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Introduction to Commutative Algebra

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Introduction to Commutative Algebra INTRODUCTION TO COMMUTATIVE ALGEBRA Mel Hochster Department of Mathematics, University of Michigan, Ann Arbor, MI 48109{1043, USA E-mail address: [email protected] INTRODUCTION TO COMMUTATIVE ALGEBRA Prof. Mel Hochster, instructor [email protected] (734)764{4924 (office1) These are lecture notes for Math 614, Fall 2020. The individual lectures are organized into sections. c Mel Hochster 2020 1However, I will not be using my office in Fall, 2020. Contents Chapter 1. Overview, notation, review of topology, categories and Spec 5 1. Lecture of August 31 5 1.1. Algebraic sets 8 2. Lecture of September 2 10 2.1. Filters and ultrafilters 11 2.2. Categories 11 2.3. Metric spaces. 14 3. Lecture of September 4 15 Chapter 2. Equivalence of categories, products and coproducts, free modules, semigroup rings, and localization 19 1. Lecture of September 9 19 2. Lecture of September 11 23 3. Lecture of September 14 26 4. Lecture of September 16 30 5. Lecture of September 18 32 Chapter 3. Embedding in products of domains, integral and module-finite extensions, height, and Krull dimension 37 1. Lecture of September 21 37 2. Lecture of September 23 43 3. Lecture of September 25 46 4. Lecture of September 28 50 5. Lecture of September 30 53 Chapter 4. Noether normalization, Hilbert's Nullstellensatz, and dimension in finitely generated algebras over a field 57 1. Lecture of October 2 57 1.1. Algebraic independence and transcendence bases 58 2. Lecture of October 5 62 3. Lecture of October 7 67 Chapter 5. Chain conditions on rings and modules 71 1. Lecture of October 9 71 1.1. Annihilators 73 2. Lecture of October 12 76 2.1. Rings that are direct summands 77 2.2. Noetherian induction 78 3. Lecture of October 14 80 3.1. The category of closed algebraic sets 81 3 4 CONTENTS 4. Lecture of October 16 84 5. Lecture of October 19 88 5.1. Formal power series rings 88 Chapter 6. Tensor products, base change, and coproducts of algebras 93 1. Lecture of October 21 93 1.1. Tensor products of modules 93 2. Lecture of October 23 98 3. Lecture of October 26 102 4. Lecture of October 28 105 4.1. Coproducts and epimorphisms 108 4.2. Flatness and finite intersection 108 Chapter 7. Properties of flatness, uses of localization, the functor Hom, and projective modules 109 1. Lecture of October 30 109 2. Lecture of November 2 111 2.1. Adjointness of tensor and Hom 114 3. Lecture of November 4 115 Chapter 8. Primary Decomposition 121 1. Lecture of November 6 121 2. Lecture of November 9 124 3. Lecture of November 11 127 4. Lecture of November 13 131 Chapter 9. Artin Rings 137 1. Lecture of November 16 137 Chapter 10. Krull's principal ideal theorem and the dimension theory of Noetherian rings 141 1. Lecture of November 18 141 2. Lecture of November 20 145 Chapter 11. Algebraic sets, products, and the local nature of elements and regular maps 149 1. Lecture of November 20 149 2. Lecture of November 30 153 Chapter 12. Normal Noetherian domains, Dedekind domains, and divisor class groups 157 1. Lecture of December 2 157 2. Lecture of December 4 162 Chapter 13. Direct and inverse limits, completion, and the Artin-Rees Theorem 173 1. Lecture of December 7 173 Bibliography 183 Index 185 CHAPTER 1 Overview, notation, review of topology, categories and Spec 1. Lecture of August 31 f R −! S We assume familiarity with the notions of ring, ideal, module, and with the polynomial ring in one or finitely many variables over a commutative ring, as well as with homomorphisms of rings and homomorphisms of R-modules over the ring R. As a matter of notation, N ⊆ Z ⊆ Q ⊆ R ⊆ C are the non-negative integers, the integers, the rational numbers, the real numbers, and the complex numbers, respectively, throughout this course. Unless otherwise specified, all rings are commutative, associative, and have a multiplicative identity 1 (when precision is needed we write 1R for the identity in the ring R). It is possible that 1 = 0, in which case the ring is f0g, since for every r 2 R, r = r · 1 = r · 0 = 0. We shall assume that a homomorphism h of rings R ! S preserves the identity, i.e., that h(1R) = 1S. We shall also assume that all given modules M over a ring R are unital, i.e., that 1R · m = m for all m 2 M. The submodule of an R-module M generated by a family