DOCSLIB.ORG

Twenty-Four Hours of Local Cohomology This Page Intentionally Left Blank Twenty-Four Hours of Local Cohomology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 To our teachers This page intentionally left blank Contents Preface Introduction Lecture 1. Basic Notions §1. Algebraic sets §2. Krull dimension of a ring §3. Dimension of an algebraic set §4. An extended example §5. Tangent spaces and regular rings §6. Dimension of a module Lecture 2. Cohomology §1. Sheaves §2. Cech cohomology §3. Calculus versus topology §4. Cech cohomology and derived functors Lecture 3. Resolutions and Derived Functors §1. Free, projective, and flat modules §2. Complexes §3. Resolutions §4. Derived functors Lecture 4. Limits §1. An example from topology Vlll Contents §2. Direct limits §3. The category of diagrams §4. Exactness §5. Diagrams over diagrams §6. Filtered posets §7. Diagrams over the pushout poset §8. Inverse limits Lecture 5. Gradings, Filtrations, and Grobner Bases §1. Filtrations and associated graded rings §2. Hilbert polynomials §3. Monomial orders and initial forms §4. Weight vectors and flat families §5. Buchberger's algorithm §6. Grobner bases and syzygies Lecture 6. Complexes from a Sequence of Ring Elements §1. The Koszul complex §2. Regular sequences and depth: a first look §3. Back to the Koszul complex §4. The Cech complex Lecture 7. Local Cohomology §1. The torsion functor §2. Direct limit of Ext modules §3. Direct limit of Koszul cohomology §4. Return of the Cech complex Lecture 8. Auslander-Buchsbaum Formula and Global Dimension §1. Regular sequences and depth redux §2. Global dimension §3. Auslander-Buchsbaum formula §4. Regular local rings §5. Complete local rings Lecture 9. Depth and Cohomological Dimension §1. Depth §2. Cohomological dimension Contents IX §3. Arithmetic rank Lecture 10. Cohen-Macaulay Rings §1. Noether normalization §2. Intersection multiplicities §3. Invariant theory §4. Local cohomology Lecture 11. Gorenstein Rings §1. Bass numbers §2. Recognizing Gorenstein rings §3. Injective resolutions of Gorenstein rings §4. Local duality §5. Canonical modules Lecture 12. Connections with Sheaf Cohomology §1. Sheaf theory §2. Flasque sheaves §3. Local cohomology and sheaf cohomology Lecture 13. Projective Varieties §1. Graded local cohomology §2. Sheaves on projective varieties §3. Global sections and cohomology Lecture 14. The Hartshorne-Lichtenbaum Vanishing Theorem Lecture 15. Connectedness §1. Mayer-Vietoris sequence §2. Punctured spectra Lecture 16. Polyhedral Applications §1. Polytopes and faces §2. Upper bound theorem §3. The h-vector of a simplicial complex §4. Stanley-Reisner rings §5. Local cohomology of Stanley-Reisner rings §6. Proof of the upper bound theorem Lecture 17. D-modules X Contents §1. Rings of differential operators §2. The Weyl algebra §3. Holonomic modules §4. Grobner bases Lecture 18. Local Duality Revisited §1. Poincare duality §2. Grothendieck duality §3. Local duality §4. Global canonical modules Lecture 19. De Rham Cohomology §1. The real case: de Rham's theorem §2. Complex manifolds §3. The algebraic case §4. Local and de Rham cohomology Lecture 20. Local Cohomology over Semigroup Rings §1. Semigroup rings §2. Cones from semigroups §3. Maximal support: the Ishida complex §4. Monomial support: Zd-graded injectives §5. Hartshorne's example Lecture 21. The Frobenius Endomorphism §1. Homological properties §2. Frobenius action on local cohomology modules §3. A vanishing theorem Lecture 22. Curious Examples §1. Dependence on characteristic §2. Associated primes of local cohomology modules Lecture 