Appendix 1 Introduction to Local Cohomology
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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
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}. -
Twenty-Four Hours of Local Cohomology This Page Intentionally Left Blank Twenty-Four Hours of Local Cohomology
http://dx.doi.org/10.1090/gsm/087 Twenty-Four Hours of Local Cohomology This page intentionally left blank Twenty-Four Hours of Local Cohomology Srikanth B. Iyengar Graham J. Leuschke Anton Leykln Claudia Miller Ezra Miller Anurag K. Singh Uli Walther Graduate Studies in Mathematics Volume 87 |p^S|\| American Mathematical Society %\yyyw^/? Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V.Ivanov Steven G. Krantz The book is an outgrowth of the 2005 AMS-IMS-SIAM Joint Summer Research Con• ference on "Local Cohomology and Applications" held at Snowbird, Utah, June 20-30, 2005, with support from the National Science Foundation, grant DMS-9973450. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 2000 Mathematics Subject Classification. Primary 13A35, 13D45, 13H10, 13N10, 14B15; Secondary 13H05, 13P10, 13F55, 14F40, 55N30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-87 Library of Congress Cataloging-in-Publication Data Twenty-four hours of local cohomology / Srikanth Iyengar.. [et al.]. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 87) Includes bibliographical references and index. ISBN 978-0-8218-4126-6 (alk. paper) 1. Sheaf theory. 2. Algebra, Homological. 3. Group theory. 4. Cohomology operations. I. Iyengar, Srikanth, 1970- II. Title: 24 hours of local cohomology. QA612.36.T94 2007 514/.23—dc22 2007060786 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. -
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 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. -
Characterizing Gorenstein Rings Using Contracting Endomorphisms
CHARACTERIZING GORENSTEIN RINGS USING CONTRACTING ENDOMORPHISMS BRITTNEY FALAHOLA AND THOMAS MARLEY Dedicated to Craig Huneke on the occasion of his 65th birthday. Abstract. We prove several characterizations of Gorenstein rings in terms of vanishings of derived functors of certain modules or complexes whose scalars are restricted via contracting endomorphisms. These results can be viewed as analogues of results of Kunz (in the case of the Frobenius) and Avramov- Hochster-Iyengar-Yao (in the case of general contracting endomorphisms). 1. Introduction In 1969 Kunz [17] proved that a commutative Noetherian local ring of prime char- acteristic is regular if and only if some (equivalently, every) power of the Frobenius endomorphism is flat. Subsequently, the Frobenius map has been employed to great effect to study homological properties of commutative Noetherian local rings; see [20], [10], [23], [16], [4] and [3], for example. In this paper, we explore characteriza- tions of Gorenstein rings using the Frobenius map, or more generally, contracting endomorphisms. Results of this type have been obtained by Iyengar and Sather- Wagstaff [14], Goto [9], Rahmati [21], Marley [18], and others. Our primary goal is to obtain characterizations for a local ring to be Gorenstein in terms of one or more vanishings of derived functors in which one of the modules is viewed as an R-module by means of a contracting endomorphism (i.e., via restriction of scalars). A prototype of a result of this kind for regularity is given by Theorem 1.1 below. Let R be a commutative Noetherian local ring with maximal ideal m and φ : R ! R an endomorphism which is contracting, i.e., φi(m) ⊆ m2 for some i. -
Bertini's Theorem on Generic Smoothness
