DOCSLIB.ORG

THE F-DEPTH of an IDEAL on a MODULE 1. Introduction Let (R, M)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 7, Pages 1905{1912 S 0002-9939(01)06269-4 Article electronically published on December 27, 2001 THE F-DEPTH OF AN IDEAL ON A MODULE RENCAI LU¨ AND ZHONGMING TANG (Communicated by Wolmer V. Vasconcelos) Abstract. Let I be an ideal of a Noetherian local ring R and M afinitely generated R-module. The f-depth of I on M is the least integer r such that r the local cohomology module HI (M) is not Artinian. This paper presents some part of the theory of f-depth including characterizations of f-depth and a relation between f-depth and f-modules. 1. Introduction Let (R; m) be a (commutative) Noetherian local ring, I a proper ideal of R and M a finitely generated R-module. It is well known that the depth, depth(I;M), of I on M, i.e., the length of a maximal M-regular sequence in I, is the least integer r 6 r such that the local cohomology module HI (M) = 0. Faltings [2] proved that the r least integer r such that HI (M) is not finitely generated is minfdepth(Mp) + ht((I + p)=p)jp 6⊇ Ig: r Now we consider the problem of what is the least integer r such that HI (M)is not Artinian. In [4], Melkersson showed that, when Supp(M=IM) 6⊆ fmg,theleast r integer r such that HI (M) is not Artinian is minfdepth(IRp;Mp)jp 2 Supp(M=IM) nfmgg; and called this integer the f-depth (filter depth) of I on M. In this paper, for any proper ideal I, we define the f-depth of I on M as the length of a maximal M-filter regular sequence in I. Then it turns out that our f-depth of I on M is the least r integer r such that HI (M) is not Artinian for any proper ideal I and, in the case Supp(M=IM) 6⊆ fmg, our f-depth coincides with the one of Melkersson. After summarizing some results about filter regular sequences in section 2, we define f-depth in section 3. Then, we characterize f-depth by Ext, Koszul complexes and local cohomology modules. Section 4 contains an equivalent condition using f-depth for an R-module to be an f-module which is similar to a Cohen-Macaulay module. Received by the editors July 26, 2000 and, in revised form, January 16, 2001. 2000 Mathematics Subject Classification. Primary 13C15, 13D45, 14B15. Key words and phrases. f-depth, f-modules, local cohomology modules. This work was supported by the National Natural Science Foundation of China. c 2001 American Mathematical Society 1905 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1906 RENCAI LU¨ AND ZHONGMING TANG 2. Preliminaries on filter regular sequences Throughout the paper, let (R; m) be a (commutative) Noetherian local ring and M a finitely generated R-module. For any submodule N of M,weuseN :M hmi to denote the submodule fm 2 Mjmnm ⊆ N for some n>0g. Definition 2.1. Let x1;::: ;xn 2 m. If, for i =1;::: ;n, (x1;::: ;xi−1)M :M xi ⊆ (x1;::: ;xi−1)M :M hmi; then we say that x1;::: ;xn is an M-filter regular sequence. [ Notice that x 2 m is M-filter regular if and only if x 62 p and p2AssR(M)nfmg x1;x2;::: ;xn is an M-filter regular sequence if and only if x1 is M-filter regular and x2;::: ;xn is an M=x1M-filter regular sequence. For a finitely generated R- module N, the length of N is denoted by `(N). Then `(N) < 1 if and only if dim(N) ≤ 0. Thus x1;::: ;xn is an M-filter regular sequence is equivalent to `((x1;::: ;xi−1)M :M xi=(x1;::: ;xi−1M)) < 1;i=1;::: ;n: On the other hand, we are reminded that y1;::: ;ys 2 R is a poor M-regular sequence if (y1;::: ;yi−1)M :M yi =(y1;::: ;yi−1)M;i =1;::: ;s; and, if furthermore (y1;::: ;ys)M =6 M,wecally1;::: ;ys an M-regular sequence. Then x1;::: ;xn is an M-filter regular sequence if and only if, for any p 2 Supp(M)n fmg;x1=1;::: ;xn=1isapoorMp-regular sequence, and, if x1;::: ;xn is an M- filter regular sequence, then x1=1;::: ;xn=1isanMp-regular sequence for any p 2 Supp(M) nfmg with x1;::: ;xn 2 p. Since, for any poor