23. Algorithmic Aspects of Local Cohomology §1. Holonomicity of localization §2. Local cohomology as a D-module §3. Bernstein-Sato polynomials §4. Computing with the Frobenius morphism Lecture 24. Holonomic Rank and Hyper geometric Systems Contents xi §1. GKZ A-hypergeometric systems 247 §2. Rank vs. volume 250 §3. Euler-Koszul homology 251 §4. Holonomic families 254 Appendix. Injective Modules and Matlis Duality 257 §1. Essential extensions 257 §2. Noetherian rings 260 §3. Artinian rings 263 §4. Matlis duality 265 Bibliography 269 Index 277 This page intentionally left blank Preface This book is an outgrowth of the summer school Local cohomology and its interactions with algebra, geometry, and analysis that we organized in June 2005 in Snowbird, Utah. This was a joint program under the AMS-IMS- SIAM Summer Research Conference series and the MSRI Summer Graduate Workshop series. The school centered around local cohomology, and was in• tended for graduate students interested in various branches of mathematics. It consisted of twenty-four lectures by the authors of this book, followed by a three-day conference. We thank our co-authors for their support at all stages of the workshop. In addition to preparing and delivering the lectures, their enthusiastic par• ticipation, and interaction with the students, was critical to the success of the event. We also extend our hearty thanks to Wayne Drady, the AMS conference coordinator, for cheerful and superb handling of various details. We profited greatly from the support and guidance of David Eisenbud and Hugo Rossi at MSRI, and Jim Maxwell at AMS. We express our thanks to them, and to our Advisory Committee: Mel Hochster, Craig Huneke, Joe Lipman, and Paul Roberts. We are also indebted to the conference speakers: Markus Brodmann, Ragnar-Olaf Buchweitz, Phillippe Gimenez, Gennady Lyubeznik, Paul Roberts, Peter Schenzel, Rodney Sharp, Ngo Viet Trung, Kei-ichi Watanabe, and Santiago Zarzuela. Finally, we thank the AMS and the MSRI for their generous support in hosting this summer school, and the AMS for publishing this revised version of the "Snowbird notes". Anurag K. Singh and Uli Walt her xiii This page intentionally left blank Introduction Local cohomology was invented by Grothendieck to prove Lefschetz-type theorems in algebraic geometry. This book seeks to provide an introduction to the subject which takes cognizance of the breadth of its interactions with other areas of mathematics. Connections are drawn to topological, geo• metric, combinatorial, and computational themes. The lectures start with basic notions in commutative algebra, leading up to local cohomology and its applications. They cover topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, connections to sheaf cohomology and to de Rham cohomology, Grobner bases in the com• mutative setting as well as for D-modules, the Probenius morphism and characteristic p methods, finiteness properties of local cohomology modules, semigroup rings and polyhedral geometry, and hypergeometric systems aris• ing from semigroups. The subject can be introduced from various perspectives. We start from an algebraic one, where the definition is elementary: given an ideal a in a Noetherian commutative ring, for each module consider the submodule of elements annihilated by some power of a. This operation is not exact, in
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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}.
    [Show full text]
  • Program of the Sessions San Diego, California, January 9–12, 2013
    Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes.
