Derived Categories and Their Uses
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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. -
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. -
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. -
Derived Categories of Modules and Coherent Sheaves
Derived Categories of Modules and Coherent Sheaves Yuriy A. Drozd Abstract We present recent results on derived categories of modules and coherent sheaves, namely, tame{wild dichotomy and semi-continuity theorem for derived categories over finite dimensional algebras, as well as explicit calculations for derived categories of modules over nodal rings and of coherent sheaves over projective configurations of types A and A~. This paper is a survey of some recent results on the structure of derived categories obtained by the author in collaboration with Viktor Bekkert and Igor Burban [6, 11, 12]. The origin of this research was the study of Cohen{ Macaulay modules and vector bundles by Gert-Martin Greuel and myself [27, 28, 29, 30] and some ideas from the work of Huisgen-Zimmermann and Saor´ın [42]. Namely, I understood that the technique of \matrix problems," briefly explained below in subsection 2.3, could be successfully applied to the calculations in derived categories, almost in the same way as it was used in the representation theory of finite-dimensional algebras, in study of Cohen{ Macaulay modules, etc. The first step in this direction was the semi-continuity theorem for derived categories [26] presented in subsection 2.1. Then Bekkert and I proved the tame{wild dichotomy for derived categories over finite di- mensional algebras (see subsection 2.2). At the same time, Burban and I described the indecomposable objects in the derived categories over nodal rings (see Section 3) and projective configurations of types A and A~ (see Sec- tion 4). Note that it follows from [23, 29] that these are the only cases, where such a classification is possible; for all other pure noetherian rings (or projec- tive curves) even the categories of modules (respectively, of vector bundles) are wild. -
Agnieszka Bodzenta
June 12, 2019 HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY AGNIESZKA BODZENTA Contents 1. Categories, functors, natural transformations 2 1.1. Direct product, coproduct, fiber and cofiber product 4 1.2. Adjoint functors 5 1.3. Limits and colimits 5 1.4. Localisation in categories 5 2. Abelian categories 8 2.1. Additive and abelian categories 8 2.2. The category of modules over a quiver 9 2.3. Cohomology of a complex 9 2.4. Left and right exact functors 10 2.5. The category of sheaves 10 2.6. The long exact sequence of Ext-groups 11 2.7. Exact categories 13 2.8. Serre subcategory and quotient 14 3. Triangulated categories 16 3.1. Stable category of an exact category with enough injectives 16 3.2. Triangulated categories 22 3.3. Localization of triangulated categories 25 3.4. Derived category as a quotient by acyclic complexes 28 4. t-structures 30 4.1. The motivating example 30 4.2. Definition and first properties 34 4.3. Semi-orthogonal decompositions and recollements 40 4.4. Gluing of t-structures 42 4.5. Intermediate extension 43 5. Perverse sheaves 44 5.1. Derived functors 44 5.2. The six functors formalism 46 5.3. Recollement for a closed subset 50 1 2 AGNIESZKA BODZENTA 5.4. Perverse sheaves 52 5.5. Gluing of perverse sheaves 56 5.6. Perverse sheaves on hyperplane arrangements 59 6. Derived categories of coherent sheaves 60 6.1. Crash course on spectral sequences 60 6.2. Preliminaries 61 6.3. Hom and Hom 64 6.4. -
Intersection Homology Duality and Pairings: Singular, PL, and Sheaf
Intersection homology duality and pairings: singular, PL, and sheaf-theoretic Greg Friedman and James E. McClure July 22, 2020 Contents 1 Introduction 2 2 Conventions and some background 11 2.1 Pseudomanifolds and intersection homology . .. 11 3 Hypercohomology in the derived category 13 4 Some particular sheaf complexes 16 4.1 The Verdier dualizing complex and Verdier duality . 16 4.2 TheDelignesheaf................................. 17 4.3 The sheaf complex of intersection chains . .. 19 4.4 Thesheafofintersectioncochains . .. 20 ∗ 4.4.1 Ip¯C is quasi-isomorphic to PDp¯ ..................... 22 5 Sheafification of the cup product 24 5.1 Applications: A multiplicative de Rham theorem for perverse differential forms and compatibility with the blown-up cup product . 