every ideal containing (f1,...,fr) is maximal. Second proof of Lemma 10.15, not using Nullstellensatz. We use induction on r. ′ Let k = κ(Q). If r = 1, f1 is a polynomial in one variable with non-zero deriva- tive, hence Q is generated by one of the irreducible factors of f1 and the result is immediate. Suppose r > 1 and the Lemma is known for r 1. Let −

k[x1,...,xr−1] k˜ = Q∩k[x1,...,xr−1] . (Q k[x ,...,x ]) ∩ 1 r−1

∂fr We may assume that / Q. Then ht(fr)k˜[xr] = 1 and ∂xr ∈

Qk˜[x ] =(f )k˜[x ] . r Q∩k˜[xr] r r Q∩k˜[xr ]

Hence

(10.2) (f ,...,f )k[x ,...,x ] = ((f ) + Q k[x ,...,x ])k[x ,...,x ] . 1 r 1 r Q r ∩ 1 r−1 1 r Q 48 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Since the rank of the matirx ∂fi is clearly independent of the choice of the base of ∂xj the ideal (f ,...,f ), (10.2) implies that there exist f˜ ,...,f˜ Q k[x ,...,x ] 1 r 1 r−1 ∈ ∩ 1 r−1 ∂f˜i such that det ∂x / Q. By the induction assumption, j 1≤i,j≤r−1 ∈

ht(Q k[x1,...,xr−1]) = r 1, ∩ −

hence ht Q = r = dim k[x1,...,xr], which proves that Q is maximal. This completes the proof of Theorem 10.13. Corollary 10.16. Let k be a perfect field, A a regular ring, which is a localization of a finitely generated k-algebra, P a prime ideal of A. Then AP is regular. Remark 10.17. Corollary 10.16 is true for any regular local ring, whether or not it is a localization of a finitely generated k-algebra, but we will not prove it here. See [3, (19.B), Theorem 48, p. 142] for a proof. Example 10.18. Let k be an imperfect field, a k kp. Let x be an independent k[x] ∈ \ variable and put K = (xp−a) . Then K is a field (Proposition 4.2 and Lemma 4.4), ∂(xp−a) that is, a regular local ring of dimension 0. At the same time, ∂x 0. We have K = κ(0) for (0) SpecK. This shows that the hypothesis that κ(P )≡ be separably generated over k ∈is necessary in Theorem 10.13 (3). Of course, a zero dimensional local ring is regular if and only if it is a field. We end this section by considering the case of regular one-dimensional local rings. Problem 10.6 says that any regular local ring is normal. We will now show that the converse is true for one-dimensional local rings. Theorem 10.19. Let A be a one-dimensional Noetherian local ring. Then A is regular if and only if A is normal. Proof. Although “only if” is a special case of Problem 10.6, we will prove it directly. Suppose A is regular. Then the maximal ideal m of A is principal, that is, generated ∞ by a single element x. Let y be any non-zero element of m. Since mn = (0) in n=0 any Noetherian local ring (Exercise 10.5 or use Nakayama’s Lemma),T there exists N N such that y / mN . y can be written in the form y = xnz, where z / (x) = m. Of∈ course, this representation∈ is unique. This implies that A is a UFD, hence∈ normal. “If” is more subtle. We assume that A is normal and want to prove that m is principal. We define the notion of I−1 for an ideal I in a domain A. Notation. Let A be a domain, K the field of fractions of A, I an ideal of A. We define I−1 = x K xa A for all a I . I−1 is an A-submodule{ ∈ | of K∈. Of course,∈II−}1 is an ideal of A. Lemma 10.20. Assume that (A, m) is local. Then I is principal if and only if II−1 = A. Proof. If I =(x) is principal, then I−1 = Ax−1 K. Hence II−1 = A. ⊂ INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 49

Conversely, suppose II−1 = A. Then

n (10.2) 1= aixi, i=1 X −1 where ai I, xi I . Since aixi A for all i, (10.2) implies that aixi / m for some i. Say,∈ x a ∈/ m. This means∈ that a x is invertible, so that 1 I−∈1. Now 1 1 1 1 x1 ∈ 1 −1 b ∈ take any other b I. Since I , A, hence b (x1). This proves that ∈ x1 ∈ x1 ∈ ∈ I =(x1), so that I is principal. Exercise 10.21. Let A be any domain (i.e. not necessarily local), I A and ideal. −1 ⊂ Prove that II = A if and only if I is locally principal, that is, IAP is principal for any P Spec A. Locally principal ideals together with their inverses generate a group (under∈ the operation of multiplication). In geometry, elements of this group are called line bundles on Spec A. We continue with the proof of Theorem 10.19. By Lemma 10.20, it remains to prove that mm−1 = A for a local 1-dimensional normal Noetherian ring (A, m). Suppose mm−1 $ A. Since mm−1 is an ideal, we have mm−1 m. Since 1 m−1, ⊂ ∈ mm−1 = m. By induction on n, m(m−1)n = m for any n N. ∈ Lemma 10.22. Let A be a domain with ﬁeld of fractions K. Take x K. Let ∞ ∈ A : x = a A ax A . If x is integral over A, then (A : xn) = (0). The { ∈ | ∈ } n=1 6 converse holds if A is Noetherian. T n−1 n−1 Proof. If x is integral over A, then xn Axi for some n N. Then xl Axi ∈ i=0 ∈ ∈ i=0 P n−1 P for all l n by induction on l. Then A : xl (A : xi) for all l n. This proves ≥ ⊃ i=1 ≥ the ﬁrst statement. T Conversely, assume that A is Noetherian and take a non-zero element

∞ y (A : xn). ∈ n=1 \ n Then Ay−1 is isomorphic to A as an A-module, hence is Noetherian. Axn form i=0 an ascending chain of submodules of Ay−1. By the Noetherian propertyP this chain n−1 must stabilize. Hence for some n, xn Axi, which implies that x satisﬁes an ∈ i=0 integral equation of degree n over A. P Coming back to the proof of Theorem 10.19, take any x m−1. Since ∈ m(m−1)n = m for all n N, ∈ 50 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

we have A : xn m for all n N. By Lemma 10.22, x is integral over A, hence x A by normality⊃ of A. This∈ proves that ∈ (10.3) m−1 = A.

We now use the fact that A is one dimensional to derive a contradiction. Take a non- zero element x m. Since A is a domain, x is not a zero divisor, hence ht(x)=1. ∈ Since dim A = 1, this means that (x) is m-primary, so that m = (x). Then there exists n N such that mn−1 (x) but mn (x). Take any y mn−1 (x). Then p y m−1∈ A, which contradicts6⊂ (10.3). This6⊂ completes the proof.∈ \ x ∈ \ Let A be an integral domain with field of fractions K. We may consider the -integral closure A¯ of A in K. The inclusion A ֒ A¯ gives rise to the natural mor phism of integral schemes π : Spec A¯ Spec A→. If A is Noetherian and satisfies certain mild additional conditions (they→ will be specified later, but they hold for any algebra of finite type over a field and any quotient or localization of a formal or convergent power series ring), A¯ is also Noetherian. The canonical morphism π is called normalization of the scheme Spec A. It was defined (surprisingly late – in the nineteen forties) by Oscar Zariski. This is an example of the usefulness of the algebraic language in geometry: this notion, extremely important as it turned out to be, did not occur to anyone until the algebraic language was developed. Normalization will be of interest to us when we discuss resolution of singularities. Geometrically, Theorem 10.19 says that normalization resolves the singularities of curves. More generally, it says that for an arbitrary Noetherian scheme normalization resolves the singularities in codimension 1. When normalization was defined, the theorem of resolution of singularities was known for almost a century, yet it was quite a surprise that it had such a simple and elegant proof and that the procedure for desingularization had such a simple description.

11. Sheaves. § Until now, we only considered affine schemes. However, inasmuch as we are interested in studying global questions in algebraic geometry (for instance, if we want to define the analogue of compact complex manifolds), we are led to consider more general schemes, for which affine schemes are basic buidling blocks. Arbitrary schemes are glued from affine ones in the same way that manifolds are glued from open subsets of Rn. The notion which expresses the idea of glueing global objects from local data in modern algebraic geometry is that of a sheaf. The definition of sheaves is the subject of this section. Before making the general definitions, we consider the case of an affine scheme Spec A, where A is a domain (as we will see, the absence of zero divisors makes life easier). We saw that open affine subschemes of Spec A of the special form Spec Af , f A, form a basis for the Zariski topology. It is natural to regard the ring Af as the∈ “coordinate ring” or the “ring of regular functions” on the affine open scheme Spec Af . However, these basic affine open sets do not, in general, exhaust all the INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 51 open subsets of Spec A. For the purposes of glueing affine schemes together, we are forced to consider these more general affine open subsets. In particular, we must define what we mean by the ring of regular functions on an arbitrary open set U. The main ideas are illustrated by the following exercises.

Exercise 11.1. Let A be a domain. Prove that A = AP = Am, P ∈Spec A m∈Max(A) where we view all the AP as subsets of the ﬁeld of fractionsT of A. T The point of this exercise is that an aﬃne scheme can be reconstructed in a straightforward way from the collection of all its local rings. Exercise 11.2. Let A be a domain and let U Spec A be an open set. Let ⊂ (11.1) = A . OU P P\∈U Prove that:

(1) For every P U, ( U )P ∩OU = AP . (2) There are natural∈ inclusionsO

ι .U ֒ Spec ֒ Spec A (11.2) → OU → (3) If A is noetherian then so is . OU

If ι of (11.2) is a set theoretic bijection then it is natural to regard U = Spec U as an affine open subscheme of Spec A. Of course, it may happen that ι isO not surjective, in which case U is open but not affine. An example of that is provided by letting U be the plane without the origin. In other words, let A = k[x,y], U = Spec A (x,y) . Then = A. The conclusion we can draw from all of the \{ } OU above is that it is natural to think of U as the ring of regular functions on the set U. O It is not immediately obvious how to generalize the above definitions to the case when A is not a domain, because it is not clear what should replace AP (if A is P ∈U not a domain, the different local rings AP are not naturally subringsT of any single ring of fractions. To overcome this difficulty, we will develop the powerful machine of sheaf theory. Now for the general definitions. Let X be a topological space. Definition 11.3. A pre-sheaf of sets on X is the data consisting of a set F FU for each open set U X and a map ρUV : U V for each pair U, V of open sets such that V U⊂, satisfying the followingF conditions:→ F ⊂ (1) is a one-element set. F∅ (2) ρUU is the identity map for all U. (3) Given any three open sets W V U, we have ρ = ρ ρ . ⊂ ⊂ UW VW ◦ UV 52 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

The set U is called the set of sections of over U; the maps ρUV are called restrictionF maps. F

