DOCSLIB.ORG

1. Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1. Introduction BRAUER GROUPS AND QUOTIENT STACKS DAN EDIDIN BRENDAN HASSETT ANDREW KRESCH AND ANGELO VISTOLI Abstract A natural question is to determine which algebraic stacks are quotient stacks In this pap er we give some partial answers and relate it to the old question of whether for a scheme X the natural map from the Brauer group equivalence classes of Azumaya algebras to the cohomolog 2 ical Brauer group the torsion subgroup of H X G is surjective m Introduction Quotients of varieties by algebraic groups arise in many situations for in stance in the theory of mo duli where mo duli spaces are often naturally con structed as quotients of parameter spaces by linear algebraic groups The quotient of a scheme by a group need not exist as a scheme or even as an algebraic space and even when a quotient exists the quotient morphism may not have exp ected prop erties For example if Z and G are smo oth then the morphism Z Z G need not b e smo oth To overcome this diculty it is often helpful to consider quotients as stacks rather than as schemes or algebraic spaces If G is a at group scheme acting on an algebraic space Z G must b e separated and nitely presented over some base scheme with the space Z and the action map dened over this base then a quotient Z G always exists as a stack and this stack is algebraic Knowing that an algebraic stack has a presentation as a quotient Z G with G a linear algebraic group say can make the stack easier to study for then the geometry of the stack is the Gequivariant geometry on the space Z A natural question is to determine which algebraic stacks are quotient stacks In this pap er we give some partial answers to this question and relate it to the old question of whether for a scheme X the natural map from the Brauer group classes of Azumaya algebras mo dulo an equivalence relation to the cohomological Brauer group the torsion subgroup of etale H X G is sur m jective Edidin received supp ort from the NSA NSF and the University of Missouri Research Board while preparing this pap er Hassett and Kresch were partially supp orted by NSF Postdo ctoral Research Fellowships Hassett received additional supp ort from the Institute of Mathematical Sciences of the Chinese University of Hong Kong and NSF Vistoli was partially supp orted by the University of Bologna funds for selected research topics D EDIDIN B HASSETT A KRESCH AND A VISTOLI Some quick answers to this natural question are the rst two are folklore i all orbifolds are quotient stacks Theorem ii all regular Deligne Mumford stacks of dimension are quotient stacks Example iii there exists a DeligneMumford stack normal and of nite typ e over the com plex numb ers but singular and nonseparated which is not a quotient stack Example In fact the example in iii is a stack with stabilizer group Z at every p oint it is a gerb e over a normal but nonseparated scheme of dimension over the complex numb ers Theorem says such a stack is a quotient stack if and only if a certain class in the cohomological Brauer group asso ciated with it lies in the image of the map from the Brauer group So iii yields an example of indep endent interest of nonsurjectivity of the Brauer map for a nitetyp e normal but nonseparated scheme Corollary This stands in contrast with the recent result of S Schroer Sch which says that the Brauer map is surjective for any separated geometrically normal algebraic surface The pap er is organized as follows In Section we review the denition of algebraic stacks and state accompanying results relative to quotient stacks Additional results concern nite covers of stacks by schemes In Section we review gerb es and Brauer groups and state the result relating the Brauer map to gerb es b eing quotient stacks Finally in Section we give pro ofs Acknowledgements The authors thank Andrei Caldararu Bill Graham and Amnon Yekutieli for helpful discussions They are also grateful to Laurent MoretBailly and the referee for a numb er of corrections and suggestions Stacks and quotient stacks Stacks Here we give a brief review of stacks Some references are DM Vi and LMB Stacks are categories b ered in group oids satisfying descenttyp e axioms the stacks of interest to us will b e algebraic and hence admit descriptions in the form of group oid schemes First recall that a group oid is a small category C in which all arrows are isomorphisms Write R HomC and X Ob jC There are two maps s t R X sending a morphism to its source and target resp ectively a map e X R taking an ob ject to the identity morphism of itself a map i R R taking a morphism to its inverse and a map m R R R taking a pair of comp osable morphisms to their tX s comp osition Write j t s R X X There are obvious compatibilities b etween