U.F.R. Mathematiques´ et Informatique Universite´ Bordeaux 1 351, Cours de la Liberation´ Master Thesis in Mathematics BERTINI1S THEOREM ON GENERIC SMOOTHNESS Academic year 2011/2012 Supervisor: Candidate: Prof.Qing Liu Andrea Ricolfi ii Introduction Bertini was an Italian mathematician, who lived and worked in the second half of the nineteenth century. The present disser- tation concerns his most celebrated theorem, which appeared for the first time in 1882 in the paper [5], and whose proof can also be found in Introduzione alla Geometria Proiettiva degli Iperspazi (E. Bertini, 1907, or 1923 for the latest edition). The present introduction aims to informally introduce Bertini’s Theorem on generic smoothness, with special attention to its re- cent improvements and its relationships with other kind of re- sults. Just to set the following discussion in an historical perspec- tive, recall that at Bertini’s time the situation was more or less the following: ¥ there were no schemes, ¥ almost all varieties were defined over the complex numbers, ¥ all varieties were embedded in some projective space, that is, they were not intrinsic. On the contrary, this dissertation will cope with Bertini’s the- orem by exploiting the powerful tools of modern algebraic ge- ometry, by working with schemes defined over any field (mostly, but not necessarily, algebraically closed). In addition, our vari- eties will be thought of as abstract varieties (at least when over a field of characteristic zero). This fact does not mean that we are neglecting Bertini’s original work, containing already all the rele- vant ideas: the proof we shall present in this exposition, over the complex numbers, is quite close to the one he gave. -
Cohen-Macaulay Modules Over Gorenstein Local Rings
COHEN{MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS RYO TAKAHASHI Contents Introduction 1 1. Associated primes 1 2. Depth 3 3. Krull dimension 6 4. Cohen{Macaulay rings and Gorenstein rings 9 Appendix A. Proof of Theorem 1.12 11 Appendix B. Proof of Theorem 4.8 13 References 17 Introduction Goal. The structure theorem of (maximal) Cohen{Macaulay modules over commutative Gorenstein local rings Throughout. • (R : a commutative Noetherian ring with 1 A = k[[X; Y ]]=(XY ) • k : a field B = k[[X; Y ]]=(X2;XY ) • x := X; y := Y 1. Associated primes Def 1.1. Let I ( R be an ideal of R (1) I is a maximal ideal if there exists no ideal J of R with I ( J ( R (2) I is a prime ideal if (ab 2 I ) a 2 I or b 2 I) (3) Spec R := fPrime ideals of Rg Ex 1.2. Spec A = f(x); (y); (x; y)g; Spec B = f(x); (x; y)g Prop 1.3. I ⊆ R an ideal (1) I is prime iff R=I is an integral domain (2) I is maximal iff R=I is a field 2010 Mathematics Subject Classification. 13C14, 13C15, 13D07, 13H10. Key words and phrases. Associated prime, Cohen{Macaulay module, Cohen{Macaulay ring, Ext module, Gorenstein ring, Krull dimension. 1 2 RYO TAKAHASHI (3) Every maximal ideal is prime Throughout the rest of this section, let M be an R-module. Def 1.4. p 2 Spec R is an associated prime of M if 9 x 2 M s.t p = ann(x) AssR M := fAssociated primes of Mg Ex 1.5. -
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:1802.08409V2 [Math.AC]
CORRESPONDENCE BETWEEN TRACE IDEALS AND BIRATIONAL EXTENSIONS WITH APPLICATION TO THE ANALYSIS OF THE GORENSTEIN PROPERTY OF RINGS SHIRO GOTO, RYOTARO ISOBE, AND SHINYA KUMASHIRO Abstract. Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties. Contents 1. Introduction 1 2. Correspondence between trace ideals and birational extensionsofthebasering 5 3. The case where R is a Gorenstein ring of dimension one 9 4. Modules in which every submodule is a trace module 13 5. Surjectivity of the correspondence ρ in dimension one 17 6. Anti-stable rings 21 References 25 1. Introduction This paper aims to explore the structure of (not necessarily Noetherian) commutative arXiv:1802.08409v2 [math.AC] 7 Dec 2018 rings in connection with their trace ideals. Let R be a commutative ring. For R-modules M and X, let τM,X : HomR(M,X) ⊗R M → X denote the R-linear map defined by τM,X (f ⊗ m) = f(m) for all f ∈ HomR(M,X) and m ∈ M.