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Introduction to Commutative Algebra

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Introduction to Commutative Algebra Introduction to Commutative Algebra December 20, 2019 About This Document This document was typeset by Jason McCullough. It is based on course notes from a course taught by Professor S.P. Dutta at the University of Illinois texed by Jason McCullough and Bart Snapp. The notes have been redone in 2019 for Math 619 at Iowa State taught by Jason McCullough. Special thanks to: Michael Dewar, Jordan Disch, Dan Lior, Christian McRoberts, Elizabeth Sprangel, and Patrick Szuta, for making many comments and corrections con- cerning these notes. Please report corrections, suggestions, gripes, complaints, and criticisms to: [email protected] Contents 0 Background 1 0.1 OperationsonIdeals ......................... 1 0.2 ChainConditions........................... 4 0.3 FlatModules ............................. 8 0.4 Localization.............................. 9 1 Primary Decomposition 13 1.1 PrimaryandCoprimaryModules . 13 1.2 ThePrimaryDecompositionTheorem . 15 1.2.1 Primary Decomposition and Localization . 18 1.2.2 AssociatedPrimes ...................... 21 1.3 ArbitraryModules .......................... 25 2 Filtrations and Completions 28 2.1 Limits ................................. 28 2.1.1 Direct Limits . 28 2.1.2 InverseLimits......................... 30 2.2 Filtrations and Completions . 32 2.2.1 Topology and Algebraic Structures . 32 2.2.2 Filtered Rings and Modules . 32 2.2.3 The Topology Corresponding to a Filtration . 33 2.2.4 GradedRingsandModules . 36 2.3 Adic Completions and Local Rings . 39 2.4 Faithfully Flat Modules . 44 3 Dimension Theory 52 3.1 TheGradedCase........................... 52 3.1.1 The Hilbert Polynomial . 54 3.2 TheHilbert-SamuelPolynomial . 59 3.3 TheTopologicalApproach. 66 3.3.1 Basic Definitions . 66 3.3.2 The Zariski Topology and the Prime Spectrum . 68 3.4 Systems of Parametersand the Dimension Theorem . 72 4 Integral Extensions 79 4.1 BasicProperties ........................... 79 4.2 ValuationRings............................ 90 4.3 NormalDomainsandDVRs. 92 4.4 DedekindDomains .......................... 95 4.5 TheKrull-AkizukiTheorem . 100 4.6 UniqueFactorizationandClassGroups . 103 4.7 Noether’sNormalizationLemma . 105 5 Homological Methods 113 5.1 ComplexesandHomology . 113 5.1.1 Projective Resolutions . 118 5.1.2 Injective Resolutions . 124 5.2 TorandExt.............................. 129 5.2.1 Tor...............................129 5.2.2 Ext ..............................134 5.3 Homological Notions of Dimension . 138 5.3.1 Projective Dimension . 138 5.3.2 Injective Dimension . 141 5.3.3 Global Dimension . 142 5.4 TheLocalCase............................ 143 5.4.1 Minimal Free Resolutions . 143 5.5 Regular Rings and Global Dimension . 146 5.5.1 RegularLocalRings . 146 5.5.2 RegularRings......................... 151 6 DepthandCohen-MacaulayModules 153 6.1 Depth .................................153 6.2 TheKoszulComplex......................... 167 6.3 Cohen-MacaulayModulesandRings . 174 7 InjectiveModulesoverNoetherianRings 186 7.1 Essential Extensions and Injective Hulls . 186 7.2 Matlis Duality . 191 7.3 Minimal Injective Resolutions . 196 7.4 GorensteinRings ........................... 201 7.5 Grothendieck’sFunctorialApproach . 210 8 Local Cohomology 216 8.1 Grothendieck’s Definition and Basic Properties . 216 8.2 LocalCohomologyandtheKoszulComplex . 219 8.3 LocalDuality ............................. 223 8.4 Set-TheoreticCompleteIntersections . 226 8.5 Grothendieck’sFunctorialApproach . 234 9 ApplicationsandGeneralizations 239 9.1 Hartshorne-LichtenbaumVanishing Theorem . 239 9.2 CanonicalModulesofLocalRings . 245 9.3 Mayer-VietorisSequence for Local Cohomology . 250 9.4 Faltings’ConnectednessTheorem . 252 9.5 TheKrullSchmidtTheorem. 253 9.6 S2-ification ..............................254 Bibliography 257 Index 259 ADiagramandExamplesofDomains 259 B Table of Invariances 262 Notation and Terminology All rings are commutative and contain multiplicative identity, moreover we will always insist that ring homomorphisms respect the multiplicative identity ele- ment. Local rings are assumed to be Noetherian. Additionally, all modules are unitary modules. We have made an attempt to be consistent with our notation: (1) Rings are often denoted by A, B, R, S. (2) Modules are often denoted by M or N. (3) Fields are often denoted by k, K, L, or F . (4) Ideals are denoted by I, J, a, and b, with m usually reserved for maximal ideals. We will try to reserve p, q, P , and Q for prime ideals. (5) X is often used to denote indeterminants and in general X := X1,...,Xn and x := x1,...,xn with the value of n (which is possibly infinite) being given by the context. (6) The symbol 1 will denote the identity map 1 : M M. M M → (7) The letter η will be often used to denote the canonical or natural map. (8) If ϕ is a map, ϕ will often stand for the map induced by ϕ. (9) We use the notation for set inclusion and use ( for strict inclusion. e ⊆ -The notion ֒ is used to denote an injective map and ։ denotes a sur (10) jective map.