these maps A group oid scheme consists of schemes R and X dened over a xed base scheme L together with maps s t e i m satisfying the same compatibility conditions as ab ove A group oid scheme is called etale resp ectively smo oth resp ectively at if the maps s and t are etale resp smo oth resp faithfully at and lo cally of nite presentation The stabilizer of a group oid scheme is BRAUER GROUPS AND QUOTIENT STACKS the scheme S j here X X is the diagonal This is a group X X scheme over X Let L b e a xed ground scheme and let F b e a category together with a functor p F Sch L For a xed Lscheme B let F B denote the sub category of F consisting of ob jects mapping to B and morphisms mapping to Roughly a category bered in groupoids over L is pair consisting of a B category F and a functor p F Sch L such that i For all Lschemes B F B is a group oid 0 ii For any morphism of Lschemes f B B and any ob ject x F B 0 there is an ob ject f x in F B unique up to canonical isomorphism together with a morphism f x x lying over f For the precise denition see eg DM Sec A morphism of categories b ered in group oids is simply a functor commuting with the pro jection functors to Sch L An isomorphism of categories b ered in group oids is a morphism which is an equivalence of categories Any contravariant functor Sch L sets determines a category b ered in group oids We say that a category b ered in group oids over L is represented by a scheme resp algebraic space if it is equivalent to the functor of p oints of a scheme resp algebraic space An imp ortant construction is the b er pro duct Given morphisms f F F and f F F the b er pro duct F F is the category b ered in F group oids dened as follows Ob jects are triples x x where x is an ob ject of F x is an ob ject of F and f x f x is an isomorphism lying over an identity morphism of Sch L A morphism is sp ecied by a pair of morphisms compatible with the induced isomorphism in F Denition A category b ered in group oids F p is a stack if it satises two descent prop erties For ob jects x y in F B the functor Iso x y Sch B sets assigning B 0 to a B scheme f B B the set of isomorphisms b etween f x and f y is a sheaf for the etale top ology F has eective descent for etale morphisms Denition A morphism of stacks is representable if for any morphism of 0 an algebraic space B F the b er pro duct B F is represented by an F algebraic space A morphism is strongly representable if for any morphism of 0 a scheme B F the b er pro duct B F is represented by a scheme F Let P b e a prop erty of morphism of schemes which is preserved by base 0 change and is lo cal for the smo oth top ology A representable morphism F F has prop erty P if for all morphisms B F of algebraic spaces the induced 0 morphism B F B has prop erty P F Stein factorization holds for algebraic spaces and implies Kn I I that if f X Y is a separated quasinite morphism of algebraic spaces and if Y D EDIDIN B HASSETT A KRESCH AND A VISTOLI is a scheme then X is a scheme Hence a representable separated quasinite morphism is always strongly representable Denition A stack F is algebraic or is an Artin stack if There exists a representable smo oth surjective morphism X F from a scheme The diagonal morphism F F F is representable quasicompact and L separated Remark The representability of the diagonal implies that any morphism from an algebraic space is representable For stacks with quasinite diagonal any morphism from a scheme is strongly representable Remark A stack F is called a DeligneMumford stack if there exists an etale cover of F by a scheme By LMB this happ ens if and only if the diagonal F F F is unramied A DeligneMumford stack has in L particular quasinite diagonal The geometric b ers of the diagonal are group schemes so if all the residue elds of L have characteristic then conversely any algebraic stack with quasinite diagonal is a DeligneMumford stack Finally we describ e very briey group oid presentations or atlases of al gebraic stacks see LMB for a full treatment By denition any algebraic stack F admits a smo oth surjective map from a scheme X X F is called a smooth atlas In this case the b er pro duct R X X is an algebraic F space However for stacks with quasinite diagonal the diagonal is strongly representable so R is in fact a scheme The smo oth group oid scheme R X is called a presentation for F Conversely any smo oth group oid scheme R X with separated nitetyp e relative diagonal R X X determines an al gebraic stack R X A theorem of Artin cf LMB says that any faithfully at group oid scheme R X with separated nitetyp e relative di agonal determines an algebraic stack In this case the group oid scheme R X is called a faithful ly at presentation
Load more

Recommended publications
  • 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 .
