Geometricity for Derived Categories of Algebraic Stacks 3
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Categorical Enhancements of Triangulated Categories
On the uniqueness of ∞-categorical enhancements of triangulated categories Benjamin Antieau March 19, 2021 Abstract We study the problem of when triangulated categories admit unique ∞-categorical en- hancements. Our results use Lurie’s theory of prestable ∞-categories to give conceptual proofs of, and in many cases strengthen, previous work on the subject by Lunts–Orlov and Canonaco–Stellari. We also give a wide range of examples involving quasi-coherent sheaves, categories of almost modules, and local cohomology to illustrate the theory of prestable ∞-categories. Finally, we propose a theory of stable n-categories which would interpolate between triangulated categories and stable ∞-categories. Key Words. Triangulated categories, prestable ∞-categories, Grothendieck abelian categories, additive categories, quasi-coherent sheaves. Mathematics Subject Classification 2010. 14A30, 14F08, 18E05, 18E10, 18G80. Contents 1 Introduction 2 2 ∞-categorical enhancements 8 3 Prestable ∞-categories 12 arXiv:1812.01526v3 [math.AG] 18 Mar 2021 4 Bounded above enhancements 14 5 A detection lemma 15 6 Proofs 16 7 Discussion of the meta theorem 21 8 (Counter)examples, questions, and conjectures 23 8.1 Completenessandproducts . 23 8.2 Quasi-coherentsheaves. .... 27 8.3 Thesingularitycategory . .... 29 8.4 Stable n-categories ................................. 30 1 2 1. Introduction 8.5 Enhancements and t-structures .......................... 33 8.6 Categorytheoryquestions . .... 34 A Appendix: removing presentability 35 1 Introduction This paper is a study of the question of when triangulated categories admit unique ∞- categorical enhancements. Our emphasis is on exploring to what extent the proofs can be made to rely only on universal properties. That this is possible is due to J. Lurie’s theory of prestable ∞-categories. -
Representation Stability, Configuration Spaces, and Deligne
Representation Stability, Configurations Spaces, and Deligne–Mumford Compactifications by Philip Tosteson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2019 Doctoral Committee: Professor Andrew Snowden, Chair Professor Karen Smith Professor David Speyer Assistant Professor Jennifer Wilson Associate Professor Venky Nagar Philip Tosteson [email protected] orcid.org/0000-0002-8213-7857 © Philip Tosteson 2019 Dedication To Pete Angelos. ii Acknowledgments First and foremost, thanks to Andrew Snowden, for his help and mathematical guidance. Also thanks to my committee members Karen Smith, David Speyer, Jenny Wilson, and Venky Nagar. Thanks to Alyssa Kody, for the support she has given me throughout the past 4 years of graduate school. Thanks also to my family for encouraging me to pursue a PhD, even if it is outside of statistics. I would like to thank John Wiltshire-Gordon and Daniel Barter, whose conversations in the math common room are what got me involved in representation stability. John’s suggestions and point of view have influenced much of the work here. Daniel’s talk of Braids, TQFT’s, and higher categories has helped to expand my mathematical horizons. Thanks also to many other people who have helped me learn over the years, including, but not limited to Chris Fraser, Trevor Hyde, Jeremy Miller, Nir Gadish, Dan Petersen, Steven Sam, Bhargav Bhatt, Montek Gill. iii Table of Contents Dedication . ii Acknowledgements . iii Abstract . .v Chapters v 1 Introduction 1 1.1 Representation Stability . .1 1.2 Main Results . .2 1.2.1 Configuration spaces of non-Manifolds . -
The Nori Fundamental Gerbe of Tame Stacks 3
THE NORI FUNDAMENTAL GERBE OF TAME STACKS INDRANIL BISWAS AND NIELS BORNE Abstract. Given an algebraic stack, we compare its Nori fundamental group with that of its coarse moduli space. We also study conditions under which the stack can be uniformized by an algebraic space. 1. Introduction The aim here is to show that the results of [Noo04] concerning the ´etale fundamental group of algebraic stacks also hold for the Nori fundamental group. Let us start by recalling Noohi’s approach. Given a connected algebraic stack X, and a geometric point x : Spec Ω −→ X, Noohi generalizes the definition of Grothendieck’s ´etale fundamental group to get a profinite group π1(X, x) which classifies finite ´etale representable morphisms (coverings) to X. He then highlights a new feature specific to the stacky situation: for each geometric point x, there is a morphism ωx : Aut x −→ π1(X, x). Noohi first studies the situation where X admits a moduli space Y , and proceeds to show that if N is the closed normal subgroup of π1(X, x) generated by the images of ωx, for x varying in all geometric points, then π1(X, x) ≃ π1(Y,y) . N Noohi turns next to the issue of uniformizing algebraic stacks: he defines a Noetherian algebraic stack X as uniformizable if it admits a covering, in the above sense, that is an algebraic space. His main result is that this happens precisely when X is a Deligne– Mumford stack and for any geometric point x, the morphism ωx is injective. For our purpose, it turns out to be more convenient to use the Nori fundamental gerbe arXiv:1502.07023v3 [math.AG] 9 Sep 2015 defined in [BV12]. -