If the sets U have some additional structure, such as rings or groups, we may talk about a presheafF of ring (resp. groups). In this case, we require, in addition, that the maps ρUV be homomorphisms of the structure in question. Also, Definition 11.3 (1) has to be modified to say that ∅ is the zero ring (resp. the trivial group). Let be a presheaf. F F Definition 11.4. We say that is a sheaf if the following two conditions (identity F and gluability) are satisfied for any open set U and any open covering U = Uλ. λ∈Λ

For λ, µ Λ, let us write Uλµ for Uλ Uµ, ρλ for ρUUλ and ρλµ for ρUλUλµS. We require ∈ ∩ (1) Identity: for f,g , if ρ (f) = ρ (g) for all λ Λ then f = g. ∈FU λ λ ∈ (2) Gluability: given sections fλ Uλ , λ Λ such that ρλµ(fλ) = ρµλ(fµ) for all λ, µ Λ, there exists f ∈Fsuch that∈ f = ρ (f) for all λ. ∈ ∈FU λ λ Remark 11.5. The meaning of Definition 11.4 is that is a sheaf if and only if each can be completely recovered from local information.F Indeed, Definition FU 11.4 amounts to saying that for any open covering U = Uλ, λ∈Λ S f ρ (f ) = ρ (f ) for all λ, µ Λ . U ∼= λ λ∈Λ Uλ λµ λ µλ µ F ( { } ∈ F ∈ ) λ∈Λ Y Examples 11.6. I. The constant sheaf F on X. Let F be a set, put = F for U all U and let ρ : F F be the identity map for all V U. F UV → ⊂ II. Let X be a topological (resp. smooth) manifold and let U be the ring of continuous (resp. C∞) functions on U. Given two open sets V F U and f , ⊂ ∈ FU define ρUV (f) to be the restriction of f to V (this motivates the name restriction maps). Then together with the maps ρ define a sheaf of rings. FU UV Exercise 11.7. Let X = Spec A be an integral affine scheme (in other words, A is a domain). For each open set U X, let denote the ring defined above. For a pair ⊂ OU of open sets V U, we have the obvious inclusion U AP AP V . ⊂ O ≡ P ∈U ⊂ P ∈V ≡ O the natural projection AP AP induces a ringT homomorphismT ρUV : P ∈U → P ∈V U V . Prove that theQ rings U Q, together with the maps ρUV , form a sheaf of Orings→ on OX. This sheaf is called theO structure sheaf of X. Now let X = Spec A be any affine scheme (i.e. we no longer require A to be a domain). Let U be an open subset of X. We define to be the following subset OU of AP . For a Q Spec A, g / Q, let ρgQ denote the natural localization map P ∈U ∈ ∈ ρgQQ: Ag AQ. We define (11.3) → = f A P U g A P,f A s.t. Q g, f = ρ (f) . OU {{ P ∈ P }P ∈U | ∀ ∈ ∃ ∈ \ ∈ g ∀ 6∋ Q gQ } INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 53

Expressing (11.3) in words, U consists of those elements fP AP P ∈U AP O { ∈ } ∈ P ∈U such that every point P is contained in some basic open set Spec Ag such thatQ all the f , Q A , are images of a single element f A . Q ∈ g ∈ g Exercise 11.8. (a) Prove that if A is a domain, (11.3) agrees with the definition of U given earlier in (11.1). O(b) Keep the above notation. Let V be an open subset of U. Show that the natural projection AP AP induces a map ρUV : U V . P ∈U → P ∈V O → O (c) ProveQ that Qwith the restriction maps ρ form a sheaf of rings on X. {OU } UV (d) Prove that if A is noetherian then U is noetherian for each open set U. (e)∗ Prove that = A (and hence O = A for each f A). OX OX\V (f) f ∈ Definition 11.9. is called the structure sheaf of X. OU We now give an alternative definition of U . The reader will appreciate, however, that all these definitions revolve around oneO basic idea: glueing global sections from local data. More precisely, (11.3) defines sections over U by glueing them from the infinitesimal data of f , P U, while in the following definition we consider an P ∈ open cover U = and glue sections over U from sections over the Uλ. λ∈Λ Again, let X =S Spec A be any affine scheme, and let U be an open subset of X.

Consider an open covering of the form U = Uλ, where each Uλ = Spec Agλ is a λ∈Λ basic aﬃne open subset. Then for each λ, µ S Λ, Uλ Uµ = Spec Ag gµ . For each ∈ ∩ λ λ, µ Λ, let ρλµ : Agλ Agλgµ denote the localization by gµ (ρλµ is nothing but the restriction∈ map of the→ structure sheaf of Spec A. Exercise 11.10. (a) Prove that the ring is naturally isomorphic to the follow- OU ing subring of Agλ : λ∈Λ Q (11.4) = f A λ, mu Λ,ρ (f ) = ρ (f ) . OU ∼ {{ λ ∈ gλ }λ∈Λ | ∀ ∈ λµ λ µλ µ }

(b) Given two open subsets of X, V U, show the restriction map ρUV can be described as follows in terms of (11.4).⊂ Choose an open cover of V by basic aﬃne

open sets: V = Vψ, where Vψ = Spec Ahψ and where for each ψ Ψ there exists ψ∈Ψ ∈ λ(ψ) Λ satisfyingS g h (in particular, V U ). Let ρ : A ∈ λ(ψ) | ψ ψ ⊂ λ(ψ) λ(ψ),ψ gλ(ψ) → Ahψ be the localization by hψ. For f U , represent f by fλ Agλ λ∈Λ, as in (11.4). Show that ρ (f) is the element∈ O of , described by{ ρ ∈ (f} ) . UV OV { λ(ψ),ψ λ(ψ) }ψ∈Ψ ι .Exercise 11.11. Show that there are natural maps U ֒ Spec U ֒ Spec A Show also that for any P U, ( ) A , so that→ we haveO a natural→ map U P ∩OU ∼= P A for each P U.∈ O OU → P ∈ If ι is a set theoretic bijection, then we say that U = Spec U is an aﬃne open subscheme of Spec A. O 54 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Stalks and germs. Let X be a topological space and a presheaf of sets on X. Let P be a point of X. Open sets containing P , togetherF with the inclusion maps V U, form an inverse system. Since, by deﬁnition, the correspondence U ⊂ → FU is a contravariant functor from the open sets of X to sets, the sets U form a direct system. F

Definition 11.12. The stalk of at P is P := lim U . For an open set U P F F −→ F ∋ U∋P and a section f , the natural image of f in , which we will sometimes denote ∈FU FP by fP , is called the germ of f at P . Exercise 11.13. Describe the stalk at a point P in each of the three examples of sheaves above. Sheafication. The following three exercises describe how, given a presheaf , one F can canonically associate to it a sheaf ˜. F Exercise 11.14. Let be a presheaf of sets (resp. rings, resp. groups). For an F open set U X, define ˜U to be the following subset of P : ⊂ F P ∈U F Q (11.5) ˜ = f P U open V, P V U and g such that FU {{ P ∈FP }P ∈U | ∀ ∈ ∃ ∈ ⊂ ∈FV Q V, f = g . ∀ ∈ Q Q} Show that is a sheaf if and only if for each open set U X, = ˜ . F ⊂ FU FU Exercise 11.15. Let be a presheaf. For each open set U X, let ˜ be defined F ⊂ FU as in (11.5). Given two open set V U, the natural projection P P ⊂ P ∈U F → P ∈V F induces a map ρ˜ : ˜ ˜ . Show that the sets ˜ , togetherQ with theQ maps UV FU → FV FU ρ˜ , form a sheaf. The resulting sheaf ˜ is called the sheafication of . UV F F Exercise 11.16. Let X = Spec A be an affine scheme and X = Uλ, with λ∈Λ

Uλ = Spec Agλ a covering of X by basic affine open sets. Show that theS structure sheaf of X is the sheafication of the presheaf , defined as follows: A = , if U is contained in one of the U AU OU λ = the zero ring otherwise

Given open set V U, the restriction map is deﬁned to be the restriction map of the ⊂ structure sheaf of Uλ if U is contained in one of the Uλ and the zero map otherwise.

12. Schemes. § Let A be a ring. The n-dimensional aﬃne space over A is, by deﬁnition, n A = Spec A[t1,...,tn]. The natural inclusion A A[t1,...,tn] gives rise to the A ⊂ INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 55

n n natural projection AA Spec A. Intuitively, one should think of AA as the direct product of Spec A with→ the usual n-dimensional affine space. Of course, this is strictly heuristic. If one tried to make this last statement precise, one would run into trouble with the definition of “the usual affine space” (affine space over what field? and why over a field? remember that A might be something like Z). The heuristic picture is precise only if A is of finite type over a field k. The example of the n-dimensional affine space points to another feature of scheme theory and algebraic geometry as a whole: the built in capacity for talking about families of varieities or schemes (or deformations of a variety or a scheme), which is such a central notion in the subject. Namely, if Spec B Spec A is a morphism of affine schemes, we may regard Spec B as a family of affine→ schemes over Spec A. Spec A is the base of the family, while the fibers V (PB), P Spec A are individual n ∈ members of the family. For example, AA can be regarded as the trivial family of affine spaces over Spec A. The fiber over P Spec A is the affine space An , ∈ κ(P ) which is the scheme-theoretic version of the algebraic variety κ(P )n. Now that we have developed the theory of sheaves, we are ready for a definition of a scheme. Definition 12.1. A scheme is a topological space X together with a sheaf of rings , such that there exists an open covering X = Uλ, where each Uλ is of the O λ∈Λ form Uλ = Spec Aλ for some ring Aλ, such that forS each λ Λ, the restriction of ∈ to Uλ is the structure sheaf of Spec Aλ. The sheaf is called the structure Osheaf of X. O We now give a more explicit description of in terms of the structure sheaves O of the Uλ. Let X be a scheme, and let V be an open subset of X. Consider an open covering of V of the form V = Vψ, where each Vψ = Spec Bψ is an affine open subset. ψ∈Ψ Moreover, for each ψ, µS Ψ, V V is an open subset of X and therefore can be ∈ ψ ∩ µ covered by open affine subschemes: Vψ Vµ = Vψµφ, where Vψµφ = Spec Bψµφ. ∩ φ∈Φψµ For each ψ, µ Ψ and φ Φ , let ρ : B S B denote the restriction map ∈ ∈ ψµ ψµφ ψ → ψµφ of the structure sheaf of Spec Bψ. Exercise 12.2. (a) Show that is naturally isomorphic to the following subring OV of Bψ: ψ∈Ψ Q (12.1) = f ψ, µ Ψ, φ φ Φ ,ρ (f ) = ρ (f ) . OV {{ ψ}ψ∈Ψ | ∀ ∈ ∈ ∈ ψµ ψµφ ψ µψφ µ } (b) Given two open subsets of X, W V , we can choose an affine open cover ⊂ W = Wδ, such that for each δ ∆ there exists ψ(δ) Ψ satisfying Wδ Vψ(δ). δ∈∆ ∈ ∈ ⊂ For f S V , represent f by fψ ψ∈Ψ, as in (12.1). Show that ρVW (f) is the element∈ of O , described by ρ{ } (f ) . OW { Uψ(δ)Vδ ψ(δ) }δ∈∆ 56 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Definition 12.3. Let X, Y be schemes. A morphism from X to Y is a map of topological spaces X Y , together with affine open coverings Y = Vλ, → λ∈Λ −1 X = Uλφ such that π (Vλ) = Uλφ and such that for each λ,S φ the λ∈Λ,φ∈Φλ φ∈Φλ map π S: U V is induced by a morphismS of affine schemes. |Uλφ λφ → λ n n We now define the projective space PA over Spec A. By definition, PA is glued n from n + 1 affine pieces, each of which is isomorphic to AA. n Define n = U , where U = Spec A x0 ,..., xi−1 , xi+1 ,..., xn . The meaning PA i i xi xi xi xi i=0 h i n of this notationS is that, in addition to saying that each Ui is isomorphic to AA, it also describes the glueing of U with U along U U . Namely, if we write i j i ∩ j Ui = Spec A[y1,...,yn], Uj = Spec A[z1,...,zn], with i < j and the variables listed in the same order as above, our notation is a concise way of writing the change of coordinates on U U : i ∩ j