M-regular sequence y1;::: ;ys and any integers ij > 0;j = i1 is 1;::: ;s, y1 ;::: ;ys is also a poor M-regular sequence, it follows that if x1;::: ;xn i1 is an M-filter regular sequence, then, for any integers ij > 0;j=1;::: ;n, x1 ;::: ; in xn is also an M-filter regular sequence. However, a permutation of a filter regular sequence is not necessarily filter regular again. For example, let K[x1;x2]bea ⊕ 2 polynomial ring and K a field. Set R = K[x1;x2](x1;x2) and M = R R=(x2). Then, ⊆ 2 as 0 :M x1 =0and0:M=x1M x2 R=(x1;x2) has finite length, it follows that x1;x2 is an M-filter regular sequence. But, as the prime ideal (x2) 2 AssR(M)nf(x1;x2)g and x2 2 (x2), we see that x2 is not M-filter regular, hence, x2;x1 is not an M-filter regular sequence. For any x1;::: ;xn 2 R,letHi(x1;::: ;xn; M)bethei-th homology module of the Koszul complex K:(x1;::: ;xn; M)ofM with respect to x1;::: ;xn. Then, we have the following: Proposition 2.2. If x1;::: ;xn is an M-filter regular sequence, then `(Hi(x1;::: ;xn; M)) < 1; for any i>0: Proof. By induction on n.Ifn =1,thenH1(x1; M)=0:M x1 has finite length by definition. Now, assume that n>1. From the long exact sequence ···!Hi(x1;::: ;xn−1; M) ! Hi(x1;::: ;xn; M) ! Hi−1(x1;::: ;xn−1; M) i−1 (−1) xn ! Hi−1(x1;::: ;xn−1; M) !···!H0(x1;::: ;xn; M) ! 0; License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE F-DEPTH OF AN IDEAL ON A MODULE 1907 we see that `(Hi(x1;::: ;xn; M)) < 1 for all i>1 from the induction assumption that `(Hi(x1;::: ;xn−1; M)) < 1 and `(Hi−1(x1;::: ;xn−1; M)) < 1.Forthe case i = 1, we have an exact sequence H1(x1;::: ;xn−1; M) ! H1(x1;::: ;xn; M) ! H0(x1;::: ;xn−1; M) xn = M=(x1;::: ;xn−1)M ! M=(x1;::: ;xn−1)M: As H1(x1;::: ;xn−1; M)and0:M=(x1;::: ;xn−1)M xn have finite length, we see that `(H1(x1;::: ;xn; M)) < 1. Hence, for all i>0, `(Hi(x1;::: ;xn; M)) < 1. Notice that the converse of Proposition 2.2 is not true because the conditions `(Hi(x1;::: ;xn; M)) < 1 for all i>0 do not depend on the order of x1;::: ;xn, but the filter regularity does. 3. Characterizations of f-depth In order to show that f-depth is well-defined, we need the following: Lemma 3.1. Let I ⊆ m be an ideal. If `(HomR(R=I; M)) < 1, then there exists x 2 I which is M-filter regular. [ Proof. Assume the contrary. Then I ⊆ p,sothatI ⊆ p for some p2AssR(M)nfmg p 2 Ass (M) nfmg. Hence p =ann (m)forsomem 2 M,sodim(Rm) > 0. But R ∼ R Im =0,soRm ⊆ 0:M I = HomR(R=I; M). Thus dim(HomR(R=I; M)) > 0, a contradiction. Proposition 3.2. Let I ⊆ m be an ideal and n>0 an integer. Then the following are equivalent: i 1 (1) `(ExtR(R=I; M)) < , for all i<n; (2) I contains an M-filter regular sequence of length n. When x1;::: ;xn 2 I is an M-filter regular sequence, n ∼ 2 nf g ExtR(R=I; M)p = HomR(R=I; M=(x1;::: ;xn)M)p; for any p Spec(R) m : i 1 Proof. Assume that `(ExtR(R=I; M)) < , for all i<n. We use induction on n to show that I contains an M-filter regular sequence of length n.Ifn =1,thenI contains an M-filter regular element by Lemma 3.1. Now assume that n>1and theresultistrueforn − 1. Then, by Lemma 3.1 again, there is x1 2 I which is M-filter regular. From the short exact sequences x1 0 ! 0:M x1 ! M ! x1M ! 0; 0 ! x1M ! M ! M=x1M ! 0; we get the long exact sequences x1 0 ! HomR(R=I; 0:M x1) ! HomR(R=I; M) ! HomR(R=I; x1M) ! 1 !··· ExtR(R=I; 0:M x1) ; 0 ! HomR(R=I; x1M) ! HomR(R=I; M) ! HomR(R=I; M=x1M) ! 1 !··· ExtR(R=I; x1M) : 1 i 1 ≥ As `(0 :M x1) < ,wehavethat`(ExtR(R=I; 0:M x1)) < for any i 0. Then i 1 we see that `(ExtR(R=I; x1M)) < for all i<nfrom the first long exact sequence. i 1 From the second long exact sequence, we get that `(ExtR(R=I; M=x1M)) < for all i<n− 1. Then, by the induction assumption, there exist x2;::: ;xn 2 I which License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1908 RENCAI LU¨ AND ZHONGMING TANG is an M=x1M-filter regular sequence.
Recommended publications
  • The Geometry of Syzygies

