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CRITERIA for FLATNESS and INJECTIVITY 3 Ring of R
CRITERIA FOR FLATNESS AND INJECTIVITY NEIL EPSTEIN AND YONGWEI YAO Abstract. Let R be a commutative Noetherian ring. We give criteria for flatness of R-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if R has characteristic p, or more generally if it has a locally contracting en- domorphism. Dualizing, we give criteria for injectivity of R-modules in terms of coassociated primes and (h-)divisibility of certain Hom-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (including a localization result), a theorem on coassociated primes across a Hom-module base change, and a local criterion for injectivity. 1. Introduction The most important classes of modules over a commutative Noetherian ring R, from a homological point of view, are the projective, flat, and injective modules. It is relatively easy to check whether a module is projective, via the well-known criterion that a module is projective if and only if it is locally free. However, flatness and injectivity are much harder to determine. R It is well-known that an R-module M is flat if and only if Tor1 (R/P,M)=0 for all prime ideals P . For special classes of modules, there are some criteria for flatness which are easier to check. For example, a finitely generated module is flat if and only if it is projective. More generally, there is the following Local Flatness Criterion, stated here in slightly simplified form (see [Mat86, Section 22] for a self-contained proof): Theorem 1.1 ([Gro61, 10.2.2]). -
Homotopical Categories: from Model Categories to ( ,)-Categories ∞
HOMOTOPICAL CATEGORIES: FROM MODEL CATEGORIES TO ( ;1)-CATEGORIES 1 EMILY RIEHL Abstract. This chapter, written for Stable categories and structured ring spectra, edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman’s categories of frac- tions to Quillen’s model categories, through Dwyer and Kan’s simplicial localiza- tions and culminating in ( ;1)-categories, first introduced through concrete mod- 1 els and later re-conceptualized in a model-independent framework. This reader is not presumed to have prior acquaintance with any of these concepts. Suggested exercises are included to fertilize intuitions and copious references point to exter- nal sources with more details. A running theme of homotopy limits and colimits is included to explain the kinds of problems homotopical categories are designed to solve as well as technical approaches to these problems. Contents 1. The history of homotopical categories 2 2. Categories of fractions and localization 5 2.1. The Gabriel–Zisman category of fractions 5 3. Model category presentations of homotopical categories 7 3.1. Model category structures via weak factorization systems 8 3.2. On functoriality of factorizations 12 3.3. The homotopy relation on arrows 13 3.4. The homotopy category of a model category 17 3.5. Quillen’s model structure on simplicial sets 19 4. Derived functors between model categories 20 4.1. Derived functors and equivalence of homotopy theories 21 4.2. Quillen functors 24 4.3. Derived composites and derived adjunctions 25 4.4. Monoidal and enriched model categories 27 4.5. -
Model Category Theory in Homological Action
New York Journal of Mathematics New York J. Math. 20 (2014) 1077{1159. Six model structures for DG-modules over DGAs: model category theory in homological action Tobias Barthel, J.P. May and Emily Riehl The authors dedicate this paper to John Moore, who pioneered this area of mathematics. He was the senior author's adviser, and his mathematical philosophy pervades this work and indeed pervades algebraic topology at its best. Abstract. In Part 1, we describe six projective-type model structures on the category of differential graded modules over a differential graded algebra A over a commutative ring R. When R is a field, the six collapse to three and are well-known, at least to folklore, but in the general case the new relative and mixed model structures offer interesting alterna- tives to the model structures in common use. The construction of some of these model structures requires two new variants of the small object argument, an enriched and an algebraic one, and we describe these more generally. In Part 2, we present a variety of theoretical and calculational cofi- brant approximations in these model categories. The classical bar con- struction gives cofibrant approximations in the relative model structure, but generally not in the usual one. In the usual model structure, there are two quite different ways to lift cofibrant approximations from the level of homology modules over homology algebras, where they are classi- cal projective resolutions, to the level of DG-modules over DG-algebras. The new theory makes model theoretic sense of earlier explicit calcu- lations based on one of these constructions. -