of elements mi 2 M f, for i 2 I, an index set is the smallest submodule of M that contains these elements. If the set is empty, it is 0. Otherwise, it consists of all elements of M of the form r1mi1 + ··· + rhmih , which are called R-linear combinations of the mi. Here, the integer h may vary, but even if the set of elements is infinite, h is finite in any given P instance. This submodule may be denoted i2I Rmi. The R-submodules of R itself are the ideals of R. The ideal generated by P elements ri for i 2 I may be denoted i2I riR or (ri : i 2 I)R or even (ri : i 2 I). The ideal generated by r1; : : : ; rn may be denoted (r1; : : : ; rn)R or (r1; : : : ; rn), although the last notation has the disadvantage that it may be confused with the n-tuple that is denoted in the same way. The ring itself is an ideal and is referred to as the unit ideal. An element of the ring is called an invertible element or a unit if it has an inverse under multiplication. When R and S are rings we write S = R[θ1; : : : ; θn] to mean that S is gener- ated as a ring over its subring R by the elements θ1; : : : ; θn. This means that S contains R and the elements θ1; : : : ; θn, and that no strictly smaller subring of S contains R and the θ1; : : : ; θn. It also means that every element of S can be written k1 kn (not necessarily uniquely) as an R-linear combination of the monomials θ1 ··· θn . 5 6 1. OVERVIEW, NOTATION, REVIEW OF TOPOLOGY, CATEGORIES AND Spec When one writes S = R[x1; : : : ; xk] it often means that the xi are indeterminates, so that S is the polynomial ring in n variables over R. But one should say this. The main emphasis in this course will be on Noetherian rings, i.e., rings in which every ideal is finitely generated. Specifically, for all ideals I ⊆ R, there exist Pk f1; : : : ; fk 2 R such that I = (f1; : : : ; fk) = (f1; : : : ; fk)R = i=1 Rfi. We shall develop a very useful theory of dimension in such rings. This will be discussed further quite soon. We shall not be focused on esoteric examples of rings. In fact, almost all of the theory we develop is of great interest and usefulness in studying the properties of polynomial rings over a field or the integers, and homomorphic images of such rings. There is a strong connection between studying systems of equations, study- ing their solutions sets, which often have some kind of geometry associated with them, and studying commutative rings. Suppose the equations involve variables X1;:::;Xn with coefficients in K. The most important case for us will be when K is an algebraically closed field such as the complex numbers C. Suppose the equations have the form Fi = 0 where the Fi are polynomials in the Xj with co- efficients in K. Let I be the ideal generated by the Fi in the polynomial ring K[X1;:::;Xn] and let R be the quotient ring K[X1;:::;Xn]=I. In R, the im- ages xj of the variables Xj give a solution of the equations, a sort of \universal" solution. The connection between commutative algebra and algebraic geometry is that algebraic properties of the ring R are reflected in geometric properties of the solution set, and conversely. Solutions of the equations in the field K give maximal ideals of R. This leads to the idea that maximal ideals of R should be thought of as points in a geometric object. Some rings have very few maximal ideals: in that case it is better to consider all of the prime ideals of R as points of a geometric object. We shall soon make this idea more formal. Before we begin the systematic development of our subject, we shall look at some very simple examples of problems, many unsolved, that are quite natural and easy to state. Suppose that we are given polynomials f and g in C[x], the polynomial ring in one variable over the complex numbers C. Is there an algorithm that enables us to tell whether f and g generate C[x] over C? This will be the case if and only if x 2 C[f; g], i.e., if and only if x can be expressed as a polynomial with complex coefficients in f and g. For example, suppose that f = x5 + x3 − x2 + 1 and g = x14 − x7 + x2 + 5. Here it is easy to see that f and g do not generate, because neither has a term involving x with nonzero coefficient. But if we change f to x5 + x3 − x2 + x + 1 the problem does not seem easy. The following theorem of Abhyankar and Moh [1] gives a method of attacking this sort of problem. Theorem 1.1 (Abhyankar-Moh).
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