Tight Closure and Elements of Small Order in Integral Extensions*
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Foundations of Tight Closure Theory
FOUNDATIONS OF TIGHT CLOSURE THEORY by Melvin Hochster These notes give an introduction to tight closure theory based on Lectures given in Math 711 in Fall, 2007, at the University of Michigan. In the current version, only the positive characteristic theory is covered. There are several novel aspects to this treatment. One is that the theory for all modules, not just finitely generated modules, is developed systematically. It is shown that all the known methods of proving the existence of test elements yield completely stable test elements that can be used in testing tight closure for all pairs of modules. The notion of strong F-regularity is developed without the hypothesis that the ring be F-finite. The definition is simply that every submodule of every module is closed, without the hypothesis of finite generation. This agrees with the usual definition in the F-finite case, which is developed first in these notes. A great deal of revision and expansion are planned. The structure of the individual lectures has been preserved. References often refer to a result as, say, \the Theorem on p. k of the Lecture of ::: ". The page numbering meant is internal to the lecture and is not shown explicitly. Thus, if the Lecture begins on p. n, the result specified would be on page n + k − 1. I have not tried to give complete references. When references are given, they occur in full in the text where they are cited. At this time there is no separate list. j Subsidiary material has been included in the notes that was not covered in detail in the lectures. -
M-Adic Perturbations in Noetherian Local Rings
Perturbations in Noetherian Local Rings m-adic Perturbations in Noetherian Local Rings Nick Cox-Steib [email protected] University of Missouri Abstract We develop new methods to study m-adic stability in an arbitrary Noetherian local ring. These tech- niques are used to prove results about the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine m-adic perturbations. 1 Introduction The topic of this paper involves comparing a fixed ideal, I =( f1,..., fc) = f , inside a Noetherian local ring, (R,m ), with ideals of the form ( f + ε , f + ε ,..., f + ε ) = f + ε , where ε ,...,ε are in R 1 1 2 2 c c 1 c some large power of the maximal ideal of R. When it is in the interest of clarity and space and the ideal I is understood, we often use the notation M M M M := = , and M := IM f M ( f +ε) f + ε M where M is an R-module. We will refer to the ideals f + ε , as mR-adic perturbations of I, and, more generally, we may also speak of the quotients M as perturbations of M. At the most basic level, we ( f +ε) want to understand the relationship between I and its perturbations. Recent interest in these problems stems from their relationship to the deformation of properties. This arXiv:2011.08044v2 [math.AC] 14 Dec 2020 relationship has been formalized in a recent paper by De Stefani and Smirnov with the concept of m-adic stability [SS20]. A property, P, of Noetherian local rings is m-adically stable if it is stable under sufficiently small perturbations of ideals generated by a regular element — that is, for every regular element f in a Noetherian local ring (R,m) such that R/( f ) has property P, there is an N ∈ N such that R/( f + ε) also has property P for all ε ∈ mN. -
Actions of Reductive Groups on Regular Rings and Cohen-Macaulay Rings
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 2, March 1974 ACTIONS OF REDUCTIVE GROUPS ON REGULAR RINGS AND COHEN-MACAULAY RINGS BY MELVIN HOCHSTER AND JOEL L. ROBERTS1 Communicated by Dock S. Rim, August 30, 1973 0. The main results. This note is an announcement of the results below, whose proofs will appear separately [7]. MAIN THEOREM. Let G be a linearly reductive affine linear algebraic group over a field K of arbitrary characteristic acting K-rationally on a regular Noetherian K-algebra S. Then the ring of invariants R=S° is Cohen-Macaulay. THEOREM. If S is a regular Noetherian ring of prime characteristic p > 0, and R is a pure subring of S (i.e. for every R-module M, M-+M (g>R S is injective), e.g. if R is a direct summand of S as R-modules, then R is Cohen-Macaulay. The proofs utilize results of interest in their own right: PROPOSITION A. Let L be a field, y0, • • • 9ym indeterminates over > L, S=L[y0, • • • ,ym], and F=Proj(5)=/ £. Let K be a subfield of L, and let R be a finitely generated graded K-algebra with R0=K. Let h : R-^S be a K-homomorphism which multiplies degrees by d. Let P be the irrelevant maximal ideal of R9 and let X=Froj(R). Let U=Y-V(h(P)S). Let (p =h* be the induced K-morphism from the quasi-projective L-variety U to the projective K-scheme X. Then (pf'.H^X, 0x)->#*(£/, Ojj) is zero fori^l. -
Algebra & Number Theory Vol. 7 (2013)
Algebra & Number Theory Volume 7 2013 No. 3 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Edward Frenkel University of California, Berkeley, USA Michael Singer North Carolina State University, USA Andrew Granville Université de Montréal, Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Virginia, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University, -
JOINTMA Volume 17, Number 5 Join the MAA and AMS at the Winter Meeting January 7-10, 1998, in Baltimore, Maryland
THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA JOINTMA Volume 17, Number 5 Join the MAA and AMS at the winter meeting January 7-10, 1998, in Baltimore, Maryland. This meeting will In this Issue exceed all your professional and personal expectations 3 Joint Mathematics with a rich program and a Meetings meeting location with great 7-10, cultural and historic ambi January ence. Baltimore, MD Experience the insight and knowledge of one of the 18NewMAA world's leading mathemati President Elected cal physicists when Edward Witten, delivers the AMS Josiah Willard Gibbs lecture. 18 Visual This Fields Medalist played a key role in discovering the Mathematics "Seiberg-Witten Equations", and is sure to be a popular 19 Personal Opinion speaker. More than fifteen additional invited addresses, featuring speakers such as 20 IMO Results Marjorie Senechal, who has worked extensively on quasicrystal theory, and 21 MathFest Review Herbert Wilf, well-known for his work on combinatorics, 22 Joint Meetings: will fill your busy schedule. What's in it Attend the invited address on "Some Exceptional Ob for Students? jects and their History" de livered by John Stillwell for a exceptional symplectic structures and attend the three AMS perspective. Then add to your history lesson Colloquium lectures by Gian-Carlo Rota. Of 27 Employment with two joint AMS/MAA special paper ses course, these are in addition to the paper ses Opportunities sions, two MAA contributed paper sessions, an sions and the MAA Student lecture with a geo AMS special session on The History of Math metric focus. ematical logic, and two MAA minicourses on These activities are just a sampling of what the the relationships between history and math Joint Mathematics Meetings in Baltimore has to ematics instruction. -
Arxiv:Math/0206250V2
CANONICAL BIG COHEN-MACAULAY ALGEBRAS AND RATIONAL SINGULARITIES HANS SCHOUTENS Abstract. We give a canonical construction of a balanced big Coh- en-Macaulay algebra for a domain of finite type over C by taking ul- traproducts of absolute integral closures in positive characteristic. This yields a new tight closure characterization of rational singularities in characteristic zero. 1. Introduction In [4], Hochster proves the existence of big Cohen-Macaulay modules for a large class of Noetherian rings containing a field. Recall that a module M over a Noetherian local ring R is called a big Cohen-Macaulay module, if there is a system of parameters of R which is M-regular (the adjective big is used to emphasize that M need not be finitely generated). He also exhibits in that paper the utility of big Cohen-Macaulay modules in answering various homo- logical questions. Often, one can even obtain a big Cohen-Macaulay module M such that every system of parameters is M-regular; these are called bal- anced big Cohen-Macaulay modules. In [5], Hochster and Huneke show that for equicharacteristic excellent local domains, one can even find a bal- anced big Cohen-Macaulay algebra, that is to say, M admits the structure of a (commutative) R-algebra. In fact, for R a local domain of positive charac- teristic, they show that the absolute integral closure of R, denoted R+, is a (balanced) big Cohen-Macaulay algebra (it is easy to see that this is false in characteristic zero). In [6], using lifting techniques similar to the ones devel- oped in the original paper of Hochster, they obtain also the existence of big Cohen-Macaulay algebras in characteristic zero. -
[Math.AC] 3 Feb 2005
LOCALIZATION OF TIGHT CLOSURE IN TWO-DIMENSIONAL RINGS KAMRAN DIVAANI-AAZAR AND MASSOUD TOUSI Abstract. It is shown that tight closure commutes with localization in any two dimensional ring R of prime characteristic if either R is a Nagata ring or R pos- sesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic. 1. Introduction Throughout this paper, R is a commutative Noetherian ring (with identity) of prime characteristic p. The theory of tight closure was introduced by Hochster and Huneke [2]. There are many applications for this notion in both commutative algebra and algebraic geometry. However, there are many basic open questions concerning tight closure. One of the essential questions is whether tight closure commutes with localization. For an expository account on tight closure, we refer the reader to [3] or [8]. In the sequel, R◦ denotes the set of elements of R which are not contained in any minimal prime ideal of R. We use the letter q for nonnegative powers pe of p. Let I be an ideal of R and I[q] the ideal generated by q-th powers of elements of I. Then I∗, tight closure of I is the set of all elements x ∈ R for which there exists c ∈ R◦ such that cxq ∈ I[q] for all q ≫ 0. Also, for a nonnegative power q′ of p an element c ∈ R◦ is called q′-weak test element, if for any ideal I of R and any element x ∈ I∗, we have cxq ∈ I[q] for all q ≥ q′. -
Commutative Algebra Provides a Big Surprise for Craig Huneke's Birthday
CÓÑÑÙØaØiÚe aÐgebÖa ÔÖÓÚide× a big ×ÙÖÔÖi×e fÓÖ CÖaig ÀÙÒeke³× biÖØhdaÝ Irena Swanson Communicated by Tom Garrity A commutative algebra conference in July, 2016, on the occasion of Craig Huneke’s 65th birthday, included a major mathematical surprise. Craig Huneke has been at the forefront of research in commutative algebra, introducing and advancing several influential notions, such as d-sequences, licci ideals, symbolic powers, homological methods, computational methods, tight closure, uniform bounds, and prime characteristic methods. He has mentored 24 PhD students and many postdocs; he has co-organized conferences, such as the Kansas-Missouri-Nebraska KUMUNU commutative algebra conference; he has served on the Executive Committee of the AMS and on the Board of Trustees of the MSRI. Almost all the talks were directly related to Huneke’s work, and most speakers started