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Commutative Algebra Andreas Gathmann

Class Notes TU Kaiserslautern 2013/14

Contents

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. Prime and Maximal Ideals. . . . . . . . . . . . . . . . . . . . . 18 3. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4. Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . 36 5. Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . 43 6. Localization . . . . . . . . . . . . . . . . . . . . . . . . . 52 7. Chain Conditions. . . . . . . . . . . . . . . . . . . . . . . . 62 8. Prime Factorization and Primary Decompositions . . . . . . . . . . . . . 70 9. Integral Ring Extensions . . . . . . . . . . . . . . . . . . . . . 80

10. Noether Normalization and Hilbert’s Nullstellensatz . . . . . . . . . . . . 91 11. Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 96 12. Valuation Rings . . . . . . . . . . . . . . . . . . . . . . . . 109 13. Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . 117

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

0. Introduction 3

0. Introduction

Commutative algebra is the study of commutative rings. In this class we will assume the basics of ring theory that you already know from earlier courses (e. g. ideals, quotient rings, the homo- morphism theorem, and unique prime factorization in principal ideal domains such as the integers or polynomial rings in one variable over a field), and move on to more advanced topics, some of which will be sketched in Remark 0.14 below. For references to earlier results I will usually use my German notes for the “Algebraic Structures” and occasionally the “Foundations of Mathematics” and “Introduction to Algebra” classes [G1, G2, G3], but if you prefer English references you will certainly have no problems to find them in almost any textbook on abstract algebra.

You will probably wonder why the single algebraic structure of commutative rings deserves a full one-semester course for its study. The main motivation for this is its many applications in both algebraic geometry and (algebraic) number theory. Especially the connection between commutative algebra and algebraic geometry is very deep — in fact, to a certain extent one can say that these two fields of mathematics are essentially the same thing, just expressed in different languages. Although some algebraic constructions and results in this class may seem a bit abstract, most of them have an easy (and sometimes surprising) translation in terms of geometry, and knowing about this often helps to understand and remember what is going on. For example, we will see that the Chinese Remainder Theorem that you already know [G1, Proposition 11.22] (and that we will extend to more general rings than the integers in Proposition 1.14) can be translated into the seemingly obvious geometric statement that “giving a function on a disconnected space is the same as giving a function on each of its connected components” (see Example 1.15 (b)).

However, as this is not a geometry class, we will often only sketch the correspondence between al- gebra and geometry, and we will never actually use algebraic geometry to prove anything. Although our “Commutative Algebra” and “Algebraic Geometry” classes are deeply linked, they are deliber- ately designed so that none of them needs the other as a prerequisite. But I will always try to give you enough examples and background to understand the geometric meaning of what we do, in case you have not attended the “Algebraic Geometry” class yet.

So let us explain in this introductory chapter how algebra enters the field of geometry. For this we have to introduce the main objects of study in algebraic geometry: solution sets of polynomial equations over some field, the so-called varieties.

Convention 0.1 (Rings and fields). In our whole course, a ring R is always meant to be a commu- tative ring with 1 [G1, Definition 7.1]. We do not require that this multiplicative unit 1 is distinct from the additive neutral element 0, but if 1 = 0 then R must be the zero ring [G1, Lemma 7.5 (c)]. Subrings must have the same unit as the ambient ring, and ring homomorphisms are always required to map 1 to 1. Of course, a ring R 6= {0} is a field if and only if every non-zero element has a multiplicative inverse.

Definition 0.2 (Polynomial rings). Let R be a ring, and let n ∈ N>0. A polynomial over R in n variables is a formal expression of the form

f = ∑ i1,...,in∈N

ai1,...,in x i1 1 · · · · · x

in n ,

with coefficients ai1,...,in ∈ R and formal variables x = (x1, . . . ,xn), such that only finitely many of the coefficients are non-zero (see [G1, Chapter 9] how this concept of “formal variables” can be defined in a mathematically rigorous way).

Polynomials can be added and multiplied in the obvious way, and form a ring with these operations. We call it the polynomial ring over R in n variables and denote it by R[x1, . . . ,xn].

4 Andreas Gathmann

Definition 0.3 (Varieties). Let K be a field, and let n ∈ N.

