INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY. The purpose of these notes is to give a quick and relatively painless introduc- tion 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 inter- est 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 geom- etry 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 combina- tions of the{ form} n 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 <l − i i i=1 ! X (where, by convention, we take deg 0 to be 1). − Now, suppose I is not finitely generated. Then l > 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 geome- try, 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 alge- braic 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.