their talks describing how Huneke affected their work as a mathematician and as a person; they talked about his mathematical productivity, extensive collaborations, productive and precious lunches with napkin notes, excellent and influential talks, his advising, mentoring, his friendly competitiveness, and so on. Claudia Polini addressed how commutative algebra in general is friendly to women (possibly due to Emmy Noether being one of us), and in particular how Huneke has been a tremendous role model as a teacher, collaborator, mentor, and organizer. His own family life, with wife Edith Clowes, Professor of Slavic Languages and Literatures, and their children Sam and Ned, has been a shining example for possibilities in home and professional life for women in academia. During the banquet, Hochster read his poem honoring Huneke. -
Contemporary Mathematics 448
CONTEMPORARY MATHEMATICS 448 Algebra/ Geometry and Their Interactions International Conference Midwest Algebra, Geometry and Their Interactions October 7-11, 2005 University of Notre Dame, Notre Dame, Indiana Alberto Corso Juan Migliore Claudia Polini Editors http://dx.doi.org/10.1090/conm/448 CoNTEMPORARY MATHEMATICS 448 Algebra, Geometry and Their Interactions International Conference Midwest Algebra, Geometry and Their Interactions October 7-11, 2005 University of Notre Dame, Notre Dame, Indiana Alberto Corso Juan Migliore Claudia Polini Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor George Andrews Andreas Blass Abel Klein 2000 Mathematics Subject Classification. Primary 05C90, 13C40, 13D02, 13D07, 13D40, 14C05, 14J60, 14M12, 14N05, 65H10, 65H20. Library of Congress Cataloging-in-Publication Data International Conference on Midwest Algebra, Geometry and their Interactions, MAGIC'05 (2005 : University of Notre Dame) Algebra, geometry and their interactions : International Conference on Midwest Algebra, Geometry and their Interactions: MAGIC'05. October 7-11, 2005, University of Notre Dame, Notre Dame, Indiana/ Albert Corso, Juan Migliore, Claudia Polini, editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; v. 448) Includes bibliographical references. ISBN 978-0-8218-4094-8 (alk. paper) 1. Algebra-Congresses. 2. Geometry-Congresses. I. Corso, Alberto. II. Migliore, Juan C. (Juan Carlos), 1956- Ill. Polini, Claudia, 1966- IV. Title. QA150.1566 2007 512-dc22 2007060846 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. -
Arxiv:1805.00492V2 [Math.AC] 12 Apr 2019 a Nt Rjciedimension
NON-COMMUTATIVE RESOLUTIONS OF TORIC VARIETIES ELEONORE FABER, GREG MULLER, AND KAREN E. SMITH Abstract. Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring EndR(A), where A is the (finite) direct sum of all (isomorphism classes of) conic R-modules, has finite global dimension equal to the dimension of R. This gives a precise version, and an elementary proof, of a theorem of Spenkoˇ and Van den Bergh implying that EndR(A) has finite global dimension. Furthermore, we show that EndR(A) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field k of prime charac- teristic, we show that the ring of differential operators Dk(R) has finite global dimension. 1. Introduction Consider a local or graded ring R which is commutative and Noetherian. A well-known theorem of Auslander-Buchsbaum and Serre states that R is regular if and only if R has finite global dimension—that is, if and only if every R-module has finite projective dimension. While the standard definition of regularity does not extend to non-commutative rings, the definition of global dimension does, so this suggests that finite global di- mension might play the role of regularity for non-commutative rings. This is an old idea going back at least to Dixmier [Dix63], though since then our understanding of connection between regularity and finite global dimension has been refined by the works of Auslander, Artin, Shelter, Van den Bergh and others. -
Tight Closure and Characteristic P Methods
TIGHT CLOSURE THEORY AND CHARACTERISTIC P METHODS by Melvin Hochster 0. Introduction Unless otherwise specified, the rings that we consider here will be Noetherian rings R containing a field. Frequently, we restrict, for simplicity, to the case of domains finitely generated over a field K. The theory of tight closure exists in much greater generality, and we refer the reader to [HH1{14] and [Ho1{3], as well as to the expository accounts [Bru], [Hu1,2] and [Leu] (in this volume) for the development of the larger theory and its applications as well as discussion of related topics such as the existence of big Cohen- Macaulay algebras. We shall give an introductory overview of the theory of tight closure, which has recently played a primary role among characteristic p methods. We shall see that such methods can be used even when the ring contains a field of characteristic 0. Here, in reverse order, are several of the most important reasons for studying tight closure theory, which gives a closure operation on ideals and on submodules. We focus mostly on the case of ideals here, although there is some discussion of modules. We shall elaborate on the themes brought forth in the list below in the sequel. 11. Tight closure can be used to shorten difficult proofs of seemingly unrelated results. The results turn out to be related after all. Often, the new results are stronger than the original results. 10. Tight closure provides algebraic proofs of several results that can otherwise be proved only in equal characteristic 0, and whose original proofs depended on analytic tech- niques. -
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.