26 arXiv:1812.10585v3 [math.GT] 22 Jul 2020 5.1.1 Perversedifferentialforms . 26 5.1.2 Blown-up intersection cohomology . 31 6 Classical duality via sheaf maps 32 6.1 Orientationsandcanonicalproducts. ... 33 6.2 ProofofTheorem6.1............................... 35 7 Compatibility of cup, intersection, and sheaf products 50 8 Classical duality and Verdier duality 55 A Co-cohomology 59 1 B A meditation on signs 60 2000 Mathematics Subject Classification: Primary: 55N33, 55N45 Secondary: 55N30, 57Q99 Keywords: intersection homology, intersection cohomology, pseudomanifold, cup product, cap product, intersection product, Poincar´e duality, Verdier dual- ity, sheaf theory Abstract We compare the sheaf-theoretic and singular chain versions of Poincar´eduality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the sin- gular intersection cohomology cup product, the hypercohomology product induced by the Goresky-MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky- MacPherson PL intersection product. -
Arxiv:1910.05574V3 [Math.AT] 21 Jun 2021 2.7
REPRESENTATION STABILITY, SECONDARY STABILITY, AND POLYNOMIAL FUNCTORS JEREMY MILLER, PETER PATZT, AND DAN PETERSEN Abstract. We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two generalizations of classical homological stability theorems with twisted coefficients. We apply our results to prove homological stability for hyperelliptic mapping class groups with twisted coefficients, prove new representation stability results for congruence subgroups, establish secondary homological stability for groups of diffeomorphisms of surfaces viewed as discrete groups, and improve the known stable range for homological stability for general linear groups of the sphere spectrum. Contents 1. Introduction 2 1.1. Homological stability with polynomial coefficients2 1.2. Representation stability with polynomial coefficients3 1.3. Secondary homological stability with polynomial coefficients4 1.4. Stability for polynomial coefficients5 1.5. Applications 6 1.6. Outline of the paper9 1.7. Acknowledgments9 2. Categorical and algebraic preliminaries9 2.1. Tensor products as coends9 2.2. Stability categories 10 2.3. Central stability homology and degree-wise coherence 12 2.4. Stability short exact sequences 15 2.5. Rings, modules, and Tor groups 15 2.6. Splitting complexes and the Koszul complex 18 arXiv:1910.05574v3 [math.AT] 21 Jun 2021 2.7. Polynomial coefficient systems 21 3. Polynomial modules and derived representation stability 23 3.1. Via central stability complexes 23 3.2. Via the Koszul resolution 28 Date: June 22, 2021. Jeremy Miller was supported in part by NSF grant DMS-1709726 and a Simons Foundation Collaboration Grant. -
Abstract Homotopy Theory, Generalized Sheaf Cohomology, Homotopical Algebra, Sheaf of Spectra, Homotopy Category, Derived Functor
TRANSACTIONSOF THE AMERICANMATHEMATICAL SOCIETY Volume 186, December 1973 ABSTRACTHOMOTOPY THEORY AND GENERALIZED SHEAF COHOMOLOGY BY KENNETHS. BROWN0) ABSTRACT. Cohomology groups Ha(X, E) are defined, where X is a topological space and £ is a sheaf on X with values in Kan's category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defined by means of the "homotopical algebra" of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen's, and this is developed in Part I of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence which generalizes the Atiyah-Hirzebruch spectral sequence (in gen- eralized cohomology theory) and the "local to global" spectral sequence (in sheaf cohomology theory). Introduction. In this paper we will study the homotopy theory of sheaves of simplicial sets and sheaves of spectra. This homotopy theory will be used to give a derived functor definition of generalized sheaf cohomology groups H^iX, E), where X is a topological space and E is a sheaf of spectra on X, subject to certain finiteness conditions. These groups include as special cases the usual generalized cohomology of X defined by a spectrum [22] and the cohomology of X with coefficients in a complex of abelian sheaves. The cohomology groups have all the properties one would expect, the most important one being a spectral sequence Ep2q= HpiX, n_ E) =■»Hp+*iX, E), which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized coho- mology) and the "local to global" spectral sequence (in sheaf cohomology), and Received by the editors June 8, 1972 and, in revised form, October 15, 1972.