Derived Categories and Their Uses
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COVERING THEORY for LINEAR CATEGORIES with APPLICATION to DERIVED CATEGORIES Introduction the Covering Technique Has Been Playin
COVERING THEORY FOR LINEAR CATEGORIES WITH APPLICATION TO DERIVED CATEGORIES RAYMUNDO BAUTISTA AND SHIPING LIU Abstract. We extend the Galois covering theory introduced by Bongartz- Gabriel for skeletal linear categories to general linear categories. We show that a Galois covering between Krull-Schmidt categories preserves irreducible morphisms and almost splits sequences. Specializing to derived categories, we study when a Galois covering between locally bounded linear categories induces a Galois covering between the bounded derived categories of finite dimensional modules. As an application, we show that each locally bounded linear category with radical squared zero admits a gradable Galois covering, which induces a Galois covering between the bounded derived categories of finite dimensional modules, and a Galois covering between the Auslander-Reiten quivers of these bounded derived categories. In a future paper, this will enable us to obtain a complete description of the bounded derived category of finite dimensional modules over a finite dimensional algebra with radical squared zero. Introduction The covering technique has been playing an important role in the representation theory of finite dimensional algebras; see, for example, [6, 8, 9, 16]. In this connec- tion, algebras are regarded as locally bounded linear categories; see [6]. To each Galois covering between such categories, Bongartz-Gabriel associated a push-down functor between their module categories, which induces a Galois covering between the Auslander-Reiten quivers in the locally representation-finite case; see [6, 8]. This technique was extended later by Asashiba by studying the induced push-down functor between the bounded homotopy categories of finitely generated projective modules; see [1]. -
Perverse Sheaves
Perverse Sheaves Bhargav Bhatt Fall 2015 1 September 8, 2015 The goal of this class is to introduce perverse sheaves, and how to work with it; plus some applications. Background For more background, see Kleiman's paper entitled \The development/history of intersection homology theory". On manifolds, the idea is that you can intersect cycles via Poincar´eduality|we want to be able to do this on singular spces, not just manifolds. Deligne figured out how to compute intersection homology via sheaf cohomology, and does not use anything about cycles|only pullbacks and truncations of complexes of sheaves. In any derived category you can do this|even in characteristic p. The basic summary is that we define an abelian subcategory that lives inside the derived category of constructible sheaves, which we call the category of perverse sheaves. We want to get to what is called the decomposition theorem. Outline of Course 1. Derived categories, t-structures 2. Six Functors 3. Perverse sheaves—definition, some properties 4. Statement of decomposition theorem|\yoga of weights" 5. Application 1: Beilinson, et al., \there are enough perverse sheaves", they generate the derived category of constructible sheaves 6. Application 2: Radon transforms. Use to understand monodromy of hyperplane sections. 7. Some geometric ideas to prove the decomposition theorem. If you want to understand everything in the course you need a lot of background. We will assume Hartshorne- level algebraic geometry. We also need constructible sheaves|look at Sheaves in Topology. Problem sets will be given, but not collected; will be on the webpage. There are more references than BBD; they will be online. -
The Simplicial Parallel of Sheaf Theory Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 10, No 4 (1968), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES YUH-CHING CHEN Costacks - The simplicial parallel of sheaf theory Cahiers de topologie et géométrie différentielle catégoriques, tome 10, no 4 (1968), p. 449-473 <http://www.numdam.org/item?id=CTGDC_1968__10_4_449_0> © Andrée C. Ehresmann et les auteurs, 1968, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE COSTACKS - THE SIMPLICIAL PARALLEL OF SHEAF THEORY by YUH-CHING CHEN I NTRODUCTION Parallel to sheaf theory, costack theory is concerned with the study of the homology theory of simplicial sets with general coefficient systems. A coefficient system on a simplicial set K with values in an abe - . lian category 8 is a functor from K to Q (K is a category of simplexes) ; it is called a precostack; it is a simplicial parallel of the notion of a presheaf on a topological space X . A costack on K is a « normalized» precostack, as a set over a it is realized simplicial K/" it is simplicial « espace 8ta18 ». The theory developed here is functorial; it implies that all >> homo- logy theories are derived functors. Although the treatment is completely in- dependent of Topology, it is however almost completely parallel to the usual sheaf theory, and many of the same theorems will be found in it though the proofs are usually quite different. -