    [Show full text]
  • Stack Completions and Morita Equivalence for Categories in a Topos Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 20, No 4 (1979), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES MARTA BUNGE Stack completions and Morita equivalence for categories in a topos Cahiers de topologie et géométrie différentielle catégoriques, tome 20, no 4 (1979), p. 401-436 <http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0> © Andrée C. Ehresmann et les auteurs, 1979, 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 Vol. XX-4 (1979) ET GEOMETRIE DIFFERENTIELLE STACK COMPLETIONS AND MORITA EQUIVALENCE FOR CATEGORIES IN A TOPOS by Marta BUNGE 1) 0. INTRODUCTION. The origin of this paper can be traced back to one of a series of lectures given by F.W. Lawvere [10, Lecture V]. In it, Lawvere dealt for the case of an arbitrary topos S , with the notion of stack, a notion which, for Grothendieck toposes, had been considered by J. Giraud [6], and which is given relative to a site. A topos S may always be regarded as a site with the regular epi- morphism topology, and the notion of stack over S is then defined with respect to this particular topology. Special as it may be, this notion of stack over a topos S plays an important role in the development of Category Theory over a base to- pos S .
    [Show full text]
  • Schematic Homotopy Types and Non-Abelian Hodge Theory
    Compositio Math. 144 (2008) 582–632 doi:10.1112/S0010437X07003351 Schematic homotopy types and non-abelian Hodge theory L. Katzarkov, T. Pantev and B. To¨en Abstract We use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor X → (X ⊗ C)sch, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on (X ⊗ C)sch. This Hodge decomposition is encoded in an action of the discrete group C×δ on the object (X ⊗ C)sch and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro- algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold. Contents Introduction 583 1 Review of the schematization functor 589 1.1 Affine stacks and schematic homotopy types ................. 589 1.2 Equivariant stacks ............................... 594 1.3 Equivariant co-simplicial algebras and equivariant affine stacks ...... 595 1.4 An explicit model for (X ⊗ C)sch ....................... 597 2 The Hodge decomposition 598 2.1 Review of the non-abelian Hodge correspondence .............
    [Show full text]
  • Vector Bundles As Generators on Schemes and Stacks
    Vector Bundles as Generators on Schemes and Stacks Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨atD¨usseldorf vorgelegt von Philipp Gross aus D¨usseldorf D¨usseldorf,Mai 2010 Aus dem Mathematischen Institut der Heinrich-Heine-Universit¨atD¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Heinrich-Heine-Universit¨atD¨usseldorf Referent: Prof. Dr. Stefan Schr¨oer Koreferent: Prof. Dr. Holger Reich Acknowledgments The work on this dissertation has been one of the most significant academic challenges I have ever had to face. This study would not have been completed without the support, patience and guidance of the following people. It is to them that I owe my deepest gratitude. I am indebted to my advisor Stefan Schr¨oerfor his encouragement to pursue this project. He taught me algebraic geometry and how to write academic papers, made me a better mathematician, brought out the good ideas in me, and gave me the opportunity to attend many conferences and schools in Europe. I also thank Holger Reich, not only for agreeing to review the dissertation and to sit on my committee, but also for showing an interest in my research. Next, I thank the members of the local algebraic geometry research group for their time, energy and for the many inspiring discussions: Christian Liedtke, Sasa Novakovic, Holger Partsch and Felix Sch¨uller.I have had the pleasure of learning from them in many other ways as well. A special thanks goes to Holger for being a friend, helping me complete the writing of this dissertation as well as the challenging research that lies behind it.