Complicial Sets, an Overture
COMPLICIAL SETS, AN OVERTURE EMILY RIEHL Abstract. The aim of these notes is to introduce the intuition motivating the notion of a complicial set, a simplicial set with certain marked \thin" simplices that witness a composition relation between the simplices on their boundary. By varying the marking conventions, complicial sets can be used to model (1; n)-categories for each n ≥ 0, including n = 1. For this reason, compli- cial sets present a fertile setting for thinking about weak infinite dimensional categories in varying dimensions. This overture is presented in three acts: the first introducing simplicial models of higher categories; the second defining the Street nerve, which embeds strict !-categories as strict complicial sets; and the third exploring an important saturation condition on the marked simplices in a complicial set and presenting a variety of model structures that capture their basic homotopy theory. Scattered throughout are suggested exercises for the reader who wants to engage more deeply with these notions. Contents 1. Introducing complicial sets 3 1.1. Quasi-categories as ( ; 1)-categories 3 1.2. Towards a simplicial1 model of ( ; 2)-categories 5 1.3. Stratified simplicial sets1 6 1.4. Complicial sets 7 1.5. n-trivialization and the n-core 9 2. The Street nerve of an !-category 10 2.1. !-categories 11 2.2. Orientals 13 2.3. The Street nerve as a strict complicial set 15 3. Saturated complicial sets 16 3.1. Weak complicial sets from strict !-categories 17 3.2. Saturation 19 3.3. Model categories of complicial sets 22 arXiv:1610.06801v1 [math.CT] 21 Oct 2016 References 23 As the objects that mathematicians study increase in sophistication, so do their natural habitats. -
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. -
The 2-Category Theory of Quasi-Categories
Advances in Mathematics 280 (2015) 549–642 Contents lists available at ScienceDirect Advances in Mathematics www.elsevier.com/locate/aim The 2-category theory of quasi-categories Emily Riehl a,∗, Dominic Verity b a Department of Mathematics, Harvard University, Cambridge, MA 02138, USA b Centre of Australian Category Theory, Macquarie University, NSW 2109, Australia a r t i c l e i n f o a b s t r a c t Article history: In this paper we re-develop the foundations of the category Received 11 December 2013 theory of quasi-categories (also called ∞-categories) using Received in revised form 13 April 2-category theory. We show that Joyal’s strict 2-category of 2015 quasi-categories admits certain weak 2-limits, among them Accepted 22 April 2015 weak comma objects. We use these comma quasi-categories Available online xxxx Communicated by the Managing to encode universal properties relevant to limits, colimits, Editors of AIM and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate MSC: form as absolute lifting diagrams in the 2-category, which primary 18G55, 55U35, 55U40 we show are determined pointwise by the existence of certain secondary 18A05, 18D20, 18G30, initial or terminal vertices, allowing for the easy production 55U10 of examples. All the quasi-categorical notions introduced here are equiv- Keywords: alent to the established ones but our proofs are independent Quasi-categories and more “formal”. In particular, these results generalise 2-Category theory Formal category theory immediately to model categories enriched over quasi-cate- gories. © 2015 Elsevier Inc. -
Emily Riehl Wins the AWM - Joan & Joseph Birman Research Prize in Topology and Geometry
PRESS RELEASE April 23, 2020 Emily Riehl wins the AWM - Joan & Joseph Birman Research Prize in Topology and Geometry The Association for Women in Mathematics category theory have become the standard (AWM) will present the fourth AWM Joan & references, and her draft book on ∞-categories is Joseph Birman Research Prize in Topology and already finding immediate use by researchers. Geometry to Emily Riehl, Associate Professor of Riehl is an internationally recognized scholar for Mathematics at Johns Hopkins University at the her important research works in category theory Joint Mathematics Meetings in Washington, DC and her innovative ideas about mentorship and in January 2021. Established in 2013, the AWM communication of mathematics. Joan & Joseph Birman Research Prize recognizes exceptional