    [Show full text]
  • Graduate Texts in Mathematics 52
    Graduate Texts in Mathematics 52 Editorial Board S. Axler F.W. Gehring K.A. Ribet Graduate Texts in Mathematics most recent titles in the GTM series 129 FULTON/HARRIS. Representation Theory: 159 CONWAY. Functions of One A First Course. Complex Variable II. Readings in Mathematics 160 LANG. Differential and Riemannian 130 DODSON/PoSTON. Tensor Geometry. Manifolds. 131 LAM. A First Course in Noncommutative 161 BORWEINIERDEL Yl. Polynomials and Rings. Polynomial Inequalities. 132 BEARDON. Iteration of Rational Functions. 162 ALPERIN/BELL. Groups and 133 HARRIS. Algebraic Geometry: A First Representations. Course. 163 DIXON/MORTIMER. Permutation 134 RoMAN. Coding and Information Theory. Groups. 135 ROMAN. Advanced Linear Algebra. 164 NATHANSON. Additive Number Theory: 136 ADKINS/WEINTRAUB. Algebra: An The Classical Bases. Approach via Module Theory. 165 NATHANSON. Additive Number Theory: 137 AxLERIBouRDoN/RAMEY. Harmonic Inverse Problems and the Geometry of Function Theory. Sumsets. 138 CoHEN. A Course in Computational 166 SHARPE. Differential Geometry: Cartan's Algebraic Number Theory. Generalization of Klein's Erlangen 139 BREDON. Topology and Geometry. Program. 140 AUBIN. Optima and Equilibria. An 167 MORANDI. Field and Galois Theory. Introduction to Nonlinear Analysis. 168 EWALD. Combinatorial Convexity and 141 BECKERIWEISPFENNING/KREDEL. Grtibner Algebraic Geometry. Ba~es. A Computational Approach to 169 BHATIA. Matrix Analysis. Commutative Algebra. 170 BREDON. Sheaf Theory. 2nd ed. 142 LANG. Real and Functional Analysis. 171 PETERSEN. Riemannian Geometry. 3rd ed. 172 REMMERT. Classical Topics in Complex 143 DOOB. Measure Theory. Function Theory. 144 DENNISIFARB. Noncommutative 173 DIESTEL. Graph Theory. Algebra. 174 BRIDGES. Foundations of Real and 145 VICK. Homology Theory. An Abstract Analysis. Introduction to Algebraic Topology.
    [Show full text]
  • The Completion and Local Cohomology Theorems for Complex Cobordism
    THE COMPLETION AND LOCAL COHOMOLOGY THEOREMS FOR COMPLEX COBORDISM FOR ALL COMPACT LIE GROUPS MARCO LA VECCHIA Abstract. We generalize the completion theorem for equivariant MU- module spectra for finite groups or finite extensions of a torus to compact Lie groups using the splitting of global functors proved by Schwede. This proves a conjecture of Greenlees and May. Contents 1. Overview 1 2. Notations, Conventions and Facts 3 3. Classical Statement 5 4. Schwede’s Splitting 8 5. The main Result 10 References 11 1. Overview 1.1. Introduction. A completion theorem establishes a close relationship between equivariant cohomology theory and its non-equivariant counterpart. It takes various forms, but in favourable cases it states ∗ ∧ ∼ ∗ (EG)JG = E (BG) ∗ where E is a G-spectrum, EG is the associated equivariant cohomology theory and JG is the augmentation ideal (Definition 3.7). The first such theorem is the Atiyah-Segal Completion Theorem for com- plex K-theory [AS69]. This especially favourable because the coefficient ring KU∗ = R(G)[v, v−1] is well understood and in particular it is Noetherian, arXiv:2107.03093v1 [math.AT] 7 Jul 2021 G and so in this case we can view the theorem as the calculation of the coho- mology of the classifying space. The good behaviour for all groups permits one to make good use of naturality in the group, and indeed [AS69] uses this to give a proof for all compact Lie groups G. The result raised the question of what other theories enjoy a completion theorem, and the case of equivari- ant complex cobordism was considered soon afterwards, with L¨offler giving ∗ a proof in the abelian case [L¨of74].