zl yl = , 0 l i or j

To restate the same deﬁnition in yet another way, Ui Uj is a basic aﬃne open set of both ∩

x x x x U = Spec A 0 ,..., i−1 , i+1 ,..., n and i x x x x i i i i x x x x U = Spec A 0 ,..., j−1 , j+1 ,..., n ; j x x x x j j j j

xj xi we have Ui Uj = Ui V ( ) = Uj V ( ), where the last equality is described ∩ \ xi \ xj algebraically by the identiﬁcation of the localizations

x0 xi−1 xi+1 xn x0 xj−1 xj+1 xn A ,..., , ,..., = A ,..., , ,..., . x ∼ xi xi xi xi j xj xj xj xj xi xi xj

We have a natural projection Pn Spec A, deﬁned to be, in each coordinate chart, A → the natural projection An Spec A: the maps A → x x A A 0 ,..., n → x x i i Spec A Spec An ← A INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 57

n patch together to give a morphism Spec A PA. The ring A[x0,...,xn] is some- ← n times called the homogeneous coordinate ring of PA. n We now give another, more canonical description of PA, independent of our choice of coordinate system and the covering by charts, directly in terms of the graded A-algebra A[x0,...,xn]. ∞ Let G = Gn be a graded algebra. An ideal I G is said to be homogeneous n=0 ⊂ L n if it can be generated by homogeneous elements ( given x = xi I, with ⇐⇒ i=0 ∈ ∞ P xi Gi, we have xi I for all i,0 i n). For example, Gn is a homogeneous ∈ ∈ ≤ ≤ n=1 ideal of G. L Definition 12.4. P roj G is defined to be the set of all the homogeneous prime ∞ ideals of G, not containing n=0 Gn, endowed with the Zariski topology: the closed sets are all the sets of the form V (I) = P P roj G I P , where I ranges over the homogeneousLideals of G. { ∈ | ⊂ } Remark 12.5. To see that P roj G is a scheme, we can cover it by affine charts as follows. Let G = G0[xλ]λ∈Λ, with xλ homogenous of degree kλ > 0. Then

P roj G = Uλ where Uλ = Spec (Gxλ )0. Here Gxλ has the obvious structure of λ∈Λ a graded algebraS and the subscript 0 denotes its degree 0 part. In other words, there is a 1-1 correspondence between homogeneous prime ideals

P G, not containing xλ, and prime ideals of (Gxλ )0. This correspondence is given by⊂P P G (G ) . Conversely, given Q Spec (G ) , it is easy to show → xλ ∩ xλ 0 ∈ xλ 0 that QGxλ is a homogeneous prime ideal, whose contraction to G is the element of P roj G corresponding to Q. Verifying the details is left as an exercise. p Take λ, µ Λ. The coordinate transformations on U U is described by ∈ λ ∩ µ

k k (Gxλ )0 x λ = (Gxµ )0 x µ = Gxλxµ µ λ 0 kµ kλ xλ xµ

The construction globalizes naturally. Namely, if X = Spec Ai, we can define i n n n PX = iPAi , where the PAi are glued together in the obviousS way. The canonical ∪ n n morphisms PA Spec Ai are also glued together to give the natural map PX X. One of the main→ subjects of study in the rest of this course, and in algebraic→ geometry at large, are birational projective morphisms, or blowings up. We end this section by defining birational and projective. In the next section we will give a more constructive definition of blowing up, rather than characterizing it by its abstract properties of being birational and projective. Definition 12.6. Let π : Y X be a morphism of schemes. We say that π is projective, or that Y is projective→ over X, if there exists an open affine cover −1 X = Spec Ai and n N with the following property. Let Vi = π (Spec Ai) i ∈ S 58 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

and let πi = π Vi . Then we require that for each i, πi have a factorization of the ιi n | αi form Vi ֒ PAi Spec Ai, where ιi is a closed embedding and αi is the natural projection.→ To say→ that π is projective is a little weaker than saying that π has a factorization of the form

,Y ֒ Pn X (12.2) → X → since the definition of projective requires that a factorization of type (12.2) exist locally on X. Definition 12.7. Let X be a scheme, ξ X a point. Let Spec A X be an open affine chart of X containing ξ and let P be∈ the prime ideal of A corresponding⊂ to ξ. The ring AP is called the local ring of ξ in X and is denoted by X,ξ. Of course, is independent of the choice of the affine open chart Spec A.O OX,ξ Definition 12.8.

(1) A scheme X = Spec Ai is said to be reduced if all the Ai are reduced. i (2) X is said to beSintegral if it is reduced and irreducible as a topological space (i.e. is not a union of two non-trivial closed sets).

Remark 12.9. If X is integral, then each Spec Ai is a dense open set (otherwise its closure and its complement would express X as a union of two closed sets). Hence each Ai is a domain, otherwise there would exist a non-trivial decomposition of Spec Ai into closed sets, which could be extended to a non-trivial decomposition of X by passing to closures. We also have that Spec Ai Spec Aj = for any i and j, otherwise we would have X =(X Spec A ) (X Spec∩ A ). Then6 ∅ the (0) ideal is \ i ∪ \ j common to both Spec Ai and Spec Aj. Let ξ denote the common point of all the Spec A which corresponds to the (0) ideal in A . Since ξ Spec A = Spec A , i i { } ∩ i i we have ξ = X. This point ξ is called the generic point of X. Definition{ } 12.10. Let X be an integral scheme and ξ its generic point. Then X,ξ = K(X) is a field, called the field of rational functions on X or the functionO field of X. K(X) is nothing but the common field of fractions of all of the Ai. Let π : X Y be a morphism of schemes. Let ξ X and η = π(ξ) Y . Then π induces a map→ π∗ : . We have m =∈m . ∈ OY,η → OX,ξ Y,η X,ξ ∩ OY,η Exercise 12.11. Let σ : A B be a ring homomorphism and σ∗ : Spec B Spec A the corresponding morphism→ of affine schemes. Show that Ker σ is nilpotent→ if and only if σ∗ is dominant, that is, σ∗(Spec B) = Spec A. Now, suppose X and Y are integral and let ξ and η be the respective generic points of X and Y . Let π : X Y be a dominant morphism. This is equivalent to saying that π maps the generic→ point of X to the generic point of Y . Hence in this .(case we get a field extension π∗ : K(Y ) ֒ K(X → INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 59

Definition 12.12. π is birational if π∗ : K(Y ) ∼ K(X) is an isomorphism. Definition 12.13. A morphism of affine schemes→ π : Spec B Spec A is said to be of finite type if B is a finitely generated A-algebra. Let→π : X Y be a morphism of arbitrary schemes. π is said to be locally of finite type if→ the affine open coverings V and U can be chosen so that for all λ Λ, φ Φ , the { λ} { λφ} ∈ ∈ λ map π Uφλ is a morphism of finite type of affine schemes. π is said to be of finite type if,| in addition, the set Φ is finite for each λ Λ (that is, there are finitely λ ∈ many Uλφ over each Vλ). For example, any projective morphism is of finite type. Remark 12.14. Let π : X Y be a morphism of finite type between integral schemes. Then π is birational→ there exist open subsets U X, V Y , such that ∼⇔ ⊂ ⊂ π induces an isomorphism U V . Proof. Suppose π is birational.→ Take an affine open set V Y and an affine open subset U π−1(V ), of finite type over V . Let U = Spec⊂A, V = Spec B. Clearly, replacing⊂ X by U and Y by V does not change the problem. Since π is

a1 an birational and of finite type, we may write A = B ,..., , where ai,bi A. b1 bn ∗ ∈ Then π induces an isomorphisms of V V (b1,...,bnh) with its pre-image.i \ 13. Blowing up. § Let A be a ring, X = Spec A, I =(f1,...,fn) a finitely generated ideal of A. Definition 13.1. The blowing up of X along I is the birational projective mor- ˜ n−1 phism π : X X, defined as follows. Consider the morphism ϕ : X V (I) PX , which is defined→ precisely below, but which should intuitively be\ thought→ of as n ˜ the map which sends every ξ X V (I) to (ξ, (f1(ξ) : : fn(ξ))) PX . X ∈ \ n−1 ··· ∈ is defined to be the closure ϕ(X V (I)) PX in the Zariski topology. More \ ⊂ n precisely, the map ϕ is described as follows. Write X V (I) = Spec Afi , \ i=1 n n−1 x xi−1 xi+1 x S = Spec A 1 ,..., , ,..., n . Then in each of the n affine charts PX xi xi xi xi i=1 ϕ is definedS by h i x x x x A A 1 ,..., i−1 , i+1 ,..., n fi ← x x x x i i i i fj xj fi ← xi It is easy to check that those n maps glue together to give a morphism X V (I) n−1 \ → PX . Remark 13.2. n−1 (1) Since the blowing up X = ϕ(X V (I)) PX , the natural projection n−1 \ ⊂ PX Spec A induces a map X Spec A. In particular, X is projective over Spec→ A. e → e e 60 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

(2) The natural map π : X Spec A is an isomorphism away from V (I) (the → inverse mapping is given by ϕ). This means that the map π : X X is birational. e → As we will see later, blowing up is characterized by the properties of beinge birational and projective.