research in topology and geometry by a woman early in her career. The biennial presentation of this prize serves to highlight to the community outstanding contributions by women in the fields of topology and geometry and to advance the careers of the prize recipients. The award is made possible by a generous contribution from Joan and Joseph Birman. Citation The 2021 Joan & Joseph Birman Research Prize in Topology and Geometry is awarded to Emily Riehl for her deep and foundational work in category theory and homotopy theory. Riehl has proved many fundamental theorems in category theory and its relations to homotopy theory and has produced a large body of exceptional research as well as expository and pedagogical work. Her work is transforming the ways we work with higher categorical objects, drawing on classical category-theory tools and constructions to illustrate and simplify higher Photo by Liz Flyntz categorical constructions. -
Categories for the Working Mathematician Emily
∞-Categories for the Working Mathematician Emily Riehl and Dominic Verity Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA Email address: [email protected] Centre of Australian Category Theory, Macquarie University, NSW 2109, Australia Email address: [email protected] This text is a rapidly-evolving work in progress — use at your own risk. The most recent version can always be found here: www.math.jhu.edu/∼eriehl/ICWM.pdf We would be delighted to hear about any comments, corrections, or confusions readers might have. Please send to: [email protected] Commenced on January 14, 2018. Last modified on June 15, 2018. Contents Preface vii Part I. Basic ∞-category theory 1 Chapter 1. ∞-Cosmoi and their homotopy 2-categories 3 1.1. Quasi-categories 3 1.2. ∞-Cosmoi 12 1.3. Cosmological functors 22 1.4. The homotopy 2-category 24 Chapter 2. Adjunctions, limits, and colimits I 31 2.1. Adjunctions and equivalences 31 2.2. Initial and terminal elements 37 2.3. Limits and colimits 39 2.4. Preservation of limits and colimits 45 Chapter 3. Weak 2-limits in the homotopy 2-category 49 3.1. Smothering functors 50 3.2. ∞-categories of arrows 53 3.3. The comma construction 58 3.4. Representable comma ∞-categories 64 3.5. Sliced homotopy 2-categories and fibered equivalences 73 Chapter 4. Adjunctions, limits, and colimits II 81 4.1. The universal property of adjunctions 81 4.2. ∞-categories of cones 84 4.3. The universal property of limits and colimits 87 4.4. Loops and suspension in pointed ∞-categories 97 Chapter 5. -
Homotopical Categories: from Model Categories to $(\Infty, 1) $-Categories
HOMOTOPICAL CATEGORIES: FROM MODEL CATEGORIES TO (∞,)-CATEGORIES EMILY RIEHL Abstract. This chapter, written for Stable categories and structured ring spectra, edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman’s categories of frac- tions to Quillen’s model categories, through Dwyer and Kan’s simplicial localiza- tions and culminating in (∞,)-categories, first introduced through concrete mod- els and later re-conceptualized in a model-independent framework. This reader is not presumed to have prior acquaintance with any of these concepts. Suggested exercises are included to fertilize intuitions and copious references point to exter- nal sources with more details. A running theme of homotopy limits and colimits is included to explain the kinds of problems homotopical categories are designed to solve as well as technical approaches to these problems. Contents . The history of homotopical categories . Categories of fractions and localization .. The Gabriel–Zisman category of fractions . Model category presentations of homotopical categories .. Model category structures via weak factorization systems .. On functoriality of factorizations .. The homotopy relation on arrows .. The homotopy category of a model category .. Quillen’s model structure on simplicial sets . Derived functors between model categories .. Derived functors and equivalence of homotopy theories .. Quillen functors .. Derived composites and derived adjunctions arXiv:1904.00886v3 [math.AT] 16 Jul 2020 .. Monoidal and enriched model categories .. Quillen equivalences between homotopy theories .. Extending homological algebra to homotopical algebra . Homotopy limits and colimits .. Weak limits and colimits in the homotopy category .. Homotopy limits and colimits of general shapes .. Homotopy limits and colimits of Reedy diagrams . -
Algebraic Model Structures