    [Show full text]
  • An Elementary Theory of Hyperfunctions and Microfunctions
    PARTIAL DIFFERENTIAL EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 27 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1992 AN ELEMENTARY THEORY OF HYPERFUNCTIONS AND MICROFUNCTIONS HIKOSABURO KOMATSU Department of Mathematics, Faculty of Science, University of Tokyo Tokyo, 113 Japan 1. Sato’s definition of hyperfunctions. The hyperfunctions are a class of generalized functions introduced by M. Sato [34], [35], [36] in 1958–60, only ten years later than Schwartz’ distributions [40]. As we will see, hyperfunctions are natural and useful, but unfortunately they are not so commonly used as distributions. One reason seems to be that the mere definition of hyperfunctions needs a lot of preparations. In the one-dimensional case his definition is elementary. Let Ω be an open set in R. Then the space (Ω) of hyperfunctions on Ω is defined to be the quotient space B (1.1) (Ω)= (V Ω)/ (V ) , B O \ O where V is an open set in C containing Ω as a closed set, and (V Ω) (resp. (V )) is the space of all holomorphic functions on V Ω (resp. VO). The\ hyper- functionO f(x) represented by F (z) (V Ω) is written\ ∈ O \ (1.2) f(x)= F (x + i0) F (x i0) − − and has the intuitive meaning of the difference of the “boundary values”of F (z) on Ω from above and below. V Ω Fig. 1 [233] 234 H. KOMATSU The main properties of hyperfunctions are the following: (1) (Ω), Ω R, form a sheaf over R. B ⊂ Ω Namely, for all pairs Ω1 Ω of open sets the restriction mappings ̺Ω1 : (Ω) (Ω ) are defined, and⊂ for any open covering Ω = Ω they satisfy the B →B 1 α following conditions: S (S.1) If f (Ω) satisfies f = 0 for all α, then f = 0; ∈B |Ωα (S.2) If fα (Ωα) satisfy fα Ωα∩Ωβ = fβ Ωα∩Ωβ for all Ωα Ωβ = , then there is∈ Ban f (Ω) such| that f = f| .
    [Show full text]
  • 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.
    [Show full text]
  • Local Cohomology and the Cohen-Macaulay Property
    Local Cohomology and the Cohen-Macaulay property Abstract These lectures represent an extended version of the contents of a one hour introductory talk prepared by Florian Enescu and Sara Faridi for the Minnowbrook workshop to assist the lectures of one of the main speakers, Paul Roberts. Part of this talk was given earlier at the 2004 Utah mini-course on Classical problems in commutative algebra by the first author. The notes for that talk have been been typed and prepared by Bahman Engheta and we used part of them quite extensively in preparing this version. We would like to thank Bahman for his work. The references listed at the end were used in depth in preparing these notes and the authors make no claim of originality. Moreover, the reader is encouraged to consult these references for more details and many more results that were omitted due to time constraints. 1 Injective modules and essential extensions Throughout, let R be a commutative Noetherian ring. Definition 1 (injective module). An R-module E is called injective if one of the following equivalent con- ditions hold. 1. Given any R-module monomorphism f : N → M, every homomorphism u : N → E can be extended to a homomorphism g : M → E, i.e. g f = u. f 0 w NMw u g u E 2. For each ideal J of R, any homomorphism u : J → E can be extended to a homomorphism R → E. 3. The functor Hom( ,E) is exact. Example 2. From part 2 of Definition 1 it follows that Q and Q/Z are injective Z-modules.
    [Show full text]
  • 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.
    [Show full text]
  • Three Lectures on Local Cohomology Modules Supported on Monomial Ideals
    THREE LECTURES ON LOCAL COHOMOLOGY MODULES SUPPORTED ON MONOMIAL IDEALS JOSEP ALVAREZ` MONTANER Abstract. These notes are an extended version of a set of lectures given at "MON- ICA: MONomial Ideals, Computations and Applications", at the CIEM, Castro Urdiales (Cantabria, Spain) in July 2011. The goal of these lectures is to give an overview of some results that have been developed in recent years about the structure of local cohomology modules supported on a monomial ideal. We will highlight the interplay of multi-graded commutative algebra, combinatorics and D-modules theory that allow us to give differ- ent points of view to this subject. We will try to preserve the informal character of the lectures, so very few complete proofs are included. Contents Introduction 2 Lecture 1: Basics on local cohomology modules 4 1. Basic definitions 4 2. The theory of D-modules 8 Lecture 2: Local cohomology modules supported on monomial ideals 17 3. Zn-graded structure 18 4. D-module structure 22 5. Building a dictionary 40 Lecture 3: Bass numbers of local cohomology modules 47 6. Bass numbers of modules with variation zero 48 r 7. Injective resolution of HI (R) 54 Appendix: Zn-graded free and injective resolutions 56 Partially supported by MTM2007-67493 and SGR2009-1284. 1 2 JOSEP ALVAREZ` MONTANER Tutorial (joint with O. Fern´andez-Ramos) 60 8. Getting started 62 9. Tutorial 65 References 67 Introduction Local cohomology was introduced by A. Grothendieck in the early 1960's and quickly became an indispensable tool in Commutative Algebra. Despite the effort of many authors in the study of these modules, their structure is still quite unknown.