CHARACTERIZING ALGEBRAIC STACKS 1. Introduction Stacks
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 4, April 2008, Pages 1465–1476 S 0002-9939(07)08832-6 Article electronically published on December 6, 2007 CHARACTERIZING ALGEBRAIC STACKS SHARON HOLLANDER (Communicated by Paul Goerss) Abstract. We extend the notion of algebraic stack to an arbitrary subcanon- ical site C. If the topology on C is local on the target and satisfies descent for morphisms, we show that algebraic stacks are precisely those which are weakly equivalent to representable presheaves of groupoids whose domain map is a cover. This leads naturally to a definition of algebraic n-stacks. We also compare different sites naturally associated to a stack. 1. Introduction Stacks arise naturally in the study of moduli problems in geometry. They were in- troduced by Giraud [Gi] and Grothendieck and were used by Deligne and Mumford [DM] to study the moduli spaces of curves. They have recently become important also in differential geometry [Bry] and homotopy theory [G]. Higher-order general- izations of stacks are also receiving much attention from algebraic geometers and homotopy theorists. In this paper, we continue the study of stacks from the point of view of homotopy theory started in [H, H2]. The aim of these papers is to show that many properties of stacks and classical constructions with stacks are homotopy theoretic in nature. This homotopy-theoretical understanding gives rise to a simpler and more powerful general theory. In [H] we introduced model category structures on different am- bient categories in which stacks are the fibrant objects and showed that they are all Quillen equivalent. -
The Orbifold Chow Ring of Toric Deligne-Mumford Stacks
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 18, Number 1, Pages 193–215 S 0894-0347(04)00471-0 Article electronically published on November 3, 2004 THE ORBIFOLD CHOW RING OF TORIC DELIGNE-MUMFORD STACKS LEVA.BORISOV,LINDACHEN,ANDGREGORYG.SMITH 1. Introduction The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring in- troduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge num- bers, of the underlying variety. The product structure is induced by the degree zero part of the quantum product; in particular, it involves Gromov-Witten invariants. Inspired by string theory and results in Batyrev [3] and Yasuda [28], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and G¨ottsche [14] and Uribe [25] verify this conjecture when the orbifold is Symn(S)whereS is a smooth projective surface n with KS = 0 and the resolution is Hilb (S). The initial motivation for this project was to compare the orbifold Chow ring of a simplicial toric variety with the Chow ring of a crepant resolution. To achieve this goal, we first develop the theory of toric Deligne-Mumford stacks. Modeled on simplicial toric varieties, a toric Deligne-Mumford stack corresponds to a combinatorial object called a stacky fan. As a first approximation, this object is a simplicial fan with a distinguished lattice point on each ray in the fan. -
Tensor Triangular Geometry
Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010 Tensor triangular geometry Paul Balmer ∗ Abstract. We survey tensor triangular geometry : Its examples, early theory and first applications. We also discuss perspectives and suggest some problems. Mathematics Subject Classification (2000). Primary 18E30; Secondary 14F05, 19G12, 19K35, 20C20, 53D37, 55P42. Keywords. Tensor triangulated categories, spectra. Contents 1 Tensor triangulated categories in nature3 2 Abstract tensor triangular geometry6 3 Examples and applications 16 4 Problems 21 Introduction Tensor triangular geometry is the study of tensor triangulated categories by algebro- geometric methods. We invite the reader to discover this relatively new subject. A great charm of this theory is the profusion of examples to be found throughout pure mathematics, be it in algebraic geometry, stable homotopy theory, modular representation theory, motivic theory, noncommutative topology, or symplectic geometry, to mention some of the most popular. We review them in Section1. Here is an early photograph of tensor triangular geometry, in the crib : b Tensor triangulated i g d o l 2 categories k r o 6 O g v y } Algebraic Stable Modular Motivic Noncomm: Symplectic geometry homot: th: repres: th: theory topology geometry ∗Research supported by NSF grant 0654397. 