    The Geometry of Syzygies

    The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points .
  • October 2013

    October 2013

    LONDONLONDON MATHEMATICALMATHEMATICAL SOCIETYSOCIETY NEWSLETTER No. 429 October 2013 Society MeetingsSociety 2013 ELECTIONS voting the deadline for receipt of Meetings TO COUNCIL AND votes is 7 November 2013. and Events Members may like to note that and Events NOMINATING the LMS Election blog, moderated 2013 by the Scrutineers, can be found at: COMMITTEE http://discussions.lms.ac.uk/ Thursday 31 October The LMS 2013 elections will open on elections2013/. Good Practice Scheme 10th October 2013. LMS members Workshop, London will be contacted directly by the Future elections page 15 Electoral Reform Society (ERS), who Members are invited to make sug- Friday 15 November will send out the election material. gestions for nominees for future LMS Graduate Student In advance of this an email will be elections to Council. These should Meeting, London sent by the Society to all members be addressed to Dr Penny Davies 1 page 4 who are registered for electronic who is the Chair of the Nominat- communication informing them ing Committee (nominations@lms. Friday 15 November that they can expect to shortly re- ac.uk). Members may also make LMS AGM, London ceive some election correspondence direct nominations: details will be page 5 from the ERS. published in the April 2014 News- Monday 16 December Those not registered to receive letter or are available from Duncan SW & South Wales email correspondence will receive Turton at the LMS (duncan.turton@ Regional Meeting, all communications in paper for- lms.ac.uk). Swansea mat, both from the Society and 18-21 December from the ERS. Members should ANNUAL GENERAL LMS Prospects in check their post/email regularly in MEETING Mathematics, Durham October for communications re- page 11 garding the elections.
  • Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings

    Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings

    City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2005 Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings Mahdi Majidi-Zolbanin Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/1765 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings by Mahdi Majidi-Zolbanin A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of NewYork. 2005 UMI Number: 3187456 Copyright 2005 by Majidi-Zolbanin, Mahdi All rights reserved. UMI Microform 3187456 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 ii c 2005 Mahdi Majidi-Zolbanin All Rights Reserved iii This manuscript has been read and accepted for the Graduate Faculty in Mathematics in satisfaction of the dissertation requirements for the degree of Doctor of Philosophy. Lucien Szpiro Date Chair of Examining Committee Jozek Dodziuk Date Executive Officer Lucien Szpiro Raymond Hoobler Alphonse Vasquez Ian Morrison Supervisory Committee THE CITY UNIVERSITY OF NEW YORK iv Abstract Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings by Mahdi Majidi-Zolbanin Advisor: Professor Lucien Szpiro In this dissertation we study splitting of vector bundles of small rank on punctured spectrum of regular local rings.
  • Depth, Dimension and Resolutions in Commutative Algebra