    [Show full text]
  • Local Cohomology: an Algebraic Introduction with Geometric Applications,Bym.P
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 3, Pages 409{411 S 0273-0979(99)00785-5 Article electronically published on April 23, 1999 Local cohomology: An algebraic introduction with geometric applications,byM.P. Brodman and R. Y. Sharp, Cambridge University Press, 1998, xv+416 pp., $69.95, ISBN 0-521-37286-0 The title refers to the local cohomology theory introduced by Grothendieck in his 1961 Harvard seminar [4] and his 1961–62 Paris seminar [5]. Since I attended the Harvard seminar and wrote the notes for it, local cohomology has always been one of my favorite subjects. To understand what it is, let us review a little the interactions between algebra and geometry that make up the subject of algebraic geometry. Of course polynomial equations have always been used to define algebraic vari- eties, but it is only in the early 20th century with the rise of algebraic structures as objects of study that algebraic geometry and local algebra (the study of local rings) have become closely linked. Much of this connection is due to the work of Zariski, who discovered that a simple point on a variety corresponds to a regular local ring. He also explored the geometrical significance of integrally closed rings in relation to normal varieties. Meanwhile, methods of homology and cohomology, which first appeared in topol- ogy, were making their way into algebra and geometry. The great work of Cartan and Eilenberg [2] formalized the techniques of cohomology into the abstract the- ory of homological algebra, including projective and injective resolutions, derived functors, and Ext and Tor.
    [Show full text]
  • The Cohomology of Coherent Sheaves
    CHAPTER VII The cohomology of coherent sheaves 1. Basic Cechˇ cohomology We begin with the general set-up. (i) X any topological space = U an open covering of X U { α}α∈S a presheaf of abelian groups on X. F Define: (ii) Ci( , ) = group of i-cochains with values in U F F = (U U ). F α0 ∩···∩ αi α0,...,αYi∈S We will write an i-cochain s = s(α0,...,αi), i.e., s(α ,...,α ) = the component of s in (U U ). 0 i F α0 ∩··· αi (iii) δ : Ci( , ) Ci+1( , ) by U F → U F i+1 δs(α ,...,α )= ( 1)j res s(α ,..., α ,...,α ), 0 i+1 − 0 j i+1 Xj=0 b where res is the restriction map (U U ) (U U ) F α ∩···∩ Uαj ∩···∩ αi+1 −→ F α0 ∩··· αi+1 and means “omit”. Forb i = 0, 1, 2, this comes out as δs(cα , α )= s(α ) s(α ) if s C0 0 1 1 − 0 ∈ δs(α , α , α )= s(α , α ) s(α , α )+ s(α , α ) if s C1 0 1 2 1 2 − 0 2 0 1 ∈ δs(α , α , α , α )= s(α , α , α ) s(α , α , α )+ s(α , α , α ) s(α , α , α ) if s C2. 0 1 2 3 1 2 3 − 0 2 3 0 1 3 − 0 1 2 ∈ One checks very easily that the composition δ2: Ci( , ) δ Ci+1( , ) δ Ci+2( , ) U F −→ U F −→ U F is 0. Hence we define: 211 212 VII.THECOHOMOLOGYOFCOHERENTSHEAVES s(σβ0, σβ1) defined here U σβ0 Uσβ1 Vβ1 Vβ0 ref s(β0, β1) defined here Figure VII.1 (iv) Zi( , ) = Ker δ : Ci( , ) Ci+1( , ) U F U F −→ U F = group of i-cocycles, Bi( , ) = Image δ : Ci−1( , ) Ci( , ) U F U F −→ U F = group of i-coboundaries Hi( , )= Zi( , )/Bi( , ) U F U F U F = i-th Cech-cohomologyˇ group with respect to .
    [Show full text]