Infinity-Tilting Theory
∞-TILTING THEORY LEONID POSITSELSKI AND JAN SˇTOVˇ ´ICEKˇ Abstract. We define the notion of an infinitely generated tilting object of infi- nite homological dimension in an abelian category. A one-to-one correspondence between ∞-tilting objects in complete, cocomplete abelian categories with an injec- tive cogenerator and ∞-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce ∞-tilting pairs, con- sisting of an ∞-tilting object and its ∞-tilting class, and obtain a bijective corre- spondence between ∞-tilting and ∞-cotilting pairs. Finally, we discuss the related derived equivalences and t-structures. Contents Introduction 1 1. The Tilted and Cotilted Abelian Categories 4 2. ∞-Tilting-Cotilting Correspondence 7 3. ∞-Tilting and ∞-Cotilting Pairs 13 4. ∞-Tilting-Cotilting Derived Equivalences 17 5. ∞-Tilting and ∞-Cotilting t-Structures 20 6. Examples 25 References 32 Introduction The phrase ‘tilting theory’ is often used to refer to a well-developed general ma- chinery for producing equivalences between triangulated categories (see [3] for an introduction, history and applications). Such equivalences are often represented by a arXiv:1711.06169v3 [math.CT] 13 Aug 2019 distinguished object, a so-called tilting object, and it is crucial to most of the theory that such a tilting object is homologically small. If A is an abelian category with exact coproducts (e.g. a category of modules over a ring or sheaves on a topological space) and T is a tilting object, the smallness typically translates at least to the assumptions that T is finitely generated and of finite projective dimension. In this paper we introduce and systematically develop ∞-tilting theory, where all homological smallness assumptions are dropped. -
I. Overview of Activities, April, 2005-March, 2006 …
MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 …......……………………. 2 Innovations ………………………………………………………..... 2 Scientific Highlights …..…………………………………………… 4 MSRI Experiences ….……………………………………………… 6 II. Programs …………………………………………………………………….. 13 III. Workshops ……………………………………………………………………. 17 IV. Postdoctoral Fellows …………………………………………………………. 19 Papers by Postdoctoral Fellows …………………………………… 21 V. Mathematics Education and Awareness …...………………………………. 23 VI. Industrial Participation ...…………………………………………………… 26 VII. Future Programs …………………………………………………………….. 28 VIII. Collaborations ………………………………………………………………… 30 IX. Papers Reported by Members ………………………………………………. 35 X. Appendix - Final Reports ……………………………………………………. 45 Programs Workshops Summer Graduate Workshops MSRI Network Conferences MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 This annual report covers MSRI projects and activities that have been concluded since the submission of the last report in May, 2005. This includes the Spring, 2005 semester programs, the 2005 summer graduate workshops, the Fall, 2005 programs and the January and February workshops of Spring, 2006. This report does not contain fiscal or demographic data. Those data will be submitted in the Fall, 2006 final report covering the completed fiscal 2006 year, based on audited financial reports. This report begins with a discussion of MSRI innovations undertaken this year, followed by highlights -
Elements of -Category Theory Joint with Dominic Verity
Emily Riehl Johns Hopkins University Elements of ∞-Category Theory joint with Dominic Verity MATRIX seminar ∞-categories in the wild A recent phenomenon in certain areas of mathematics is the use of ∞-categories to state and prove theorems: • 푛-jets correspond to 푛-excisive functors in the Goodwillie tangent structure on the ∞-category of differentiable ∞-categories — Bauer–Burke–Ching, “Tangent ∞-categories and Goodwillie calculus” • 푆1-equivariant quasicoherent sheaves on the loop space of a smooth scheme correspond to sheaves with a flat connection as an equivalence of ∞-categories — Ben-Zvi–Nadler, “Loop spaces and connections” • the factorization homology of an 푛-cobordism with coefficients in an 푛-disk algebra is linearly dual to the factorization homology valued in the formal moduli functor as a natural equivalence between functors between ∞-categories — Ayala–Francis, “Poincaré/Koszul duality” Here “∞-category” is a nickname for (∞, 1)-category, a special case of an (∞, 푛)-category, a weak infinite dimensional category in which all morphisms above dimension 푛 are invertible (for fixed 0 ≤ 푛 ≤ ∞). What are ∞-categories and what are they for? It frames a possible template for any mathematical theory: the theory should have nouns and verbs, i.e., objects, and morphisms, and there should be an explicit notion of composition related to the morphisms; the theory should, in brief, be packaged by a category. —Barry Mazur, “When is one thing equal to some other thing?” An ∞-category frames a template with nouns, verbs, adjectives, adverbs, pronouns, prepositions, conjunctions, interjections,… which has: • objects • and 1-morphisms between them • • • • composition witnessed by invertible 2-morphisms 푓 푔 훼≃ ℎ∘푔∘푓 • • • • 푔∘푓 푓 ℎ∘푔 훾≃ witnessed by • associativity • ≃ 푔∘푓 훼≃ 훽 ℎ invertible 3-morphisms 푔 • with these witnesses coherent up to invertible morphisms all the way up. -