    Depth, Dimension and Resolutions in Commutative Algebra

    Depth, Dimension and Resolutions in Commutative Algebra Claire Tête PhD student in Poitiers MAP, May 2014 Claire Tête Commutative Algebra This morning: the Koszul complex, regular sequence, depth Tomorrow: the Buchsbaum & Eisenbud criterion and the equality of Aulsander & Buchsbaum through examples. Wednesday: some elementary results about the homology of a bicomplex Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) We would like to find something like this dm dm−1 d1 · · · Fm Fm−1 · · · F1 F0 A/a with A-modules Fi as simple as possible and s.t. Im di = Ker di−1. Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) We would like to find something like this dm dm−1 d1 · · · Fm Fm−1 · · · F1 F0 A/a with A-modules Fi as simple as possible and s.t. Im di = Ker di−1. We say that F· is a resolution of the A-module A/a Claire Tête Commutative Algebra I will begin with a little example.
  • Reflexivity Revisited

    Reflexivity Revisited

    REFLEXIVITY REVISITED MOHSEN ASGHARZADEH ABSTRACT. We study some aspects of reflexive modules. For example, we search conditions for which reflexive modules are free or close to free modules. 1. INTRODUCTION In this note (R, m, k) is a commutative noetherian local ring and M is a finitely generated R- module, otherwise specializes. The notation stands for a general module. For simplicity, M the notation ∗ stands for HomR( , R). Then is called reflexive if the natural map ϕ : M M M M is bijection. Finitely generated projective modules are reflexive. In his seminal paper M→M∗∗ Kaplansky proved that projective modules (over local rings) are free. The local assumption is really important: there are a lot of interesting research papers (even books) on the freeness of projective modules over polynomial rings with coefficients from a field. In general, the class of reflexive modules is extremely big compared to the projective modules. As a generalization of Seshadri’s result, Serre observed over 2-dimensional regular local rings that finitely generated reflexive modules are free in 1958. This result has some applications: For instance, in the arithmetical property of Iwasawa algebras (see [45]). It seems freeness of reflexive modules is subtle even over very special rings. For example, in = k[X,Y] [29, Page 518] Lam says that the only obvious examples of reflexive modules over R : (X,Y)2 are the free modules Rn. Ramras posed the following: Problem 1.1. (See [19, Page 380]) When are finitely generated reflexive modules free? Over quasi-reduced rings, problem 1.1 was completely answered (see Proposition 4.22).
  • Arxiv:1906.02669V2 [Math.AC] 6 May 2020 Ti O Nw Hte H Ulne-Etncnetr Od for Holds Conjecture Auslander-Reiten Theor the [19, Whether and Rings

    Arxiv:1906.02669V2 [Math.AC] 6 May 2020 Ti O Nw Hte H Ulne-Etncnetr Od for Holds Conjecture Auslander-Reiten Theor the [19, Whether and Rings

    THE AUSLANDER-REITEN CONJECTURE FOR CERTAIN NON-GORENSTEIN COHEN-MACAULAY RINGS SHINYA KUMASHIRO Abstract. The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay local ring R with a parameter ideal Q, the Auslander-Reiten conjecture holds for R if and only if it holds for the residue ring R/Q. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring R/Qℓ in connection with that for R, and prove the equivalence of them for the case where R is Gorenstein and ℓ ≤ dim R. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our results, we see that the Auslander-Reiten conjecture holds if there exists an Ulrich ideal whose residue ring is a complete intersection. We also explore the Auslander-Reiten conjecture for determinantal rings. 1. Introduction The Auslander-Reiten conjecture and several related conjectures are problems about the vanishing of Ext modules. As is well-known, the vanishing of cohomology plays a very important role in the study of rings and modules. For a guide to these conjectures, one can consult [8, Appendix A] and [7, 16, 24, 25]. These conjectures originate from the representation theory of algebras, and the theory of commutative ring also greatly contributes to the development of the Auslander-Reiten conjecture; see, for examples, [1, 19, 20, 21]. Let us recall the Auslander-Reiten conjecture over a commutative Noetherian ring R. i Conjecture 1.1. [3] Let M be a finitely generated R-module.
  • The Depth Theory of Hopf Algebras and Smash Products

