Local Cohomology
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Injective Modules: Preparatory Material for the Snowbird Summer School on Commutative Algebra
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to a small extent, what one can do with them. Let R be a commutative Noetherian ring with an identity element. An R- module E is injective if HomR( ;E) is an exact functor. The main messages of these notes are − Every R-module M has an injective hull or injective envelope, de- • noted by ER(M), which is an injective module containing M, and has the property that any injective module containing M contains an isomorphic copy of ER(M). A nonzero injective module is indecomposable if it is not the direct • sum of nonzero injective modules. Every injective R-module is a direct sum of indecomposable injective R-modules. Indecomposable injective R-modules are in bijective correspondence • with the prime ideals of R; in fact every indecomposable injective R-module is isomorphic to an injective hull ER(R=p), for some prime ideal p of R. The number of isomorphic copies of ER(R=p) occurring in any direct • sum decomposition of a given injective module into indecomposable injectives is independent of the decomposition. Let (R; m) be a complete local ring and E = ER(R=m) be the injec- • tive hull of the residue field of R. The functor ( )_ = HomR( ;E) has the following properties, known as Matlis duality− : − (1) If M is an R-module which is Noetherian or Artinian, then M __ ∼= M. -
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 . -
Local Cohomology and Matlis Duality – Table of Contents 0Introduction
Local Cohomology and Matlis duality { Table of contents 0 Introduction . 2 1 Motivation and General Results . 6 1.1 Motivation . 6 1.2 Conjecture (*) on the structure of Ass (D( h (R))) . 10 R H(x1;:::;xh)R h 1.3 Regular sequences on D(HI (R)) are well-behaved in some sense . 13 1.4 Comparison of two Matlis Duals . 15 2 Associated primes { a constructive approach . 19 3 Associated primes { the characteristic-free approach . 26 3.1 Characteristic-free versions of some results . 26 dim(R)¡1 3.2 On the set AssR(D(HI (R))) . 28 4 The regular case and how to reduce to it . 35 4.1 Reductions to the regular case . 35 4.2 Results in the general case, i. e. h is arbitrary . 36 n¡2 4.3 The case h = dim(R) ¡ 2, i. e. the set AssR(D( (k[[X1;:::;Xn]]))) . 38 H(X1;:::;Xn¡2)R 5 On the meaning of a small arithmetic rank of a given ideal . 43 5.1 An Example . 43 5.2 Criteria for ara(I) · 1 and ara(I) · 2 ......................... 44 5.3 Di®erences between the local and the graded case . 48 6 Applications . 50 6.1 Hartshorne-Lichtenbaum vanishing . 50 6.2 Generalization of an example of Hartshorne . 52 6.3 A necessary condition for set-theoretic complete intersections . 54 6.4 A generalization of local duality . 55 7 Further Topics . 57 7.1 Local Cohomology of formal schemes . 57 i 7.2 D(HI (R)) has a natural D-module structure . -
The Structure Theory of Complete Local Rings
The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
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. -
Lecture 3. Resolutions and Derived Functors (GL)
Lecture 3. Resolutions and derived functors (GL) This lecture is intended to be a whirlwind introduction to, or review of, reso- lutions and derived functors { with tunnel vision. That is, we'll give unabashed preference to topics relevant to local cohomology, and do our best to draw a straight line between the topics we cover and our ¯nal goals. At a few points along the way, we'll be able to point generally in the direction of other topics of interest, but other than that we will do our best to be single-minded. Appendix A contains some preparatory material on injective modules and Matlis theory. In this lecture, we will cover roughly the same ground on the projective/flat side of the fence, followed by basics on projective and injective resolutions, and de¯nitions and basic properties of derived functors. Throughout this lecture, let us work over an unspeci¯ed commutative ring R with identity. Nearly everything said will apply equally well to noncommutative rings (and some statements need even less!). In terms of module theory, ¯elds are the simple objects in commutative algebra, for all their modules are free. The point of resolving a module is to measure its complexity against this standard. De¯nition 3.1. A module F over a ring R is free if it has a basis, that is, a subset B ⊆ F such that B generates F as an R-module and is linearly independent over R. It is easy to prove that a module is free if and only if it is isomorphic to a direct sum of copies of the ring. -
Analytic Sheaves of Local Cohomology