Remark 13.3. For all 1 i, j n, the equation fixj fjxi A[x1,...,xn] vanishes ≤ ≤ − ∈ n−1 on X. This describes the full set of defining equations of X in PX in the case when A is Noetherian and V (I) is a complete intersection (which means ht I = n), bute not in general (a proof of this assertion is postponed).e In general, the defining n−1 equations f X in PX = Proj A[x1,...,xn] are not easy to describe. If A is an integral domain, we can think of X as e n f1 e fn (13.1) X = Spec A ,..., . f f i=1 i i [ e If fact, even if A is not a domain, we can still make sense of the description (13.1) (see Example 13.13 below). Example 13.4. 1) Blowing up the plane at a point. Let A = k[x,y], I = (x,y), 1 so that Spec A is the affine plane and I is the origin. Then PA = Proj A[u1, u2] 1 should intuitively be thought of as the direct product of Pk with the plane. We 1 have the map Spec A 0 PA, described locally in each of the two coordinate u1 \{u1} →x u2 u2 y charts by A[ ] Ay, and A[ ] Ax, . u2 → u2 → y u1 → u1 → x 1 X is defined in P by the equation xu2 yu1 = 0. For example, if k = R, then A − X is nothing but the Möbius band. Perhapse the most useful way of thinking about the blowing up X is that it is a schemee glued together from two coordinate charts e u2 u1 X = Spec k u1, Spec k u2, , u1 u2 [ where the glueing is implicite in the notation. 2) Similarly, we can blow up the affine n-space at the origin. Let

A = k[x1,...,xn], I =(x1,...,xn).

n−1 n−1 Then P = Proj A[u1, . . . ,un]. X P is the subscheme deﬁned by the X ⊂ X equations x u x u , 1 i, j n. Again, X is covered by n coordinate charts of i j − j i ≤ ≤ u1 ui−1 ui+1 e un the form Spec k u ,..., u , ui, u ,..., u . i i i i e 3) Finally, theh blowing up of Spec k[x1,...,xi n] along (x1,...,xl) for l

Intuitively, we may think of this last construction as first blowing up the origin in Spec k[x1,...,xl] and then taking the direct product of the whole situation with Spec k[xl+1,...,xn]. Another characterization of blowing up. We now give yet another description of the blowing up of Spec A along an ideal I, equivalent to the above, but without reference to any particular ideal base (f1,...,fn). First, we need a general fact about closed subschemes of P rojG for a graded algebra G. As mentioned earlier, such closed subschemes are defined by homogeneous ideals of G. Surprisingly, while for any ring A, by definition, distinct ideals of A define distinct closed subschemes of Spec A, it is possible for two distinct homogeneous ideals of G to define the same closed subscheme of P rojG. The next exercise gives an exact characterization of pairs of homogeneous ideals of G define the same closed subscheme of P rojG. Let G be a graded algebra. Let x be a set of homogeneous, positive degree { λ}λ∈Λ generators of G over G0. Let I be a homogeneous ideal of G. Definition 13.5. The saturation of I is the homogeneous ideal of G, defined by

¯ nλ I = y G λ Λ nλ N s.t. yx I . { ∈ | ∀ ∈ ∃ ∈ λ ∈ } I is said to be saturated if I = I¯. Exercise 13.6. (1) Show that the definition of I¯ does not depend on the choice of the set of generators xλ λ∈Λ. (2) show that I¯{is saturated.} (3) Show that for any homogeneous ideal I, √I = √I¯ (this shows that at least I and I¯ define the same closed subset of P rojG. (4) Let I and J be two homogeneous ideals of G. Show that the closed subschemes V (I) and V (J) of P rojG coincide if and only if I¯ = J¯. In particular, I¯ is the greatest homogeneous ideal of G which defines the same closed subscheme as I.

We come back to our invariant characterization of blowing up. Let A be a ring, I a ﬁnitely generated ideal, as above. Then X ∼= P roj GrI A. Indeed, let I = (f1,...,fn). We have the obvious surjective homomorphism φ : A[u1,...,un] GrI A, given by ui fi, so thate P roj GrI A is a closed subscheme n−1 → 7→ of PA . Furthermore, let J = Ker φ. Then

α J = g(u) = aαu homogeneous polynomials such that g(f1,...,fm)=0 . n o X Let J ′ denote the greates homogeneous ideal having the property ϕ(Spec A V (I)) \ ⊂ V (J ′) (by deﬁnition, X = V (J ′).

e 62 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

′ Exercise 13.7. Show that J = J¯, so that P roj GrI A = X. Now let X be any scheme, not necessarily affine. We would like to define the e notion of blowing up X along an ideal. However, we run into a difficulty: it is not clear what “ideal” means in this context. The natural thing to do is to think about sheaves of ideals (which are a special case of sheaves of modules). This brings us to another central notion in the theory of schemes — that of quasi-coherent and coherent sheaves of modules. Sheaves of modules. Coherence and quasi-coherence. Roughly speaking, quasi-coherent sheaves can be thought of as a generalization of vector bundles in topology and complex analysis, with coherent sheaves corresponding to vector bundles of finite rank. First, we need to define tensor product for modules over a ring. For a ring A, a free A-module is a module which is isomorphic to a (possibly infinite) direct sum of copies of A.

Definition 13.8. Let A be a ring and M1, M2 A-modules. The tensor product M1 A M2 is defined to be the free module generated by the symbols x1 x2 modulo ⊗ ′ ′ ⊗ the submodule generated by all the relations of the form (a1x1 +a1x1) x2 a1(x1 x ) a′ (x′ x ) and x (a x + a′ x′ ) a (x x ) a′ (x x′⊗). − ⊗ 2 − 1 1 ⊗ 2 1 ⊗ 2 2 2 2 − 2 1 ⊗ 2 − 2 1 ⊗ 2 Note that if M is an A-module and B an A-algebra, then M A B has a natural structure of a B-module. Note also that if φ : M M is a⊗ homomorphism of 1 → 2 A-modules, it makes sense to talk about the induced homomorophism φ A B : M B M B. ⊗ 1 ⊗A → 2 ⊗A Definition 13.9. Let A be a ring, M an A-module and S A a multiplicative set. ⊂x The localization MS of M by S is defined to be MS = s x M, s S / , where is the usual equivalence relation: x y whenever{ there| ∈ exists u∈ S}such∼ ∼ s ∼ t ∈ that u(xt ys)=0. MS has a natural structure of an AS-module (hence also of an A-module).− Exercise 13.10. Keep the notation of Definition 12.6. Show that M = M A . S ⊗A S Let X be a topological space. Let be a sheaf of rings on X. O Definition 13.11. A sheaf of -modules is a sheaf of abelian groups on X, O F such that for each open set U X, U has the structure of an U -module, and for any two open sets V U, the⊂ restrictionF map is a homomorphismO of ⊂ FU →FV U -modules (note that V , and hence also V are naturally U -modules via the Orestriction map O ). F O OU → OV Example 13.12. Let X = Spec A be an affine scheme. As usual, we denote by the sturcture sheaf of X. Let M be an A-module. We define a sheaf of O M -modules as follows: for each open set U, put U = M A U . The restriction mapsO are induced in the obvious wayM from the⊗ restrictionO maps of . MU →MV O INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 63

Definition 13.13. Let X be a scheme and let denote the structure sheaf of X. A sheaf of -modules is said to be quasi-coherentO if there exists an affine F O open covering X = Uλ such that the restriction of to each Uλ is of the form λ∈Λ F described in ExampleS 13.12. is said to be coherent if, in addition, U is a finitely generated -module for eachF open set U. F OU

Example 13.14. The structure sheaf of X itself is a coherent sheaf of modules. O Remark 13.15. Let X be a scheme. There is a natural 1-1 correspondence between quasi-coherent ideal sheaves on X and closed subschemes of X. Namely, let be a I quasi-coherent ideal sheaf. Let X = Uλ be the aﬃne open covering of Deﬁnition

13.13. Then V ( Uλ ) deﬁnes a closed subscheme of Yλ Uλ. Yλ is a closed I S ⊂ λ subscheme of X, which we will denote by V ( ). S I Let X = Spec Ai be a scheme, a coherent ideal sheaf on X. We can now i I deﬁne the blowingS up π : X X of X along . Namely, let I denote the ideal of → I i sections of on Spec A ; I is an ideal of A . Let X Spec A denote the blowing I i i i i → i up of Spec Ai along Ii. Ite is a routine matter to show that all the Xi can be glued together in a natural way to give the blowing upeπ : X X of X along . Note → I n that there is no reason to expect π to factor through the projectivee space PX for any n; we only know that such a factorization exists locallye on X. We will now give yet another characterization of blowing up, this time by a universal mapping property.

Let π : X Y be a morphism of schemes and a quasi-coherent ideal sheaf on → I Y . Let X = Spec Bij and Y = Spec Ai be the open coverings appearing in i,j∈Φi i the definition ofS morphism (DefinitionS 12.3). For each i and each j Φi we have a ∗ ∈ homomorphism Ai Bij. Let π denote the ideal sheaf on X whose ideal sections over Spec B is I B→ . π∗ is a quasi-coherentI ideal sheaf on X. ij i ij I Let X be a scheme and a coherent ideal sheaf on X. The idea, which we now I explain in detail, is that the blowing up π : X X of X along is characterized by the universal mapping property with respect→ to making invertibleI (Definition I 13.19 below). e If I = (x) is a principal ideal in a ring A, then I A as A-modules. If, ∼= Ann x in addition, x is not a zero divisor in A, then I ∼= A as A-modules. Recall that an ideal I is said to be locally principal if it is principal in every local ring AP , P Spec A. The next Lemma and its Corollary show that given a locally principal ideal∈ in a ring A (not necessarily Noetherian), Spec A can be covered by finitely many affine open charts of the form Spec Ag, such that IAg is principal for each g. Example 13.16. Let A be a Dedekind domain (that is, a one-dimensional regular ring, such as Z[√5i]), and I an ideal of A of height 1. Then I is locally principal (since A is a regular 1-dimensional ring), but it need not be principal (Remark 64 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

1.15).