TILTING up ITERATED TILTED ALGEBRAS 1. Introduction Throughout This Paper, We Let K Denote a Fixed Algebraically Closed Field. B
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 8, Pages 2223{2232 S 0002-9939(99)05230-2 Article electronically published on November 29, 1999 TILTING UP ITERATED TILTED ALGEBRAS IBRAHIM ASSEM, DIETER HAPPEL, AND SONIA TREPODE (Communicated by Ken Goodearl) Abstract. We show that, if A is a representation-finite iterated tilted algebra of euclidean type Q, then there exist a sequence of algebras A = A0;A1;A2;:::; (i) (i) Am, and a sequence of modules T ,where0≤ i<m,suchthateachT is Ai Ai (i) an APR-tilting Ai-module, or an APR-cotilting Ai-module, End T = Ai+1 Ai and Am is tilted representation-finite. 1. Introduction Throughout this paper, we let k denote a fixed algebraically closed field. By algebra is always meant a finite dimensional associative k-algebra with an identity, which we assume moreover to be basic and connected, and by module is meant a finitely generated right A-module. Tilting theory is by now an established tool in the representation theory of algebras. In particular, the classes of tilted and iterated tilted algebras, introduced by means of the tilting process, were very useful, for instance, in the classification of the self-injective algebras of polynomial growth (see [1], [7], [11]). We recall their definition. Let Q be a finite, connected quiver without oriented cycles and let kQ denote the path algebra of Q.AnalgebraA is said to be iterated tilted of type Q if there exists a going-up tilting series from A to kQ, that is, a sequence of algebras A = A0;A1;:::;Am = kQ and a sequence of tilting modules T (i),where0≤ i<m, such that End T (i) = A and each T (i) Ai Ai i+1 Ai is separating, that is, such that each indecomposable Ai-module M satisfies either (i) 1 (i) HomAi (T ;M)=0orExtAi (T ;M) = 0 (see [2]). -
Generalized Tilting Theory in Functor Categories
Table of content Motivation and Introduction Main results Applications Generalized tilting theory in functor categories Xi Tang April 25, 2019 logo Table of content Motivation and Introduction Main results Applications Table of content 1 Motivation and Introduction 2 Main results Equivalences induced by T A cotorsion pair 3 Applications An isomorphism of Grothendieck groups An abelian model structure A t-structure induced by T logo Table of content Motivation and Introduction Main results Applications References S. Bazzoni, The t-structure induced by an n- tilting module, Trans. Amer. Math. Soc. (to ap- pear). R. Mart´ınez-Villa and M. Ortiz-Morales, Tilting theory and functor categories I. Classical tilt- ing, Appl. Categ. Struct. 22 (2014), 595–646. R. Mart´ınez-Villa and M. Ortiz-Morales, Tilting theory and functor categories II. Generalized tilting, Appl. Categ. Struct. 21 (2013), 311– logo 348. Table of content Motivation and Introduction Main results Applications Motivation K algebraically closed field Λ finite dimensional K-algebra TΛ tilting module Γ := End(T)op Brenner-Butler Tilting Theorem The following statements hold. (1) (T (T); F(T)) is a torsion theory, where 1 T (T) := fM 2 mod Λ j ExtΛ(T; M) = 0g; logo F(T) := fM 2 mod Λ j HomΛ(T; M) = 0g: Table of content Motivation and Introduction Main results Applications Motivation (2) (X (T); Y(T)) is a torsion theory, where X (T) := fN 2 mod Γ j N ⊗Γ T = 0g; Γ Y(T) := fN 2 mod Γ j Tor1 (N; T) = 0g. (3) There are two category equivalences: T (T) F(T) g 7 s s w ' X (T) Y(T): logo Table of content Motivation and Introduction Main results Applications R associative ring TR n-tilting module S := End(T)op Miyashita Theorem There are category equivalences: e ExtR(T;−) n / n KE (TR) KT (ST); where e o s e S Tore (−;T) n i KEe (TR) := fM j ExtR(T; M) = 0; 0 6 i 6= e 6 ng; n S logo KTe (ST) := fN j Tori (N; T) = 0; 0 6 i 6= e 6 ng. -
THE DERIVED CATEGORY of a GIT QUOTIENT Contents 1. Introduction 871 1.1. Author's Note 874 1.2. Notation 874 2. the Main Theor