transactions of the american mathematical society Volume 148, April 1970 ANALYTIC SHEAVES OF LOCAL COHOMOLOGY BY YUM-TONG SIU In this paper we are interested in the following problem: Suppose Fis an analytic subvariety of a (not necessarily reduced) complex analytic space X, IF is a coherent analytic sheaf on X— V, and 6: X— V-> X is the inclusion map. When is 6JÍJF) coherent (where 6q(F) is the qth direct image of 3F under 0)? The case ¿7= 0 is very closely related to the problem of extending rF to a coherent analytic sheaf on X. This problem of extension has already been dealt with in Frisch-Guenot [1], Serre [9], Siu [11]—[14],Thimm [17], and Trautmann [18]-[20]. So, in our investigation we assume that !F admits a coherent analytic extension on all of X. In réponse to a question of Serre {9, p. 366], Tratumann has obtained the following in [21]: Theorem T. Suppose V is an analytic subvariety of a complex analytic space X, q is a nonnegative integer, andrF is a coherent analytic sheaf on X. Let 6: X— V-*- X be the inclusion map. Then a sufficient condition for 80(^\X— V),..., dq(^\X— V) to be coherent is: for xeV there exists an open neighborhood V of x in X such that codh(3F\U-V)^dimx V+q+2. Trautmann's sufficiency condition is in general not necessary, as is shown by the following example. Let X=C3, F={z1=z2=0}, ^=0, and J^the analytic ideal- sheaf of {z2=z3=0}. -
Nilpotent Elements Control the Structure of a Module
Nilpotent elements control the structure of a module David Ssevviiri Department of Mathematics Makerere University, P.O BOX 7062, Kampala Uganda E-mail: [email protected], [email protected] Abstract A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural properties. A general form is given of an example used in literature to distinguish: 1) completely prime modules from prime modules, 2) classical prime modules from classical completely prime modules, and 3) a module which satisfies the complete radical formula from one which is neither 2-primal nor satisfies the radical formula. Keywords: Semisimple module; Reduced module; Nil module; K¨othe conjecture; Com- pletely prime module; Prime module; and Reduced ring. MSC 2010 Mathematics Subject Classification: 16D70, 16D60, 16S90 1 Introduction Primeness and nilpotency are closely related and well studied notions for rings. We give instances that highlight this relationship. In a commutative ring, the set of all nilpotent elements coincides with the intersection of all its prime ideals - henceforth called the prime radical. A popular class of rings, called 2-primal rings (first defined in [8] and later studied in [23, 26, 27, 28] among others), is defined by requiring that in a not necessarily commutative ring, the set of all nilpotent elements coincides with the prime radical. In an arbitrary ring, Levitzki showed that the set of all strongly nilpotent elements coincides arXiv:1812.04320v1 [math.RA] 11 Dec 2018 with the intersection of all prime ideals, [29, Theorem 2.6]. -
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. -
SS-Injective Modules and Rings Arxiv:1607.07924V1 [Math.RA] 27
SS-Injective Modules and Rings Adel Salim Tayyah Department of Mathematics, College of Computer Science and Information Technology, Al-Qadisiyah University, Al-Qadisiyah, Iraq Email: [email protected] Akeel Ramadan Mehdi Department of Mathematics, College of Education, Al-Qadisiyah University, P. O. Box 88, Al-Qadisiyah, Iraq Email: [email protected] March 19, 2018 Abstract We introduce and investigate ss-injectivity as a generalization of both soc-injectivity and small injectivity. A module M is said to be ss-N-injective (where N is a module) if every R-homomorphism from a semisimple small submodule of N into M extends to N. A module M is said to be ss-injective (resp. strongly ss-injective), if M is ss-R- injective (resp. ss-N-injective for every right R-module N). Some characterizations and properties of (strongly) ss-injective modules and rings are given. Some results of Amin, Yuosif and Zeyada on soc-injectivity are extended to ss-injectivity. Also, we provide some new characterizations of universally mininjective rings, quasi-Frobenius rings, Artinian rings and semisimple rings. Key words and phrases: Small injective rings (modules); soc-injective rings (modules); SS- Injective rings (modules); Perfect rings; quasi-Frobenius rings. 2010 Mathematics Subject Classification: Primary: 16D50, 16D60, 16D80 ; Secondary: 16P20, 16P40, 16L60 . arXiv:1607.07924v1 [math.RA] 27 Jul 2016 ∗ The results of this paper will be part of a MSc thesis of the first author, under the supervision of the second author at the University of Al-Qadisiyah. 1 Introduction Throughout this paper, R is an associative ring with identity, and all modules are unitary R- modules. -
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.