    [Show full text]
  • No Transcendental Brauer-Manin Obstructions on Abelian Varieties
    THERE ARE NO TRANSCENDENTAL BRAUER-MANIN OBSTRUCTIONS ON ABELIAN VARIETIES BRENDAN CREUTZ Abstract. Suppose X is a torsor under an abelian variety A over a number field. We show that any adelic point of X that is orthogonal to the algebraic Brauer group of X is orthogonal to the whole Brauer group of X. We also show that if there is a Brauer-Manin obstruction to the existence of rational points on X, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that A has finite Tate-Shafarevich group. Our results are unconditional. 1. Introduction Let X be a smooth projective and geometrically integral variety over a number field k. In order that X possesses a k-rational point it is necessary that X has points everywhere locally, i.e., that the set X(Ak) of adelic points on X is nonempty. The converse to this statement is called the Hasse principle, and it is known that this can fail. When X(k) is nonempty one can ask if weak approximation holds, i.e., if X(k) is dense in X(Ak) in the adelic topology. Manin [Man71] showed that the failure of the Hasse principle or weak approximation can, in many cases, be explained by a reciprocity law on X(Ak) imposed by the Brauer group, 2 Br X := Hét(X, Gm). Specifically, each element α ∈ Br X determines a continuous map, α∗ : X(Ak) → Q/Z, between the adelic and discrete topologies with the property that the subset X(k) ⊂ X(Ak) of rational points is mapped to 0.
    [Show full text]
  • 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].
    [Show full text]
  • Arxiv:1610.06871V4 [Math.AT] 13 May 2019 Se[A H.7506 N SG E.B.6.2.7]): Rem
    THE MOREL–VOEVODSKY LOCALIZATION THEOREM IN SPECTRAL ALGEBRAIC GEOMETRY ADEEL A. KHAN Abstract. We prove an analogue of the Morel–Voevodsky localization theorem over spectral algebraic spaces. As a corollary we deduce a “derived nilpotent invariance” result which, informally speaking, says that A1-homotopy invariance kills all higher homotopy groups of a connective commutative ring spectrum. 1. Introduction ´et Let R be a connective E∞-ring spectrum, and denote by CAlgR the ∞-category of ´etale E∞-algebras over R. The starting point for this paper is the following fundamental result of J. Lurie, which says that the small ´etale topos of R is equivalent to the small ´etale topos of π0(R) (see [HA, Thm. 7.5.0.6] and [SAG, Rem. B.6.2.7]): Theorem 1.0.1 (Lurie). For any connective E∞-ring R, let π0(R) denote its 0-truncation E ´et ´et (viewed as a discrete ∞-ring). Then restriction along the canonical functor CAlgR → CAlgπ0(R) ´et induces an equivalence from the ∞-category of ´etale sheaves of spaces CAlgπ0(R) → Spc to the ´et ∞-category of ´etale sheaves of spaces CAlgR → Spc. Theorem 1.0.1 can be viewed as a special case of the following result (see [SAG, Prop. 3.1.4.1]): ′ Theorem 1.0.2 (Lurie). Let R → R be a homomorphism of connective E∞-rings that is ´et ´et surjective on π0. Then restriction along the canonical functor CAlgR → CAlgR′ defines a fully ´et faithful embedding of the ∞-category of ´etale sheaves CAlgR′ → Spc into the ∞-category of ´etale ´et sheaves CAlgR → Spc.
    [Show full text]
  • Fundamental Algebraic Geometry
    http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.
