Local Cohomology: an Algebraic Introduction with Geometric Applications,Bym.P
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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}. -
Semi-Dualizing Complexes and Their Auslander Categories
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 5, Pages 1839{1883 S 0002-9947(01)02627-7 Article electronically published on January 4, 2001 SEMI-DUALIZING COMPLEXES AND THEIR AUSLANDER CATEGORIES LARS WINTHER CHRISTENSEN Abstract. Let R be a commutative Noetherian ring. We study R{modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for R on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes \lying between" these extremes is incentive. Introduction In this paper all rings are commutative and Noetherian, in particular, R always denotes such a ring. We study R{complexes (that is, complexes of R{modules) with certain excellent duality properties. The canonical example is the ring itself, considered as a complex concentrated in degree zero. Another example is dualizing complexes for R; these were introduced by A. Grothendieck in [RAD] and have proved to be a powerful tool, as demonstrated by P. Roberts in [PRb76], and by C. Peskine and L. Szpiro in [CPsLSz73]. Modules with excellent duality properties have been studied by H.{B. Foxby in [HBF72], and in [EGl84] by E.S. Golod, who used the name suitable1 for these modules. Other complexes of the kind considered here were used tacitly by L.L. Avramov and H.{B. -
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. -
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). -
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]. -
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 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 . -
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| .