Lemma 13.17. Let I be a locally principal ideal in a ring A. Let f = fλ λ∈Λ be a set of generators of I. Then for every P Spec A, IA =(f )A for{ some} λ. ∈ P λ P Proof. Replacing A by AP , we may assume that A is local with maximal ideal P and that I is principal. Let x be a generator of I. Then for every λ Λ there exists ∈ aλ A such that fλ = aλx. If aλ P for all λ Λ, then x I = fA P (x), hence∈ x = yx for some y P = (1∈ y)x = 0 =∈ x = 0 = ∈I = (0) and⊂ there ∈ ⇒ − ⇒ ⇒ is nothing to prove. Hence we may assume that one of the aλ is a unit in A. Then −1 x = aλ fλ and J =(fλ), as desired. Corollary 13.18. Let I be a locally principal ideal in a ring A and let f = f { λ}λ∈Λ be a set of generators of I. Then there exists a ﬁnite subset f1,...,fn f and a n { } ⊂

ﬁnite open aﬃne cover Spec A = Spec Agi such that IAgi =(fi)Agi , 1 i n. i=1 ≤ ≤ Proof. By Lemma 13.17, S

(13.2) ((fλ): I) = A λX∈Λ

(otherwise, localizing at a maximal ideal containing ((fλ) : I), we would find λ∈Λ a localization A of A such that ((f ) : I)A = A Pfor any λ, that is, I is not P λ P 6 P generated by any of the fλ). By (13.2), 1 belongs to some finite sum of ideals of the form (fλ) : I, λ Λ. Hence there exists a finite subset f1,...,fn f and ∈ n { } ⊂ gi (fi) : I, 1 i n, such that gi = 1. Then IAgi = (fi)Agi and no prime ∈ ≤ ≤ i=1 P n of A can contain all of the gi, so that Spec A = Spec Agi is the desired open i=1 cover. S Definition 13.19. Let I be a locally principal ideal in a ring A. I is said to be invertible if Spec A can be covered by affine charts of the form Ag, g A, such that for each g, IA =(x )A , where x A is not a zero divisor. ∈ g g g g ∈ g Of course, if A is a domain, then invertible and locally principal are the same thing (in that case, both properties are equivalent to saying that II−1 = A by Exercise 10.21, hence the name “invertible”). Note also that if I is invertible, then we can strngthen Corollary 13.18 to say, in addition, that for each i,1 i n, f is ≤ ≤ i not a zero divisor in Agi . This fact is obtained by the same argument as Corollary 13.18, if we replace (13.2) by the stronger equality (Ann(Ann fλ))((fλ): I) = A. λ∈Λ The point is that an element x A is not a zeroP divisor if and only if Ann x = (0) Ann(Ann x) = A. ∈ Definition⇐⇒ 13.20. An ideal sheaf on a scheme X is locally principal if there I exists an affine open cover X = Ui, Ui = Spec Ai, such that Ui is a principal i I S INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 65

ideal of Ai for all i. is said to be invertible if each of the Ui is principal and generated by an elementI which is not a zero divisor. I Again, if X is integral, then invertible and locally principal are the same thing.

Example 13.21. Let A be a ring and I =(f1,...,fn) a ﬁnitely generated ideal of n A. Let X = GrI A be the blowing up of Spec A along I. By deﬁnition, X = Ui, i=1

where Ui = P P roj GrI A fi / P . Then Ui = Spec ((GrI A)fi )0, whereSfi is e { ∈ | ∈ } e viewed as an element of degree 1 in GrI A. Localization by fi annihilates Ann(fi), so that f is not a zero divisor in ((Gr A) ) . Then we may identify ((Gr A) ) i I fi 0 I fi 0 ≡ A f1 ,..., fi−1 , fi+1 ,..., fn , where f in the denominator makes sense since Ann(fi) fi fi fi fi i fi is noth a zero divisor in this chart.i Thus A f f f f U = Spec 1 ,..., i−1 , i+1 ,..., n . i Ann(f ) f f f f i i i i i We have A f f f f I 1 ,..., i−1 , i+1 ,..., n = Ann(f ) f f f f i i i i i A f f f f =(f ,...,f ) 1 ,..., i−1 , i+1 ,..., n 1 n Ann(f ) f f f f i i i i i A f f f f =(f ) 1 ,..., i−1 , i+1 ,..., n , i Ann(f ) f f f f i i i i i so that π∗I is invertible on X. Since we are dealing with a local property, this statement remains valid even if X is not affine. In other words, if π : X X is the blowing up of a coherent ideale sheaf , then π∗ is invertible. → I I e We now point out that this property is also sufficient to characterize blowing up. Namely, the blowing up π of is the smallest (in the sense explained in Theorem 13.22 below) projective morphismI such that π∗ is invertible. More precisely, we have the following theorem. I Theorem 13.22 (the universal mapping property of blowing up [2, Propo- sition II.7.14, p. 164]). Let X be a scheme, a coherent ideal sheaf on X and I π : X X the blowing up of . Let ρ : Z X be another morphism such that ρ∗ → I → I is invertible. Then ρ factors through X in a unique way. e The Noetherian hypothesis appearing in the statement of this result in [2] is e superfluous. Proof of Theorem 13.22. Because of the asserted uniqueness of the factorization, the problem is local on X: once we solve the problem over each affine chart of X, uniqueness will guarantee that the resulting morphisms will patch together nicely. 66 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Thus we may assume that X = Spec A is affine and I = (f1,...,fn) A is a finitely generated ideal. Similarly, according to Corollary 13.18, we may⊂ assume that Z = Spec B is affine, IB = (fi)B for some i, 1 i n, and that fi is not a zero divisor in B. Write ≤ ≤ n A f f f f X = Spec 1 ,..., j−1 , j+1 ,..., n . Ann(f ) f f f f j=1 j j j j j [ By assumption,e we are given a homomorphism ρ∗ : A B. It is sufficient to check → that ρ∗ factors in a unique way through A f1 ,..., fi−1 , fi+1 ,..., fn , where i Ann(fi) fi fi fi fi is as above. Since IB = (fi)B for some i, for eachh j = i there exists bj i B such 6 ∈ that fj = bjfi. Since fi is not a zero divisor in B, bj is unique. Hence the unique homomorphism ψ : A f1 ,..., fi−1 , fi+1 ,..., fn B, factoring ρ∗ is given by Ann(fi) fi fi fi fi → h i f (13.3) ψ j = b . f j i The fact that (13.3) gives a well defined homomorphism ψ follows easily from the fact that fi is not a zero divisor. Let X be an integral scheme. We saw that every blowing up of X X is a birational and projective morphism. Next, we show that, conversely, every→ birational, projective morphism X X is locally the blowing up of some coherente ideal sheaf, so that blowings up and→ birational projective morphisms are essentially the same thing. e Theorem 13.23. Let X be an integral scheme and π : X X a birational projec- → tive morphism. Then there exists an open affine cover X = Spec Ai and finitely e i generated ideals Ii Ai such that for each i, π π−1(A ) is theS blowing up of Ii. ⊂ | i Proof. Since the problem is local on X, we may assume that X = Spec A is affine n−1 and that X is a closed subscheme of PA for some n. Let x1,...,xn be the homogeneous coordinates on Pn−1. For 1 i n, let ≤ ≤ e x x x x U = Spec A 1 ,..., i−1 , i+1 ,..., n . i x x x x i i i i Since V = X U is a closed subscheme of U and an open subscheme of X, i ∩ i i Spec is an affine open subscheme of X and is a homomorphic image of OVi OVi x1 xi−e1 xi+1 xn xej A x ,..., x , x ,..., x . For 1 i, j n, j = i, let fji denote the image of x i i i i ≤ ≤e 6 i inh Vi . Let K denote the fieldi of fractions of A. Since π is birational, all the fij O are elements of K. By definition, we have the following relations among the fij, for all the allowed values of i, j, and l:

fijfji =1 (13.4) fijfjlfli =1. INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 67

To complete the proof of the Theorem, it is suﬃcient to ﬁnd f1,...,fn A such f ∈ that f = i for all i, j (for then π is the blowing up of I = (f ,...,f )). To ij fj 1 n construct f1,...,fn, let g be a common denominator of the fij, that is, an element of A such that gfij A for all i, j. Put f1 = g, fi = gfi1 for 2 i n. Then for ∈ fi fi1 ≤ ≤ all the allowed i, j, we have = = fij by (13.4), as desired. fj fj1

Let X be a scheme, a coherent ideal sheaf on X, π : X X the blowing up along . The closedI subscheme Y = V ( ) is called the center→ of the blowing I I up π. We will sometimes refer to π as “the blowing up alonge Y ”. The coherent ideal sheaf π∗ defines the closed subscheme π−1(Y ) X. Since π∗ is invertible, I ⊂ I π−1(Y ) has codimension 1 in X in the case when X is Noetherian (Corollary 8.11). Codimension 1 subschemes are called divisors. The subschemee π−1(Y ) is called the exceptional divisor ofe the birational mapeπ. By definition, π induces an isomorphism X π−1(Y ) = X Y . \ ∼ \ Strict Transforms. Let Z be a scheme, a coherent ideal sheaf on Z. Let e I ι : X ֒ Z be a closed subscheme of Z with its natural inclusion ι. Let π : Z˜ Z be the→ blowing up of . Let X := π−1(X V ( )) Z, where ”¯” denotes the closure→ in the Zariski topology.I \ I ⊂ e e Definition 13.24. X is called the strict transform of X under π.

−1 −1 Of course, X πe (X) X π (V ( )). To distinguish it from the strict transform, π−1(X⊂) is sometimes⊂ called∪ the totalI transform of X under π. e e Theorem 13.25. X together with the induced morphism ρ : X X is nothing but the blowing up of the coherent ideal sheaf ι∗ on X. → I e e Proof. Again, the problem is local on Z, so that we may assume that Z = Spec B is affine and I B is a finitely generated ideal. It is sufficient to prove that X has ⊂ the universal mapping property of blowing up (that is, X satisfies the conclusion of Theorem 13.22). Here we are using a general fact about universal mappinge properties: any object satisfying the conclusion of Theoree m 13.22 is necessarily unique up to a canonical isomorphism, since given any two such objects Theorem 13.22 guarantees the existence of a canonical morphism in both directions. Let J denote the defining ideal of X in B, A = B , I¯ = IA. By definition, ι∗ J I is the coherent ideal sheaf ¯ on Spec A given by the ideal I¯. Let ˜ι be the inclusion I map ˜ι : X Z˜. → First, we show that ρ∗ ¯ is invertible. Let I = (f ,...,f ). Let f¯ denote the I 1 n i natural imagee of fi in A. Then I¯ =(f¯1,...,f¯n). Cover Z˜ by open affine charts: Z˜ = n Spec B , where B = B f1 ,..., fn . For 1 i n, let V = X Spec B . i i Ann fi fi fi i i i=1 ≤ ≤ ∩ ThenS Vi is a closed subscheme ofh Spec Bi;i let Ji denote the defining ideal of Vi in e B . Let A = Spec Bi , so that V = Spec A . For each i we have a commutative i i Ji i i 68 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

diagram B A i −−−−→ i

x x B A  −−−−→  Now, since IBi = (fi)Bi for each i, IA¯ i = IAi = (f¯i)Ai is principal. To show that f¯ is not a zero divisor in A , suppose f¯x¯ = 0 for somex ¯ A . Let x i i i ∈ i be any representative ofx ¯ in Bi. Then fix Ji, so that x Ji(Bi)fi . Now, ∗ ∈ ∈ ∗ Spec(Bi)fi = Ui V (π I) and JiBfi is the defining ideal of Vi V (π I) in Bfi . \ ˜ ∗ \∗ Since π induces an isomorphism Z V (π I) Z V (I), X V (π I) ∼= X V (I). \ → \ ∗ \ \ Hence Vi = X Ui is the Zariski closure of Vi V (π I). Now, since x vanishes on ∗ ∩ \ Vi V (π I) in Ui, it belongs to the ideal of its Zariski closuree in Ui. This proves \ ∗ that x Ji, hencee x ¯ = 0, as desired. We have shown that ρ ¯ is invertible. Now,∈ let α : Y X be another morphism such that α∗ ¯ isI invertible. It remains → I to show that α has a unique factorization β : Y X; then we can conclude that X is the blowing up of X along ¯ by Theorem 13.22.→ Since α∗ ¯ = α∗ι∗ ¯ =(ι α)∗ , I I I ◦ I ι α has a unique factorization γ : Y Z˜ by Theoreme 13.22. In view of the assertede uniqueness◦ of β, we may assume that→Y = Spec C is affine and that α(Y ) is entirely contained in one of the coordinate charts of Z˜, say in the first one, Spec B1. Thus we have ring homomorphisms