(a) We call AnK := {(c1, . . . ,cn) : ci ∈ K for i = 1, . . . ,n}

the affine n-space over K. If the field K is clear from the context, we will write AnK also as An. Note that AnK is just Kn as a set. It is customary to use two different notations here since Kn is also a K-vector space and a ring. We will usually use the notation AnK if we want to ignore these additional structures: for example, addition and scalar multiplication are defined on Kn, but not on AnK . The affine space AnK will be the ambient space for our zero loci of polynomials below.

(b) For a polynomial f ∈ K[x1, . . . ,xn] as above and a point c = (c1, . . . ,cn) ∈ AnK we define the value of f at c to be

f (c) = ∑ i1,...,in∈N

ai1,...,in c i1 1 · · · · · c

in n ∈ K.

If there is no risk of confusion we will sometimes denote a point in AnK by the same letter x as we used for the formal variables, writing f ∈ K[x1, . . . ,xn] for the polynomial and f (x) for its value at a point x ∈ AnK .

(c) Let S⊂ K[x1, . . . ,xn] be a set of polynomials. Then

V (S) := {x ∈ AnK : f (x) = 0 for all f ∈ S} ⊂ AnK is called the zero locus of S. Subsets of AnK of this form are called (affine) varieties. If S = ( f1, . . . , fk) is a finite set, we will write V (S) =V ({ f1, . . . , fk}) also as V ( f1, . . . , fk).

Example 0.4. Varieties, say over the field R of real numbers, can have many different “shapes”. The following picture shows a few examples in A2R and A3R.

(a) V (x21 + x 2 2−1)⊂ A2 (b) V (x22− x31)⊂ A2 (c) V (x31− x1)⊂ A2

(d) V (x61 + x 6 2 + x

6 3−1)⊂ A3 (e) V (x1x3,x2x3)⊂ A3 (f) V (x22 + x33− x43− x21x23)⊂ A3

Of course, the empty set /0 and all of An are also varieties in An, since /0 =V (1) and An =V (0). It is the goal of algebraic geometry to find out the geometric properties of varieties by looking at the corresponding polynomials from an algebraic point of view (as opposed to an analytical or numerical approach). However, it turns out that it is not a very good idea to just look at the defining polynomials given initially — simply because they are not unique. For example, the variety (a) above was given as the zero locus of the polynomial x21 + x

2 2−1, but it is equally well the zero locus

of (x21 + x 2 2− 1)2, or of the two polynomials (x1− 1)(x21 + x22− 1) and x2 (x21 + x22− 1). In order to

0. Introduction 5

remove this ambiguity, it is therefore useful to consider all polynomials vanishing on X at once. Let us introduce this concept now.

Construction 0.5 (Rings and ideals associated to varieties). For a variety X ⊂ AnK (and in fact also for any subset X of AnK) we consider the set

I(X) := { f ∈ K[x1, . . . ,xn] : f (x) = 0 for all x ∈ X} of all polynomials vanishing on X . Note that this is an ideal of K[x1, . . . ,xn] (which we write as I(X)EK[x1, . . . ,xn]): it is clear that 0 ∈ I(X), and if two polynomials f and g vanish on X , then so do f +g and f ·h for any polynomial h. We call I(X) the ideal of X . With this ideal we can construct the quotient ring

A(X) := K[x1, . . . ,xn]/I(X)

in which we identify two polynomials f ,g ∈ K[x1, . . . ,xn] if and only if f −g is the zero function on X , i. e. if f and g have the same value at every point x∈ X . So one may think of an element f ∈ A(X) as being the same as a function

X → K, x 7→ f (x) that can be given by a polynomial. We therefore call A(X) the ring of polynomial functions or coordinate ring of X . Often we will simply say that A(X) is the ring of functions on X since functions in algebra are always given by polynomials. Moreover, the class of a polynomial f ∈ K[x1, . . . ,xn] in such a ring will usually also be written as f ∈ A(X), dropping the explicit notation for equivalence classes if it is clear from the context that we are talking about elements in the quotient ring.

Remark 0.6 (Polynomials and polynomial functions). You probably know that over some fields there is a subtle difference between polynomials and polynomial functions: e. g. over the field K = Z2 the polynomial f = x2 + x ∈ K[x] is certainly non-zero, but it defines the zero function on A1K [G1, Remark 9.16 (b)]. In our current notation this means that the ideal I(A1K) of functions vanishing at every point of A1K is non-trivial, in fact that I(A1K) = (x2 + x), and that