    The Depth Theory of Hopf Algebras and Smash Products

    The Depth Theory of Hopf Algebras and Smash Products Christopher J. Young Supervised by Dr. Lars Kadison November 25, 2016 To all my loved ones. Abstract The work done during this doctoral thesis involved advancing the theory of algebraic depth. Subfactor depth is a concept which had already existed for decades, but research papers discovered a purely algebraic analogue to the concept. The main uses of algebraic depth, which applies to a ring and subring pair, have been in Hopf-Galois theory, Hopf algebra actions and to some extent group theory. In this thesis we consider the application of algebraic depth to finite dimensional Hopf algebras, and smash products. This eventually leads to a striking discovery, a concept of module depth. Before explaining this work the thesis will go through many know results of depth up to this point historically. The most important result of the thesis: we discover a strong connection between the algebraic depth of a smash product A#H and the module depth of an H-module algebra A. In separate work L. Kadison discovered a connection between algebraic depth R ⊆ H for Hopf algebras and the module depth of V ∗, another important H-module algebra. The three concepts are related. Another important achievement of the work herein, we are able to calculate for the first time depth values of polynomial algebras, Taft algebras with certain subgroups and specific smash products of the Taft algebras. We also give a bound for the depth of R=I ⊆ H=I, which is extremely useful in general. Contents Background iii Contribution iii 1 The Depth Theory of a Ring Extension 1 1.1 Tensor Products .
  • Commutative Algebra

    Commutative Algebra

    Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions .
  • Gorenstein Rings Examples References

    Gorenstein Rings Examples References

    Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Gorenstein Rings Chau Chi Trung Bachelor Thesis Defense Presentation Supervisor: Dr. Tran Ngoc Hoi University of Science - Vietnam National University Ho Chi Minh City May 2018 Email: [email protected] Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Content 1 Origin 2 Aim of the Thesis 3 Structure of Minimal Injective Resolution 4 Gorenstein Rings 5 Examples 6 References Origin ∙ Grothendieck introduced the notion of Gorenstein variety in algebraic geometry. ∙ Serre made a remark that rings of finite injective dimension are just Gorenstein rings. The remark can be found in [9]. ∙ Gorenstein rings have now become a popular notion in commutative algebra and given birth to several definitions such as nearly Gorenstein rings or almost Gorenstein rings. Aim of the Thesis This thesis aims to 1 present basic results on the minimal injective resolution of a module over a Noetherian ring, 2 introduce Gorenstein rings via Bass number and 3 answer elementary questions when one inspects a type of ring (e.g. Is a subring of a Gorenstein ring Gorenstein?). Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Structure of Minimal Injective Resolution Unless otherwise specified, let R be a Noetherian commutative ring with 1 6= 0 and M be an R-module. Theorem (E. Matlis) Let E be a nonzero injective R-module. Then we have a direct sum ∼ decomposition E = ⊕i2I Xi in which for each i 2 I, Xi = ER(R=P ) for some P 2 Spec(R).
  • Commutative Algebra Ii, Spring 2019, A. Kustin, Class Notes