γ∗ B C 1 −−−−→ (13.5) x x B A −−−−→ ∗   ∗ It remains to show that γ factors through A1, that is, γ (J1) = (0). By (13.5), γ∗(JB ) = (0). We have natural maps B B ֒ B , so that 1 → 1 → f1 (13.6) (γ∗ B )(JB )=0. ⊗ f1 f1 l Since f1 is not a zero divisor in B1 and J1Bf1 = JBf1 , we have f1J1 JB1 for ∗ ⊂ some l N, so that γ (J1) = (0) by (13.6). This completes the proof of Theorem 13.25. ∈ Example 13.26. Let k be a field, u, v independent variables. Let Z = Spec k[u, v], I =(u, v), X = Spec k[u,v] Z. (u2−v3) ⊂ The blowing up Z of Z along I is covered by two affine charts: Z = Spec k u , v v ∪ Spec k u, v . Let us denote the coordinates in the first chart U by u , v , so that u 1 1 1 v = v , u = u v .e Let u , v be the coordinates in the seconde chart U , so that 1 1 1 2 2 2 u = u2, v = u2v2. To calculate the strict transform X of U2, we first find its full inverse image. This inverse image is defined by the equation u2 v3, but wirtten in the new coordinates: − e u2 v3 = u2 u3v3 = u2(1 u v3). − 2 − 2 2 2 − 2 2 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 69

Here u2 = 0 is the equation of the exceptional divisors. To obtain the strict transform X, we must factor out the maximal power of u2 out of the equation. In this 3 case, X U2 is defined by 1 u2v2. In U1, we have e ∩ − e u2 v3 = u2v2 v3 = v2(u2 v ). − 1 1 − 1 1 1 − 1 Here v = 0 is the equation of exceptional divisor, so that X U = V (u2 v ). In 1 ∩ 1 1 − 3 particular, note that although X had a singularity at the origin, X is non-singular. Thus, in this example we started with a singular variety X ewith one singular point, blew up the singularity and found that the strict transform ofeX became non- singular. This procedure is called resolution of singularities of X. The problem of Resolution of Singularities. Definition 13.27. A scheme X is said to be Noetherian if there exists a finite n affine open cover X = Spec Ai, such that each Ai is a Noetherian ring. i=1 S Definition 13.28. A scheme X is said to be regular if for every ξ X, the local ring is regular. This is equivalent to saying that X can be covered∈ by affine OX,ξ charts Spec Ai, such that each Ai is a regular ring. k[x,y,z] Exercise 13.29. Let X = Spec z2−xy denote the non-singular quadratic cone, 3 embedded in Ak. Construct a resolution of singularities of X by blowing up a suitable ideal of k[x,y,z] and showing that the strict transform of X under this blowing up is regular. Question. Let X be a reduced Noetherian scheme. Does there exist a sequence of blowings up X X X such that X is regular? n → n−1 →···→ n It is known, though we will not prove it here, that some mild technical conditions must be imposed on the ring A in order for the answer to above question to be affirmative. We have not yet developed enough tools to state all of these conditions, but we will state one of them. This condition has to do with the closedness of singular locus, and we will discuss it right now. Let A be a Noetherian ring. The regular locus of A is, by definition, Reg(A) = P Spec A AP is regular . The singular locus is Sing A = Spec A Reg(A). {Using∈ Theorem| 10.13, it is possible} to show that if A is an algebra of finite\ type over a field, then Sing A is closed in A. For example, if A = k[x1,...,xn] is a domain and (f1,...,fl) r = ht(f1,...,fl), then Sing(A) is defined in Spec A by the ideal generated by all the r r minors of the Jacobian matrix ∂fi . Similar Jacobian criteria exist in the × ∂xj case of formal power series rings and complex-analytic local rings. However, there are some pathological Noetherian rings whose singular locus is not closed. This is only one of the pathologies which needs to be ruled out in order to have resolution of singularities. 70 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

Definition 13.30 [3, Chapter 13, (32.B)]. A Noetherian ring A is said to be (1) J-0 if Reg A contains a non-empty Zariski open set (2) J 1 if Reg A is open in Spec A (3) J − 2 if any reduced finite type A-algebra is J 1. − − Note that if A is reduced, then Reg A is always non-empty since it contains all the minimal primes of A. Thus J-1 implies J-0 for reduced rings, but not in general. As we already mentioned, finite type algebras over a field and their localizations, formal and convergent power series rings are all known to be J-2. The J-2 condition is known to be necessary for the existence of resolution of singularities. There is one other property of Noetherian rings, called the G property (for Grothendieck), which is known to be necessary, but we will state it later, in the continuation of this course. In practice, resolution of singularities is usually achieved by locally embedding the given singular scheme X in a regular scheme Z, constructing a sequence of blowings up of regular subschemes of the ambient regular scheme and studying the effect of these transformations on the strict transform of X. The following fact comes in very useful in the above setup. Problem 13.31. Let Z be a regular scheme, Y Z a regular subscheme. Let ⊂ Z Z be the blowing up along Y . Then Z is regular. → The significance of resolution of Singularities: e e (1) There are many constructions which can only be defined for non-singular schemes, or at least which are much easier to define and study for non-singular schemes or varieties. These include Hodge theory, singular and étale cohomology, the canonical divisor, etc. (2) Inasmuch as one is interested in the problem of classification of algebraic varieties, it is natural to separate all the reduced and irreduclble varieties (or integral schemes) into birational equivalence classes, that is, to group together all the schemes having the same field of rational functions. From that point of view, resolution of singularities answers a very natural question: does every birational equivalence class contain a non-singular scheme? Once this question has been an- swered affirmatively (as it will be in the continuation of this course), one may, on the one hand, look for birational invariants, that is, numbers associated to the given birational equivalence class and defined in terms of some non-singular model, and, on the other hand, address the finer questions about the relation between different non-singular models in the given birational equivalence class and also what can be said about the relation between resolution of singularities and the original singular variety or scheme which it dominates. This is a very active area of research, known as the Mori program; it has been the stage of some spectacular recent developments, particularly in the case of 3-dimensional varieties, or 3-folds. (3) Embedded desingularization is a somewhat stronger form of resolution of singularities, which is particularly useful for applications. Suppose that X is embedded in a regular scheme Z. Embedded desingularization asserts that there INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 71 exists a sequence π : Z Z of blowings up along non-singular centers, under which the total transform π−1→(X) of X becomes a divisor with normal crossings, which means that locally ate each point of Z, π−1(X) is defined by a monomial with respect to some regular system of parameters. Geometrically, it means that at every point of Z there exists a local coordinate systeme such that π−1(X) looks locally like a union of coordinate hyperplanes, counted with certain multiplicities. Thus divisors with normale crossings locally have a very simple structure. There are many situations in which it is useful to know that any closed subscheme can be turned into a divisor with normal crossings by blowing up. For example, this is used for compactifying algebraic varieties. Let X be a regular algebraic variety over a field k, embedded n in some projective space Pk . If X is not projective over Spec k, that is, if X is not n ¯ closed in Pk , we can always consider its Zariski closure X, which is, by definition, projective over Spec k. The problem is that even though we started with a regular X, X¯ may well turn out to be singular. Resolution of singularities, together with its embedded version, assures us that, after blowing up closed subschemes, disjoint from X, we may embed X in a regular projective variety X′ such that X′ X is a normal crossings divisor. \ (4) Finally, resolution of singularities is useful for studying the singularities themselves. Namely, let ξ X be a singularity and let π : X X be a desingularization. We may adopt the following∈ philosophy for studying the→ singularity ξ. All the regular points are locally the same; every singular pointe is singular in its own way. We may regard resolution of singularities as a way of getting rid of the local complexity of the singularity ξ and turning it into global complexity of the regular scheme X. Thus some global invariant of X may also be regarded as invariants of the singularity ξ. For example, if X is a surface and ξ is isolated, then π−1(ξ) is a collectione of curves on the regular surfacee X. By embedded resolution, we may −1 further assume that π (ξ) is a normal crossings divisor. If Ei , 1 i n, −1 { } ≤ ≤ are the irreducible components of π (ξ), thene the intersection matrix (Ei.Ej) (or, n equivalently, the dual graph of the configuration Ei) is an important numeri- i=1 cal invariant associated to the singularity ξ. A goodS illustration of the usefulness of replacing local difficulties by global is Mumford’s theorem which asserts that a normal surface singularity which is topologically trivial is regular. More preicsely, given a normal surface singularity ξ X over C, one may consider its link, which is the intersection of X with a small∈ Euclidean sphere centered at ξ. The link is a real 3-dimensional manifold. Mumford’s theorem asserts that if the link is simply connected, then ξ is regular. At first, it seems unlikely that anything intelligent at all can be said about the link, until one realizes that the link is nothing but the n boundary of a tubular neighbourhood of the collection Ei of non-singular curves i=1 on a non-singular surface. After that the link becomesS relatively easy to analyze. We now turn to the theory of valuations, which is a very important tool in the subject of resolution of singularities and in all of birational geometry. 72 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

14. Valuations. § We will start with the algebraic definition of valuations and will then move on to their beautiful geometric interpretation in birational geometry. Definition 14.1. An ordered group is an abelian group Γ together with a subset P Γ (here P stands for “positive elements”) which is closed under addition and such⊂ that Γ = P 0 ( P ). Remark 14.2. P {induces} − a total ordering on Γ: a 0, b > 0 = a + b > 0”. Note that an ordered group is necessarily torsion-free.⇒ Examples 14.3. Z, R with the usual ordering are ordered groups. Any subgroup Γ R is an oredred group with the induced ordering (more generally, any subgroup ⊂ of an ordered group is an ordered group). Zn with the lexicographical ordering (cf. Definition 14.4) are ordered groups. Definition 14.4. Let Γ1,...,Γr be ordered groups. The lexicographical ordering r on their direct sum Γi is defined by i=1 L (a ,...,a ) < (b ,...,b ) i, 1 i n such that 1 n 1 n ⇐⇒ ∃ ≤ ≤ a1 = b1

a2 = b2 . .