Algebraic model structures Emily Riehl Harvard University http://www.math.harvard.edu/~eriehl 18 September, 2011 Homotopy Theory and Its Applications AWM Anniversary Conference ICERM Emily Riehl (Harvard University) Algebraic model structures September 2011 1 / 16 Outline 1 Flavors of the theory of algebraic model structures 2 The monoidal algebraic model structure on sSet Emily Riehl (Harvard University) Algebraic model structures September 2011 2 / 16 Weak factorization systems Any model structure has two weak factorization systems(wfs): (cofibrations, trivial fibrations) (trivial cofibrations, fibrations) The left and right classes satisfy factorization and lifting axioms. Examples (retracts of rel. cell complexes, trivial Serre fibrations) on Top (anodyne extensions, Kan fibrations) on sSet (injections with projective cokernel, surjections) on A-mod (monomorphisms, epimorphisms) on Set Emily Riehl (Harvard University) Algebraic model structures September 2011 3 / 16 \Algebraic" perspective Thinking \algebraically" to characterize maps or objects satisfying a certain property, assign to each one a particular structure that demonstrates the property. Examples a surjective map in Set or A-mod admits a (set-based) section a relative cell complex admits a cellular decomposition for any Kan complex, can choose fillers for all horns (Co)algebras for (co)monads For all of these examples there is a monad or a comonad whose algebras or coalgebras have exactly this form. Emily Riehl (Harvard University) Algebraic model structures September 2011 4 / -
Categorical Homotopy Theory Emily Riehl
Categorical homotopy theory Emily Riehl To my students, colleagues, friends who inspired this work. what we are doing is finding ways for people to understand and think about mathematics. William P. Thurston “On proof and progress in mathematics” [Thu94] Contents Preface xi Prerequisites xiv Notational Conventions xiv Acknowledgments xv Part I. Derived functors and homotopy (co)limits 1 Chapter 1. All concepts are Kan extensions 3 1.1. Kan extensions 3 1.2. A formula 5 1.3. Pointwise Kan extensions 7 1.4. All concepts 9 1.5. Adjunctions involving simplicial sets 10 Chapter 2. Derived functors via deformations 13 2.1. Homotopical categories and derived functors 13 2.2. Derived functors via deformations 18 2.3. Classical derived functors between abelian categories 22 2.4. Preview of homotopy limits and colimits 23 Chapter 3. Basic concepts of enriched category theory 25 3.1. A first example 26 3.2. The base for enrichment 26 3.3. Enriched categories 27 3.4. Underlying categories of enriched categories 30 3.5. Enriched functors and enriched natural transformations 34 3.6. Simplicial categories 36 3.7. Tensors and cotensors 37 3.8. Simplicial homotopy and simplicial model categories 42 Chapter 4. The unreasonably effective (co)bar construction 45 4.1. Functor tensor products 45 4.2. The bar construction 47 4.3. The cobar construction 48 4.4. Simplicial replacements and colimits 49 4.5. Augmented simplicial objects and extra degeneracies 51 Chapter 5. Homotopy limits and colimits: the theory 55 5.1. The homotopy limit and colimit functors 55 5.2. -
A Type Theory for Synthetic $\Infty $-Categories
A TYPE THEORY FOR SYNTHETIC ∞-CATEGORIES EMILY RIEHL AND MICHAEL SHULMAN Abstract. We propose foundations for a synthetic theory of (∞, 1)-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary types. We define Segal types, in which binary compos- ites exist uniquely up to homotopy; this automatically ensures composition is coherently associative and unital at all dimensions. We define Rezk types, in which the categorical isomorphisms are additionally equivalent to the type- theoretic identities — a “local univalence” condition. And we define covariant fibrations, which are type families varying functorially over a Segal type, and prove a “dependent Yoneda lemma” that can be viewed as a directed form of the usual elimination rule for identity types. We conclude by studying ho- motopically correct adjunctions between Segal types, and showing that for a functor between Rezk types to have an adjoint is a mere proposition. To make the bookkeeping in such proofs manageable, we use a three-layered type theory with shapes, whose contexts are extended by polytopes within directed cubes, which can be abstracted over using “extension types” that generalize the path-types of cubical type theory. In an appendix, we describe the motivating semantics in the Reedy model structure on bisimplicial sets, in which our Segal and Rezk types correspond to Segal spaces and complete Segal spaces. Contents 1. Introduction 2 2. Type theory with shapes 7 3. Simplicial type theory 12 4. Equivalences involving extension types 19 5. Segal types 23 6.