    Commutative Algebra Ii, Spring 2019, A. Kustin, Class Notes

    COMMUTATIVE ALGEBRA II, SPRING 2019, A. KUSTIN, CLASS NOTES 1. REGULAR SEQUENCES This section loosely follows sections 16 and 17 of [6]. Definition 1.1. Let R be a ring and M be a non-zero R-module. (a) The element r of R is regular on M if rm = 0 =) m = 0, for m 2 M. (b) The elements r1; : : : ; rs (of R) form a regular sequence on M, if (i) (r1; : : : ; rs)M 6= M, (ii) r1 is regular on M, r2 is regular on M=(r1)M, ::: , and rs is regular on M=(r1; : : : ; rs−1)M. Example 1.2. The elements x1; : : : ; xn in the polynomial ring R = k[x1; : : : ; xn] form a regular sequence on R. Example 1.3. In general, order matters. Let R = k[x; y; z]. The elements x; y(1 − x); z(1 − x) of R form a regular sequence on R. But the elements y(1 − x); z(1 − x); x do not form a regular sequence on R. Lemma 1.4. If M is a finitely generated module over a Noetherian local ring R, then every regular sequence on M is a regular sequence in any order. Proof. It suffices to show that if x1; x2 is a regular sequence on M, then x2; x1 is a regular sequence on M. Assume x1; x2 is a regular sequence on M. We first show that x2 is regular on M. If x2m = 0, then the hypothesis that x1; x2 is a regular sequence on M guarantees that m 2 x1M; thus m = x1m1 for some m1.
  • Lectures on Local Cohomology

    Lectures on Local Cohomology

    Contemporary Mathematics Lectures on Local Cohomology Craig Huneke and Appendix 1 by Amelia Taylor Abstract. This article is based on five lectures the author gave during the summer school, In- teractions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Grothendieck concerning the connectedness of the spectrum of certain rings. Several applications are given, including a theorem of Fulton and Hansen concern- ing the connectedness of intersections of algebraic varieties. In an appendix written by Amelia Taylor, an another application is given to prove a theorem of Kalkbrenner and Sturmfels about the reduced initial ideals of prime ideals. Contents 1. Introduction 1 2. Local Cohomology 3 3. Injective Modules over Noetherian Rings and Matlis Duality 10 4. Cohen-Macaulay and Gorenstein rings 16 d 5. Vanishing Theorems and the Structure of Hm(R) 22 6. Vanishing Theorems II 26 7. Appendix 1: Using local cohomology to prove a result of Kalkbrenner and Sturmfels 32 8. Appendix 2: Bass numbers and Gorenstein Rings 37 References 41 1. Introduction Local cohomology was introduced by Grothendieck in the early 1960s, in part to answer a conjecture of Pierre Samuel about when certain types of commutative rings are unique factorization 2000 Mathematics Subject Classification. Primary 13C11, 13D45, 13H10. Key words and phrases. local cohomology, Gorenstein ring, initial ideal. The first author was supported in part by a grant from the National Science Foundation, DMS-0244405.
  • Cohen-Macaulay Rings and Schemes

    Cohen-Macaulay Rings and Schemes

    Cohen-Macaulay rings and schemes Caleb Ji Summer 2021 Several of my friends and I were traumatized by Cohen-Macaulay rings in our commuta- tive algebra class. In particular, we did not understand the motivation for the definition, nor what it implied geometrically. The purpose of this paper is to show that the Cohen-Macaulay condition is indeed a fruitful notion in algebraic geometry. First we explain the basic defini- tions from commutative algebra. Then we give various geometric interpretations of Cohen- Macaulay rings. Finally we touch on some other areas where the Cohen-Macaulay condition shows up: Serre duality and the Upper Bound Theorem. Contents 1 Definitions and first examples1 1.1 Preliminary notions..................................1 1.2 Depth and Cohen-Macaulay rings...........................3 2 Geometric properties3 2.1 Complete intersections and smoothness.......................3 2.2 Catenary and equidimensional rings.........................4 2.3 The unmixedness theorem and miracle flatness...................5 3 Other applications5 3.1 Serre duality......................................5 3.2 The Upper Bound Theorem (combinatorics!)....................6 References 7 1 Definitions and first examples We begin by listing some relevant foundational results (without commentary, but with a few hints on proofs) of commutative algebra. Then we define depth and Cohen-Macaulay rings and present some basic properties and examples. Most of this section and the next are based on the exposition in [1]. 1.1 Preliminary notions Full details regarding the following standard facts can be found in most commutative algebra textbooks, e.g. Theorem 1.1 (Nakayama’s lemma). Let (A; m) be a local ring and let M be a finitely generated A-module.