2 Paul Balmer Before climbing into vertiginous abstraction, it is legitimate to enquire about the presence of oxygen in the higher spheres. For instance, some readers might wonder whether tensor triangulated categories do not lose too much information about the more concrete mathematical objects to which they are associated. Our first answer is Theorem 54 below, which asserts that a scheme can be reconstructed from the associated tensor triangulated category, whereas a well-known result of Mukai excludes such reconstruction from the triangular structure alone. -
Local Cohomology and Support for Triangulated Categories
LOCAL COHOMOLOGY AND SUPPORT FOR TRIANGULATED CATEGORIES DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE To Lucho Avramov, on his 60th birthday. Abstract. We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small co- products. This approach is based on a construction of local cohomology func- tors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buch- weitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support differ. In the case of group representations, this leads to a counterexample to a conjecture of Benson. Resum´ e.´ Nous proposons une fa¸con nouvelle de d´efinirune notion de support pour les objets d’une cat´egorieavec petits coproduit, engendr´eepar des objets compacts. Cette approche est bas´eesur une construction des foncteurs de co- homologie locale sur les cat´egoriestriangul´eesrelativement `aun anneau central d’op´erateurs.Comme cas particuliers, on retrouve la th´eoriepour les anneaux noeth´eriensde Foxby et Neeman, la th´eoried’Avramov et Buchweitz pour les anneaux locaux d’intersection compl`ete,ou les vari´et´espour les repr´esentations des groupes finis selon Benson, Carlson et Rickard. Nous donnons des exem- ples explicites d’objets dont le support triangul´eet le support cohomologique sont diff´erents. Dans le cas des repr´esentations des groupes, ceci nous permet de corriger et d’´etablirune conjecture de Benson. -
Handbook of Moduli
Equivariant geometry and the cohomology of the moduli space of curves Dan Edidin Abstract. In this chapter we give a categorical definition of the integral cohomology ring of a stack. For quotient stacks [X=G] the categorical co- ∗ homology ring may be identified with the equivariant cohomology HG(X). Identifying the stack cohomology ring with equivariant cohomology allows us to prove that the cohomology ring of a quotient Deligne-Mumford stack is rationally isomorphic to the cohomology ring of its coarse moduli space. The theory is presented with a focus on the stacks Mg and Mg of smooth and stable curves respectively. Contents 1 Introduction2 2 Categories fibred in groupoids (CFGs)3 2.1 CFGs of curves4 2.2 Representable CFGs5 2.3 Curves and quotient CFGs6 2.4 Fiber products of CFGs and universal curves 10 3 Cohomology of CFGs and equivariant cohomology 13 3.1 Motivation and definition 14 3.2 Quotient CFGs and equivariant cohomology 16 3.3 Equivariant Chow groups 19 3.4 Equivariant cohomology and the CFG of curves 21 4 Stacks, moduli spaces and cohomology 22 4.1 Deligne-Mumford stacks 23 4.2 Coarse moduli spaces of Deligne-Mumford stacks 27 4.3 Cohomology of Deligne-Mumford stacks and their moduli spaces 29 2000 Mathematics Subject Classification. Primary 14D23, 14H10; Secondary 14C15, 55N91. Key words and phrases. stacks, moduli of curves, equivariant cohomology. The author was supported by NSA grant H98230-08-1-0059 while preparing this article. 2 Cohomology of the moduli space of curves 1. Introduction The study of the cohomology of the moduli space of curves has been a very rich research area for the last 30 years. -
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. -
The Classification of Triangulated Subcategories 3