→ If a commutative diagram is drawn, the induced map will be dashed. (11) If A is a domain, Frac(A) will stand for the field of fractions of A. (12) If (A, m) is a local ring, A will often stand the m-adic completion of A. (13) If A is a ring, A will oftenb stand for the integral closure of A. (14) If k is a field, kewill often stand for the algebraic closure of k. Chapter 0 Background 0.1 Operations on Ideals Definition Given two ideals I,J A, the sum of I and J is defined as ⊆ I + J = x + y : x I and y J . { ∈ ∈ } Exercise 0.1 Show that if I and J are ideas in a ring A, then I +J is an ideal. Definition Given two ideals I,J A, the product of I and J is defined as ⊆ n I J = x y : x I and y J . · i · i i ∈ i ∈ (i=1 ) X Exercise 0.2 Show that if I and J are ideas in a ring A, then I J is an ideal. · Definition Given two ideals I,J A, the intersection of I and J is defined as the set-theoretic intersection of I⊆and J. Exercise 0.3 Show that if I and J are ideas in a ring A, then I J is an ideal. ∩ Exercise 0.4 If a, b, c are ideals of A, show that a(b + c)= ab + ac. Exercise 0.5 If a and b are ideals of A and M is an A-module, show that (a + b)M a(M/bM)= . bM Exercise 0.6 Assuming that a, b, c are ideals of A and that a b or a c, prove the modular law: ⊇ ⊇ a (b + c)= a b + a c. ∩ ∩ ∩ Definition Two ideals I,J A are called comaximal if I + J = (1). ⊆ 1 0.1. OPERATIONS ON IDEALS Remark Sometimes people use the term coprime for comaximal. We will refrain from doing this to avoid confusion later on with coprimary ideals. Exercise 0.7 Show that if I and J are comaximal ideals of A, then IJ = I J. ∩ Definition An element x of a ring is called nilpotent if there exists n N such that xn = 0. ∈ Definition The set of nilpotent elements of a ring A is called the nilradical of A. We will use √0 to denote this set. Note that √0 is an ideal. We can generalize the idea of the nilradical as follows: Definition The radical of an ideal I is denoted by √I and is defined to be the set √I = x A : xn I for some n N . { ∈ ∈ ∈ } Note that √I is an ideal. Proposition 0.8 Given a ring A, the radical of an ideal I is equal to the intersection of all the prime ideals which contain I. Proof ( ) Suppose that x √I. Then xn I and for each prime ideal containing⊆I, xn p. Since p is∈ prime, x p. Thus∈ ∈ ∈ √I p. ⊆ p \⊇I ( ) By the Correspondence Theorem, the prime ideals of A containing I correspond⊇ bijectively to the ideals of A/I, hence we reduce to the case where I = (0). Suppose that x is not nilpotent. We’ll show that x / p. ∈ p⊇\(0) Consider the set of ideals of A: = a : xi / a for i> 0 S { ∈ } Note that (0) and that may be ordered by inclusion. Now let be any chain of ideals∈ in S . This chainS has an upper bound in , namely theC ideal: S S a a [∈C Hence by Zorn’s Lemma, has a maximal element, call it p. We claim that p is prime. Suppose that a,b /S p. Hence (a)+ p and (b)+ p are ideals not contained in . Thus for some m,n∈ N: S ∈ xm (a)+ p and xn (b)+ p ∈ ∈ 2 CHAPTER 0. BACKGROUND Moreover, xm+n (ab)+ p ∈ and so we see that ab / p. Hence p is prime and x / p. Thus x is not in the intersection of the prime∈ ideals of A. ∈ Corollary 0.9 (Properties of Radicals) If I,J are ideals of A, the following hold: (1) I √I. ⊆ (2) √I = √I. (3) √IJ =p√I J = √I √J. ∩ ∩ (4) √pn = p. WARNING 0.10 The union of two ideals is not generally an ideal. Despite this fact there are some things we can say about unions of ideals. Lemma 0.11 (Prime Avoidance) Let A be a ring and I be an ideal of A. If p1,..., pn are prime ideals such that I p for all i, 6⊆ i then n I p . 6⊆ i i=1 [ Proof We proceed by induction on n. The base case n = 1 is trivial. So assume n 2. By induction, there exist for each 1 i n an element a I ≥ p . If any a avoid p we are done; hence≤ we may≤ assume that i ∈ − j6=i j i i ai pi for all i. Consider b = a1a2 an−1 + an. Since an / pi for all i<n ∈ S ··· ∈ and a1 an−1 pi for all i<n, we have b / pi for all i<n.
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