ai−1 = bi−1

r All the ordered groups which will appear in this course will be of the form Γi, i=1 where Γi R for all i and the total order is lexicographic. L We are⊂ now ready to define valuations. Let K be a field, Γ an ordered group. Let K∗ denote the multiplicative group of K. Definition 14.5. A valuation of K with value group Γ is a surjective group homomorphism ν : K∗ Γ, such that for any x,y K∗, → ∈ (14.1) ν(x + y) min ν(x), ν(y) and ≥ { } Exercise 14.6. Let K be a field, ν a valuation of K and x,y non-zero elements of K such that ν(x) = ν(y). Show that in this case equality must hold in (14.1), that is, 6 (13.2) ν(x + y) = min ν(x), ν(y) . { }

Example 14.7. Let X be an integral scheme, K = K(X), ξ X such that X,ξ is a regular local ring. As usual, let m be the maximal ideal∈ of . O X,ξ OX,ξ INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 73

∗ Define νξ : k(x) Z by → ν (f) = max n f mn , f . ξ { | ∈ X,ξ} ∈ OX,ξ f νξ extends from X,ξ to all of K in the obvious way by additivity: νξ( g ) = νξ(f) O n − mX,ξ νξ(g). νξ induces a group homomorphism because n+1 is an integral domain. mX,ξ In the above example, note that ξ is any scheme-theoreticL point; for example, it could stand for the generic point of an integral codimension 1 subscheme. In that case, the condition that X,ξ is non-singular holds automatically whenever X is normal (Theorem 10.19).O Remark 14.8. Let A be a domain, K its field of fractions, I A an ideal. We can generalize the above example as follows. Define ⊂

ν (f) = max n f In , for f A. I { | ∈ } ∈

In general, νI is a pseudo-valuation, which means that the condition of additivity in the definition of valuation is replaced by the inequality νI (xy) νI (x) + νI (y). νI In ≥ is a valuation if and only if n is an integral domain (a condition which always ⊕ I +1 holds if I is maximal and AI is regular). Valuations of the form ν , I Spec A are called divisorial. The reason for this I ∈ name is that even if dim AI > 1, we can always blow up Spec A along I. Let π : X Spec A be the blowing up along I. Then K(X) = K(X). → n n The property that I is a domain means that D˜ := V (π∗I) = P roj I ⊕ In+1 ⊕ In+1 is irreducible.e Then is a regular local ring of dim1 ande ν = ν ˜ . This OX,D˜ I D example illustrates an important philosophical point about valuations: a valuation is an object associated to the field K, that is, to an entire birational equivalence e class, not to a particular model in that birational equivalence class. Thus to study a given valuation, one is free to perform blowings up until one arrives at a model which is particularly convenient for understanding the given valuation. The next several examples will be located on the plane. Let k be a field, K = k(u, v) a pure transcendental extension of k of degree 2. Surprisingly, valuations of K are already highly non-trivial and interesting to study. They display almost all of the same properties as valuations of arbitrary fields, at least those arising in algebraic geometry. Example 14.9. Let Γ = (1,π) = a + bπ a,b Z R. Let { | ∈ } ⊂ ν(v)=1, (14.3) ν(u) = π.

A very important fact for us is the observation that (14.3) determines ν completely. Indeed, by additivity, specifying ν(u) and ν(v) determines the value of every α β α β monomial: for any α, β N0, we have ν(u v ) = απ +β. Now, let f = cαβu v . ∈ P 74 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

By (14.2), ν(f) = min πα + β cαβ6=0 . Now it is clear that ν extends uniquely to all of K by additivity.{ | }

Example 14.10. Let K be as in the previous example. Let Γ = Z2 with lexicographical ordering. Let ν(v)=(0, 1), ν(u)=(1, 0). As above, this data determines ν completely, that is, ν(uαvβ)=(α, β) and ν( c uαvβ) = min (α, β) c =0 . α,β { | α,β 6 } ExerciseP 14.11. Generalize Examples 14.7 and 14.10 as follows. Let X be an integral scheme. Consider a collection X1 $ X2 $ $ Xn $ X of closed integral subschemes such that the local ring is regular··· for each i. Deﬁne a valuation OXi,Xi−1 ν : K(X)∗ Zn with lexicographical ordering in a way which yields Example 14.7 if n = 1 and→ Example 14.10 in the case n = 2, X = Spec k[u, v], X = (u, v) and 1 { } X2 = V (u).

Since an abelian group is automatically a Z-module, it makes sense to talk about elements of Γ being linearly independent over Z. In view of the above examples, it is very easy to show that any valuation of k(u, v) such that ν(u) and ν(v) are linearly independent is completely determined by ν(v) and ν(u). We now give an example of a valuation with ν(u) and ν(v) linearly dependent over Z, such that ν is not determined by ν(u) and ν(v).

Example 14.12. Take K = k(u, v), as before. We define a valuation ν : K∗ Q as follows. → 3 Let Q−1 = v, Q0 = u. Let ν(Q−1)=1, ν(Q0) = 2 . Let pn be the n-th prime number, assume that Qn is defined and that ν(Qn) = qn βn = for some qn N. pn ∈ pn qn 1 Put Qn+1 = Qn v and define ν(Qn+1) = qn + . We define a valuation ν − pn+1 with value group Q as follows. Let f k[u, v]. For α Q, let Pα denote the ideal ∈ n ∈ n γi of k[u, v] generated by all the products of the form Qi with γiβi α. i=1 i=1 ≥ Put ν(f) = max α f Pα . It is not too hardQ to show thatP this valuation { | ∈ } Q ν has the following alternative description. Let t be a new variable. Let k[[t + ]] denote the ring of formal power series in t with exponents in the set of non-negative rational numbers, such that the set of exponents appearing in any given series is well ordered. There is an infinite increasing sequence α1 < α2 < α3 < ... of rational numbers and an infinite sequence of constants cn k¯, such that the denominator ∈ Q+ of αn is pn and such that ν is induced from the obvious t-adic valuation on k[[t ]] via the injection

v t → ∞ u c tαi . → i i=0 X INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 75

Example 14.13. The next example may be regarded as a special case of Example 14.12, in which αi Z for all i N0. Let K = k(u, v), as before. ∈ ∈ Let Kˆ = k((u, v)) denote the ﬁeld of fractions of k[[u, v]]. Choose an element ∞ i ∗ t k[[u, v]] k(u, v) of the form t = u civ , ci k . ∈ \ − i=1 ∈ Letν ˆ : Kˆ Z2 be given byν ˆ(t) =P (1, 0),ν ˆ(v) = (0, 1), where we take the → 2 ∗ lexicographical order on Z . Put ν :=ν ˆ K . Sinceν ˆ(K ) (0) Z, we obtain ∗ | ⊂ ⊕ ν : K Z. By deﬁnition, ν(v) = 1. Sinceν ˆ(u v(c1+c2v+... )) = (1, 0) > (0, 1) = → − 2 νˆ(v(c1 +c2v +... )), we have ν(u) = ν(v) =1. Sinceν ˆ(u c1v v (c2 +c3v +... )) = 2 − − (1, 0) > (0, 2)=ν ˆ(v (c2 + c3v + ... )), we have ν(u c1v) = 2. Continuing in this way, we obtain −

ν(u) = ν(v)=1 ν(u c v)=2 − 1 ν(u c v c v2)=3 − 1 − 2 . . ν(u c v c v2 c vn) = n +1. − 1 − 2 −···− n 15. Valuation rings. § Let K be a ﬁeld, Γ an ordered group, ν : K∗ ։ Γ a valuation of K. Associated to ν is a local subring (Rν, mν) of K, having K as its ﬁeld of fractions:

R = x K ν(x) 0 ν { ∈ | ≥ } m = x K ν(x) > 0 . ν { ∈ | } Example 15.1 (Divisorial valuations). Let X be an integral scheme, D X a closed integral subscheme, ξ the generic point of D. ⊂ Assume that X,ξ is a regular local ring of dimension 1. Let t be a generator of O n mX,ξ. Then K =( X,ξ)t. Indeed, any element f X,ξ can be written as f = t u, O n∈ O where n N and u is invertible. For each f = t u as above, we have νD(f) = n. Then R ∈= . ν OX,D In this example, Rν is Noetherian; in fact, it is a regular local ring of dimension 1. We already know (Theorem 10.19) that if (R,m,k) is a 1-dimensional local noetherian ring, R is regular R is normal. In other words, regular one-dimensional local rings and normal one-dimensional⇐⇒ (Noetherian) local rings are the same thing. Below, we show that Example 15.1 is typical in two ways:

(1) Any valuation ring Rν is normal (whether or not ν is divisorial). (2) A local domain (R,M,k) is a valuation ring for a valuation with value group Z if and only if it is noetherian, regular and of dimension 1. (2) says, in other words, that every valuation with value group Z arises in the way described in Example 15.1; regular 1-dimensional local noetherian rings and 76 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

valuation rings Rν for ν with value group Z are one and the same thing. Later (Proposition 15.8), we will also prove that for any field K and any valuation ν of K, Rν is Noetherian if and only if the value group of ν is Z. Theorem 15.2.. Let K be a field and ν : K Γ any valuation of K. Let R be → ν the valuation ring of ν. Then Rν is normal. Proof. Take any x K. Suppose x is integral over R , that is, ∈ ν (14.1) xn + c xn−1 + + c =0 1 ··· n n n−1 for some ci Rν. Then ν(ci) 0 for all i. By (14.1), ν(x ) = ν(c1x + +cn) ∈n−i ≥ n ··· n−i ≥ min ν(cix ) . Hence there exists i, 1 i n, such that ν(x ) ν(cix ) = 1≤i≤n{ } ≤ ≤ ≥ ν(c )+(n i)ν(x). Then ν(x) ν(ci) 0, so that that x R . i − ≥ i ≥ ∈ ν Theorem 15.3. Let (R,m,k) be a local domain, K the field of fractions of R. The following conditions are equivalent: (1) R is Noetherian, regular of dimension 1. ∗ (2) R = Rν for some valuation ν : K Z. (3) There exists t m such that every→ ideal of R is of the form (tn), n 0. (4) Every ideal of ∈R is a power of m. ≥