Compositio Mathematica 105: 1±27, 1997. 1 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. The classi®cation of triangulated subcategories R. W. THOMASON ? CNRS URA212, U.F.R. de Mathematiques, Universite Paris VII, 75251 Paris CEDEX 05, France email: thomason@@frmap711.mathp7.jussieu.fr Received 2 August 1994; accepted in ®nal form 30 May 1995 Key words: triangulated category, Grothendieck group, Hopkins±Neeman classi®cation Mathematics Subject Classi®cations (1991): 18E30, 18F30. 1. Introduction The ®rst main result of this paper is a bijective correspondence between the strictly full triangulated subcategories dense in a given triangulated category and the sub- groups of its Grothendieck group (Thm. 2.1). Since every strictly full triangulated subcategory is dense in a uniquely determined thick triangulated subcategory, this result re®nes any known classi®cation of thick subcategories to a classi®cation of all strictly full triangulated ones. For example, one can thus re®ne the famous clas- si®cation of the thick subcategories of the ®nite stable homotopy category given by the work of Devinatz±Hopkins±Smith ([Ho], [DHS], [HS] Thm. 7, [Ra] 3.4.3), which is responsible for most of the recent advances in stable homotopy theory. One can likewise re®ne the analogous classi®cation given by Hopkins and Neeman (R ) ([Ho] Sect. 4, [Ne] 1.5) of the thick subcategories of D parf, the chain homotopy category of bounded complexes of ®nitely generated projective R -modules, where R is a commutative noetherian ring. The second main result is a generalization of this classi®cation result of Hop- kins and Neeman to schemes, and in particular to non-noetherian rings. -
KK-Theory As a Triangulated Category Notes from the Lectures by Ralf Meyer
KK-theory as a triangulated category Notes from the lectures by Ralf Meyer Focused Semester on KK-Theory and its Applications Munster¨ 2009 1 Lecture 1 Triangulated categories formalize the properties needed to do homotopy theory in a category, mainly the properties needed to manipulate long exact sequences. In addition, localization of functors allows the construction of interesting new functors. This is closely related to the Baum-Connes assembly map. 1.1 What additional structure does the category KK have? −n Suspension automorphism Define A[n] = C0(R ;A) for all n ≤ 0. Note that by Bott periodicity A[−2] = A, so it makes sense to extend the definition of A[n] to Z by defining A[n] = A[−n] for n > 0. Exact triangles Given an extension I / / E / / Q with a completely posi- tive contractive section, let δE 2 KK1(Q; I) be the class of the extension. The diagram I / E ^> >> O δE >> > Q where O / denotes a degree one map is called an extension triangle. An alternate notation is δ Q[−1] E / I / E / Q: An exact triangle is a diagram in KK isomorphic to an extension triangle. Roughly speaking, exact triangles are the sources of long exact sequences of KK-groups. Remark 1.1. There are many other sources of exact triangles besides extensions. Definition 1.2. A triangulated category is an additive category with a suspension automorphism and a class of exact triangles satisfying the axioms (TR0), (TR1), (TR2), (TR3), and (TR4). The definition of these axiom will appear in due course. Example 1.3. -
Relative Homological Algebra and Purity in Triangulated Categories
Journal of Algebra 227, 268᎐361Ž 2000. doi:10.1006rjabr.1999.8237, available online at http:rrwww.idealibrary.com on Relative Homological Algebra and Purity in Triangulated Categories Apostolos Beligiannis Fakultat¨¨ fur Mathematik, Uni¨ersitat ¨ Bielefeld, D-33501 Bielefeld, Germany E-mail: [email protected], [email protected] Communicated by Michel Broue´ Received June 7, 1999 CONTENTS 1. Introduction. 2. Proper classes of triangles and phantom maps. 3. The Steenrod and Freyd category of a triangulated category. 4. Projecti¨e objects, resolutions, and deri¨ed functors. 5. The phantom tower, the cellular tower, homotopy colimits, and compact objects. 6. Localization and the relati¨e deri¨ed category. 7. The stable triangulated category. 8. Projecti¨ity, injecti¨ity, and flatness. 9. Phantomless triangulated categories. 10. Brown representation theorems. 11. Purity. 12. Applications to deri¨ed and stable categories. References. 1. INTRODUCTION Triangulated categories were introduced by Grothendieck and Verdier in the early sixties as the proper framework for doing homological algebra in an abelian category. Since then triangulated categories have found important applications in algebraic geometry, stable homotopy theory, and representation theory. Our main purpose in this paper is to study a triangulated category, using relative homological algebra which is devel- oped inside the triangulated category. Relative homological algebra has been formulated by Hochschild in categories of modules and later by Heller and Butler and Horrocks in 268 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. PURITY IN TRIANGULATED CATEGORIES 269 more general categories with a relative abelian structure.