Graduate Texts in Mathematics 256 Editorial Board S. Axler K.A. Ribet Gregor Kemper A Course in Commutative Algebra With 14 Illustrations 123 Gregor Kemper Technische Universitä t Zentrum Mathematik - M11 Boltzmannstr. 3 85748 Garching Germany [email protected] Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA [email protected] [email protected] ISSN 0072-5285 ISBN 978-3-642-03544-9 e-ISBN 978-3-642-03545-6 DOI 10.1007/978-3-642-03545-6 Springer Heidelberg Dordrecht London New York © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. 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Cover design: WMXDesign GmbH, Heidelberg, Germany Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Contents Introduction ................................................................ 1 Part I The Algebra–Geometry Lexicon 1 Hilbert’s Nullstellensatz............................................ 7 1.1 Maximal Ideals................................................... 8 1.2 Jacobson Rings .................................................. 12 1.3 Coordinate Rings ................................................ 16 Exercises................................................................. 19 2 Noetherian and Artinian Rings .................................. 23 2.1 The Noether and Artin Properties for Rings and Modules ..................................................... 23 2.2 Noetherian Rings and Modules................................. 28 Exercises................................................................. 30 3 The Zariski Topology ............................................... 33 3.1 Affine Varieties................................................... 33 3.2 Spectra ........................................................... 36 3.3 Noetherian and Irreducible Spaces ............................. 38 Exercises................................................................. 42 4 A Summary of the Lexicon ........................................ 45 4.1 True Geometry: Affine Varieties................................ 45 4.2 Abstract Geometry: Spectra.................................... 46 Exercises................................................................. 48 Part II Dimension 5 Krull Dimension and Transcendence Degree .................. 51 Exercises................................................................. 60 ix x Contents 6 Localization ........................................................... 63 Exercises................................................................. 70 7 The Principal Ideal Theorem ..................................... 75 7.1 Nakayama’s Lemma and the Principal Ideal Theorem........ 75 7.2 The Dimension of Fibers........................................ 81 Exercises................................................................. 87 8 Integral Extensions .................................................. 93 8.1 Integral Closure.................................................. 93 8.2 Lying Over, Going Up, and Going Down...................... 99 8.3 Noether Normalization.......................................... 104 Exercises................................................................. 111 Part III Computational Methods 9Gr¨obner Bases ........................................................ 117 9.1 Buchberger’s Algorithm......................................... 118 9.2 First Application: Elimination Ideals .......................... 127 Exercises................................................................. 133 10 Fibers and Images of Morphisms Revisited ................... 137 10.1 The Generic Freeness Lemma................................... 137 10.2 Fiber Dimension and Constructible Sets....................... 142 10.3 Application: Invariant Theory .................................. 144 Exercises................................................................. 148 11 Hilbert Series and Dimension..................................... 151 11.1 The Hilbert–Serre Theorem..................................... 151 11.2 Hilbert Polynomials and Dimension............................ 157 Exercises................................................................. 161 Part IV Local Rings 12 Dimension Theory ................................................... 167 12.1 The Length of a Module ........................................ 167 12.2 The Associated Graded Ring ................................... 170 Exercises................................................................. 176 13 Regular Local Rings................................................. 181 13.1 Basic Properties of Regular Local Rings....................... 181 13.2 The Jacobian Criterion.......................................... 185 Exercises................................................................. 193 Contents xi 14 Rings of Dimension One ........................................... 197 14.1 Regular Rings and Normal Rings .............................. 197 14.2 Multiplicative Ideal Theory..................................... 201 14.3 Dedekind Domains............................................... 206 Exercises................................................................. 212 Solutions of Some Exercises ............................................ 217 References................................................................... 235 Notation ..................................................................... 239 Index ......................................................................... 241 Solutions of exercises for Chapter 1 1 Solutions of exercises for Chapter 1 1.1. We only give examples for a few interesting cases. Let K be a field and x an indeterminate. Then • K[x]/(x2) is algebraic, but not a field. • K[x] is an integral domain (and finitely generated), but not a field, and not algebraic. • The rational function field K(x) is a field, but not algebraic. • The zero-ideal is maximal in K(x), but K[x] ∩ {0} = {0} is not maximal in K[x]. The hypothesis that B be finitely generated in Proposition 1.2 is violated here. 1.2. (a) Let f and g be nonzero elements from K[[x]], and let ai and bk be the first nonzero coefficient of f and g, respectively. Then aibk 6= 0 is the (i + k)th coefficient of fg, so fg 6= 0. (b) Define a sequence (bk)k∈N0 recursively by k−1 −1 −1 X b0 := a0 and bk := −a0 ak−ibi for k > 0. i=0 P∞ i Then i=0 bix is an inverse of f. This also shows that f is invertible if K is a ring and a0 is invertible. The converse follows from the fact that the constant coefficient of a product of formal power series is equal to the product of the constant coefficients. (c) Let m be the set of all formal power series with a0 = 0. Then m is a proper ideal, and by part (b), all proper ideals are contained in m. This implies (c). (d) By part (a), {0} is a prime ideal. But by part (c), it is not the intersection of maximal ideals. P∞ i −k (e) Let f = i=k aix ∈ L \{0}. We may assume ak 6= 0. By part (b), x f is invertible in K[[x]], so f is invertible in L. (f) Assume A := K[[x]] is finitely generated. Then so is the formal Laurent series ring L = A[x−1]. By part (e) and by Lemma 1.1(b), it follows that L is algebraic over K. But x ∈ L is not algebraic. So K[[x]] is not finitely generated. 2 Solutions of exercises for Chapter 1 1.3. Let n ∈ Specmax(R) and consider the homomorphism ϕ: R[x] → R/n, f 7→ f(0) + n. The kernel m of ϕ is a maximal ideal of R[x], and R ∩ m = n, so n ∈ Specrab(R). ∼ 1.4. We have S/(y)S = K[z], so (y)S ∈ Spec(S). Assume there exists m ∈ Specmax(S[x]) with S ∩ m = (y)S. Then S/(y)S is isomorphic to a subalgebra ∼ of S[x]/m. We have y ∈ m. Since R/(y)R = K, S[x]/m is a K-algebra. It is also a field, and finitely generated over K (by the residue classes of x and z). By Lemma 1.1(b), S[x]/m is an algebraic field extension of K. But ∼ then S/(y)S = K[z] cannot be isomorphic to a subalgebra of S[x]/m. This contradiction shows that (y)S ∈/ Specrab(S). ∼ Moreover, we have S/(z)S = R, so (z)S ∈/ Specmax(S). Consider the ideal ∼ n := (xy − 1, z)S[x] ⊆ S[x]. We have S ∩ n = (z)S. Moreover, S[x]/n = ∼ −1 −1 R[x]/(xy − 1)R[x] = R[y ]. But R[y ] is the ring of formal Laurent series, which by Exercise 1.2(e) is a field. So n ∈ Specmax(S[x]), and we conclude (z)S ∈ Specrab(S). 1.5. By hypothesis, for every P ∈ Spec(R) there is a set MP ⊆ Specmax(R) with P = T m. In particular, P ⊆ m for all m ∈ M . For an ideal m∈MP P I $ R, we have \ \ \ \ √ \ m ⊆ m = P = I ⊆ m, m∈Specmax(R), P ∈Spec(R), m∈MP P ∈Spec(R), m∈Specmax(R), I⊆m I⊆P I⊆P I⊆m where the last equality follows from Theorem 1.12, and the last inclusion follows from Lemma 1.10. 1.6. We use Exercise 1.5. The prime ideals