Proof of Theorem 15.3. (1) = (2). Take X = Spec R, D = m in Example 15.1. ⇒ { } (2) = (3). Let t be an element of R such that ν(t) = 1. Let I be any ideal of R. Let⇒n = min ν(x) x I (the minimum exists because all the ν(x) are { | ∈ } non-negative integers; here we are using very strongly that Γ = Z and not, say, Q). x For any x I, ν n 0, hence ∈ t ≥ x (15.1) R. tn ∈

n On the other hand, there exists y I such that ν(y) = n. Then ν t = 0, so ∈ y tn that y is a unit of R. Thus 

(15.2) tn (y) I. ∈ ⊂ (15.1) and (15.2) prove that I =(tn). (3) (1) is immediate, because by (3), m =(t). ⇒ We have proved that (1)–(3) are all equivalent. It remains to show that (4) is equivalent to (1)–(3). We will show that (4) (3). (3) (4). Clearly, t m. Then m =(t) and⇐⇒ (4) follows. (4) ⇒ (3). Take any t∈ m m2. Then (t) = mn for some n. Hence (t) = m and (3) follows.⇒ This completes∈ the\ proof of Theorem 15.3. . INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 77

Rings R satisfying (1)–(4) of Theorem 15.3 are called discrete valuation rings. Indeed, Theorem 15.3 says that these are precisely the valuation rings corresponding to valuations with value group Z. Now that we understand the case of discrete valuation rings, we describe the valuation rings in Examples 14.9, 14.10 and 14.12. Examples 15.4. 1. The valuation ν : k(u, v)∗ (1,π) of Example 14.9. Here → α β Rν = k[u v ]α,β∈Z; απ+β≥0 m , where m is the maximal generated by all the monomials uαvβ, απ + β > 0. 2. The valuation ν : k(u, v)∗ Z2 (with lexicographical order) of Example 14.10. α β → Rν = k[u v ]α>0 or α=0,β≥0 m, where m is the maximal ideal generated by all the monomials uαvβ such that ν(uαvβ) 0. ∗ ≥ 3. The valuation ν : k(u, v) Q of Example 14.12. Let Q−1 = v, Q0 = u, → 3 Q1,...,Qn,... be as in Example 14.12. Let β−1 = 1, β0 = 2 , βn = ν(Qn) for γ γ n N. Then Rν =(k[Q ])m, where Q ranges over all the products of the form n∈ γi Qi , with n N, γi Z and βi 0, and where m is the maximal ideal i=1 ∈ ∈ γi ≥ n n Q γi P generated by all those Qi such that γiβi > 0. i=1 i=1 Notice that none of theQ valuation rings inP these three examples are Noetherian. Theorem 15.3 gives several equivalent characterizations of discrete valuation rings. Next, we would like to characterize arbitrary valuation rings. In other words, we pose the questions: given a local domain (R, m) with ﬁeld of fractions K, when does there exist a valuation ν of K such that R = Rν? In the next theorem, we will give three criteria, each of which is equivalent to the existence of such a ν. For geometric applications, the most useful one is the criterion in terms of birational domination, which we will now deﬁne.

Deﬁnition 15.5. Let (R1, m1), (R2, m2) be two local domains with the same ﬁeld of fractions K. We say that R2 birationally dominates R1, denoted R1 subrings of K) with respect to birational domination. 78 INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY.

∗ Proof. (1) (2). Suppose R = Rν for some ν. Take any x K . Then ν(x) + ν 1 =⇒ν x 1 = ν(1) = 0. Hence either ν(x) 0 or ν 1∈ 0. This x · x ≥ x ≥ proves that either x R or 1 R , as desired. ∈ ν x ∈ ν (2) (3). Suppose the ideals of R are not totally ordered by inclusion. Then ⇒ there exist ideals I1,I2 R such that I1 I2 and I2 I1. Take a I1 I2, ⊂ a 6⊂ b 6⊂ a ∈ \ b I2 I1. By assumption, either b R or a R. Say, b R. Then b I2, a ∈ R \ a = b a I , which gives the∈ desired contradiction.∈ ∈ ∈ b ∈ ⇒ · b ∈ 2 (3) (1) We are given that the ideals of R are totally ordered by inclusion and ⇒ we must construct a valuation ν of K, such that R = Rν. First, we describe the value group of ν. Let Φ denote the set of all principal ideals of R (other than R itself), with the operation of multiplication. By assumption Φ is totally ordered by the order relation I1 < I2 if and only if I2 I1. Moreover, if α, β Φ and α < β, then there exists γ Φ such that γ + α⊂= β: indeed, write α =∈ (a), β = (b), with a,b R. α < β means∈ (b) (a), that is, there exists x R such that b = ax. If we set∈γ =(x), we have γ + ⊂α = β in Φ. ∈ Let Γ = Φ 0 ( Φ). It is easy to check that the operation of Φ extends to Γ and that Γ becomes{ } − an ordered group under this operation. For x R, define ν(x)=(x) Φ.` This` defines a map ν : R∗ Φ. Extend ν to all of K by∈ additivity. This gives the∈ desired valuation ν : K∗ Γ.→ We leave details to the reader. → (2) (4). Let (R′, m′) be a local domain with field of fractions K such that ⇒ (R, m) < (R′, m′). Suppose R $ R′. Take x R′ R. Then, by assumption, ∈ \ 1 R 1 m 1 m′ (since R

(15.3) mR[x] = R[x].

1 Applying the same reasoning to x instead of x, we obtain

1 1 (15.4) mR = R . x x (15.3)–(15.4) mean that 1 mR[x] and 1 mR[ 1 ], that is, ∈ ∈ x (15.5) 1 = a + a x + + a xn for some a m 0 1 ··· n i ∈ 1 1 (15.6) 1 = b + b + + b for some b m. 0 1 x ··· l xl i ∈ INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 79

Choose expressions (15.5)–(15.6) so that both n and l are the smallest possible. Say, l n. Multiply (15.6) by xl. We obtain: ≤ xl = b xl + b xl−1 + + b , 0 1 ··· l so that b b (15.7) 0= xl + 1 xl−1 + + l . b 1 ··· b 1 0 − 0 − n−l Multiply (15.7) by anx and subtract it from (15.5). We obtain a relation of the form (15.5), but with a smaller n. This is a contradiction, and the proof of Theorem 15.6 is complete. Remark 15.7. It is clear from the proof of (3) = (1) that the valuation ring ⇒ Rν completely determines ν. Proposition 15.8. Let ν : K∗ Γ be a valuation. Then R is Noetherian → ν ⇔ Γ Z. ≃ Proof. is contained in the (2) = (1) of Theorem 15.3. Since⇐ the ideals of R are totally⇒ ordered by inclusion, saying that R is ⇒ ν ν Noetherian is the same as saying that the set of ideals of Rν is well ordered. If Rν is Noetherian, then the set of ideals of R is well ordered the set of principal ν ⇒ ideals of Rν is well ordered. By (3) = (1) of Theorem 15.6, the set of (proper) principal ideals of Rν can be identiﬁed⇒ with the semi-group Φ of positive elements of Γ. Thus to complete the proof of the Proposition, it remains to prove the following lemma. Lemma 15.9. Let Γ be an ordered group, Φ the set of positive elements of Γ. If Φ is well ordered, then Φ N. ≃ Proof. Let 1 := min α : α Φ . For n N let n := n 1. This gives an inclusion { ∈ } ∈ · .ι : N ֒ Φ. We want to show that ι is an isomorphism → Suppose not. Take α Φ N. For any n N, we cannot have n<αn for any n N. Let β = min α Φ α>n for all n N . Then β 1 >n n. Since β 1∈< β, we have a contradiction.{ ∈ | This completes∈ the} proof of− Lemma 15.9∀ and with− it of Proposition 15.8. .

16. Numerical invariants of valuations. § In this section we deﬁne three important integer invariants associated with valuations and investigate relations between them. Let Γ be an ordered group. Deﬁnition 16.1. An isolated subgroup of Γ is a subgroup ∆ Γ, which is a segment in the ordering. This means: for any a Γ, b ∆ such that⊂ b

Example 16.2. (0) Z is an isolated subgroup of Z2 with lexicographical ordering. ⊕ Remark 16.3. Let ν : K∗ Z2 be a valuation. Then there is a one-one corre- → spondence between isolated subgroups of Γ and prime ideals of Rν , described as follows: ∆ Γ P = x R ν(x) / ∆ . ⊂ ←→ { ∈ ν | ∈ } In particular, the set of isolated subgroups of Γ is totally ordered by inclusion (this is also very easy to show directly for any ordered group Γ). Deﬁnition 16.4. For a valuation ν : K∗ Γ, the rank of ν, denoted rk ν, is deﬁned to be the number of distinct isolated→ subgroups of Γ (not counting (0), but counting Γ itself). In other words, if (0) $ ∆1 $ ∆2 $ $ ∆n = Γ is the full list of isolated subgroups of Γ, then rk ν = n. ···

Remark 16.5. rk ν = dim Rν . Examples 16.6. If Γ = Z2 with lexicographical ordering, then rk ν = 2. More generally, if Γ = Zn with lexicographical ordering, then rk ν = n. In all the other examples we have encountered so far, we had rk ν = 1. In fact, r rk ν = 1 if and only if there exists an injection Γ R. IfΓ= Γi with Γi R ⊂ i=1 ⊂ and the total order on Γ is lexicographical, then rk ν = r. L Deﬁnition 16.7. The rational rank of ν, denoted rat. rk ν, is deﬁned by rat. rk ν = dimQ Γ Z Q. ⊗ rat. rk ν is nothing but the maximal number of elements of Γ, which are linearly independent over Z. Remark 16.8. rk ν rat. rk ν for all valuations ν. ≤ Indeed, let (0) $ ∆1 $ ∆2 $ $ ∆n = Γ be the isolated subgroups of Γ. Pick ··· xi ∆i ∆i−1, i =1, 2,...,n. Then x1,x2,...,xn are linearly independent over Z in Γ.∈ \

Example 16.9. Let Γ = (1,π) R. Then rk ν =1 < 2 = rat. rk ν, since 1,π are ⊂ linearly independent over Z. Definition 16.10. Let (R,m,k) be a local domain with field of fractions K and ν a valuation of K. We say that ν is centered in R if R 0). ≥ . If ν is centered at R, then we have a natural injection k ֒ Rν → mν Definition 16.11. The transcendence degree of ν over k is

R tr. deg ν := tr. deg ν /k . k m ν Examples: INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. 81

References 1. M. Atiyah and MacDonald, Introduction to Commutative Algebra, Addison- Wesley, 1969. 2. R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977. 3. H. Matsumura, Commutative Algebra, Benjamin/Cummings Publishing Co., Reading, Mass., 1970. 4. M. Nagata, Local Rings, Krieger Publishing Co., Huntington, N.Y., 1975. 5. O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Springer-Verlag, 1960.