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 3, July 2015, Pages 871–912 S 0894-0347(2014)00815-8 Article electronically published on October 31, 2014 THE DERIVED CATEGORY OF A GIT QUOTIENT DANIEL HALPERN-LEISTNER Dedicated to Ernst Halpern, who inspired my scientific pursuits Contents 1. Introduction 871 1.1. Author’s note 874 1.2. Notation 874 2. The main theorem 875 2.1. Equivariant stratifications in GIT 875 2.2. Statement and proof of the main theorem 878 2.3. Explicit constructions of the splitting and integral kernels 882 3. Homological structures on the unstable strata 883 3.1. Quasicoherent sheaves on S 884 3.2. The cotangent complex and Property (L+) 891 3.3. Koszul systems and cohomology with supports 893 3.4. Quasicoherent sheaves with support on S, and the quantization theorem 894 b 3.5. Alternative characterizations of D (X)<w 896 3.6. Semiorthogonal decomposition of Db(X) 898 4. Derived equivalences and variation of GIT 901 4.1. General variation of the GIT quotient 903 5. Applications to complete intersections: Matrix factorizations and hyperk¨ahler reductions 906 5.1. A criterion for Property (L+) and nonabelian hyperk¨ahler reduction 906 5.2. Applications to derived categories of singularities and abelian hyperk¨ahler reductions, via Morita theory 909 References 911 1. Introduction We describe a relationship between the derived category of equivariant coherent sheaves on a smooth projective-over-affine variety, X, with a linearizable action of a reductive group, G, and the derived category of coherent sheaves on a GIT quotient, X//G, of that action. -
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. -
Derived Functors of /-Adic Completion and Local Homology
JOURNAL OF ALGEBRA 149, 438453 (1992) Derived Functors of /-adic Completion and Local Homology J. P. C. GREENLEES AND J. P. MAY Department q/ Mathematics, Unil;ersi/y of Chicago, Chicago, Illinois 60637 Communicated by Richard G. Swan Received June 1. 1990 In recent topological work [2], we were forced to consider the left derived functors of the I-adic completion functor, where I is a finitely generated ideal in a commutative ring A. While our concern in [2] was with a particular class of rings, namely the Burnside rings A(G) of compact Lie groups G, much of the foundational work we needed was not restricted to this special case. The essential point is that the modules we consider in [2] need not be finitely generated and, unless G is finite, say, the ring ,4(G) is not Noetherian. There seems to be remarkably little information in the literature about the behavior of I-adic completion in this generality. We presume that interesting non-Noetherian commutative rings and interesting non-finitely generated modules arise in subjects other than topology. We have therefore chosen to present our algebraic work separately, in the hope that it may be of value to mathematicians working in other fields. One consequence of our study, explained in Section 1, is that I-adic com- pletion is exact on a much larger class of modules than might be expected from the key role played by the Artin-Rees lemma and that the deviations from exactness can be computed in terms of torsion products. However, the most interesting consequence, discussed in Section 2, is that the left derived functors of I-adic completion usually can be computed in terms of certain local homology groups, which are defined in a fashion dual to the definition of the classical local cohomology groups of Grothendieck. -
Poisson Geometry, Deformation Quantisation and Group Representations
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hitchin, Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford OX1 3LB, United Kingdom The title below are available from booksellers, or from Cambridge University Press at www.cambridge.org 161 Lectures on block theory, BURKHARD KULSHAMMER¨ 163 Topics in varieties of group representations, S.M. VOVSI 164 Quasi-symmetric designs, M.S. SHRlKANDE & S.S. SANE 166 Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) 168 Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) 169 Boolean function complexity, M.S. PATERSON (ed) 170 Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK 171 Squares, A.R. RAJWADE 172 Algebraic varieties, GEORGE R. KEMPF 173 Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) 174 Lectures on mechanics, J.E. MARSDEN 175 Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) 176 Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) 177 Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE & A. PITTS (eds) 178 Lower K- and L-theory, A. RANlCKl 179 Complex projective geometry, G. ELLlNGSRUD et al 180 Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT 181 Geometric group theory I, G.A. NlBLO & M.A. ROLLER (eds) 182 Geometric group theory II, G.A. NlBLO & M.A. ROLLER (eds) 183 Shintani Zeta Functions, A. YUKlE 184 Arithmetical functions, W. SCHWARZ & J. SPlLKER 185 Representations of solvable groups. O. MANZ & T.R. WOLF 186 Complexity: knots, colourings and counting, D.J.A.