    [Show full text]
  • X → S Be a Proper Morphism of Locally Noetherian Schemes and Let F Be a Coherent Sheaf on X That Is flat Over S (E.G., F Is Smooth and F Is a Vector Bundle)
    COHOMOLOGY AND BASE CHANGE FOR ALGEBRAIC STACKS JACK HALL Abstract. We prove that cohomology and base change holds for algebraic stacks, generalizing work of Brochard in the tame case. We also show that Hom-spaces on algebraic stacks are represented by abelian cones, generaliz- ing results of Grothendieck, Brochard, Olsson, Lieblich, and Roth{Starr. To accomplish all of this, we prove that a wide class of relative Ext-functors in algebraic geometry are coherent (in the sense of M. Auslander). Introduction Let f : X ! S be a proper morphism of locally noetherian schemes and let F be a coherent sheaf on X that is flat over S (e.g., f is smooth and F is a vector bundle). If s 2 S is a point, then define Xs to be the fiber of f over s. If s has residue field κ(s), then for each integer q there is a natural base change morphism of κ(s)-vector spaces q q q b (s):(R f∗F) ⊗OS κ(s) ! H (Xs; FXs ): Cohomology and Base Change originally appeared in [EGA, III.7.7.5] in a quite sophisticated form. Mumford [Mum70, xII.5] and Hartshorne [Har77, xIII.12], how- ever, were responsible for popularizing a version similar to the following. Let s 2 S and let q be an integer. (1) The following are equivalent. (a) The morphism bq(s) is surjective. (b) There exists an open neighbourhood U of s such that bq(u) is an iso- morphism for all u 2 U. (c) There exists an open neighbourhood U of s, a coherent OU -module Q, and an isomorphism of functors: Rq+1(f ) (F ⊗ f ∗ I) =∼ Hom (Q; I); U ∗ XU OXU U OU where fU : XU ! U is the pullback of f along U ⊆ S.
    [Show full text]
  • Brauer Groups of Abelian Schemes
    ANNALES SCIENTIFIQUES DE L’É.N.S. RAYMOND T. HOOBLER Brauer groups of abelian schemes Annales scientifiques de l’É.N.S. 4e série, tome 5, no 1 (1972), p. 45-70 <http://www.numdam.org/item?id=ASENS_1972_4_5_1_45_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1972, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scienL EC. Norm. Sup., 4® serie, t. 5, 1972, p. 45 ^ 70. BRAUER GROUPS OF ABELIAN SCHEMES BY RAYMOND T. HOOBLER 0 Let A be an abelian variety over a field /c. Mumford has given a very beautiful construction of the dual abelian variety in the spirit of Grothen- dieck style algebraic geometry by using the theorem of the square, its corollaries, and cohomology theory. Since the /c-points of Pic^n is H1 (A, G^), it is natural to ask how much of this work carries over to higher cohomology groups where the computations must be made in the etale topology to render them non-trivial. Since H2 (A, Gm) is essentially a torsion group, the representability of the corresponding functor does not have as much geometric interest as for H1 (A, G^).
    [Show full text]
  • Arxiv:1401.2824V1 [Math.AT]
    ZMP-HH/14-2 Hamburger Beitr¨age zur Mathematik Nr. 499 January 2014 A Serre-Swan theorem for gerbe modules on ´etale Lie groupoids Christoph Schweigert a, Christopher Tropp b, Alessandro Valentino a a Fachbereich Mathematik, Universit¨at Hamburg Bereich Algebra und Zahlentheorie Bundesstraße 55, D – 20 146 Hamburg b Mathematisches Institut, WWU M¨unster Einsteinstr. 62, D – 48149 M¨unster Abstract Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact ´etale Lie groupoid M, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on M. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles. 1 Introduction The celebrated Serre-Swan theorem relates the category of vector bundles over a compact smooth manifold M to the category of finite rank projective modules over the algebra of smooth functions C∞(M, C) of M (see [GBV, Mor] for the Serre-Swan theorem in the smooth category). It relates geometric and algebraic notions and is, in particular, the starting point for the definition of vector bundles in non-commutative geometry. arXiv:1401.2824v1 [math.AT] 13 Jan 2014 A bundle gerbe on M can be seen as a geometric realization of its Dixmier-Douady class, which is a class in H3(M; Z). To such a geometric realization, a twisted K-theory group can be associated. Gerbe modules have been introduced to obtain a geometric description of twisted K-theory [BCMMS].
    [Show full text]
  • Algebraic Stacks (Winter 2020/2021) Lecturer: Georg Oberdieck
    Algebraic Stacks (Winter 2020/2021) Lecturer: Georg Oberdieck Date: Wednesday 4-6, Friday 12-2. Oral Exam: February 19 and March 26, 2021. Exam requirement: You have to hand in solutions to at least half of all exercise problems to be admitted to the exam. Summary: The course is an introduction to the theory of algebraic stacks. We will discuss applications to the construction of moduli spaces. Prerequisites: Algebraic Geometry I and II (e.g. on the level of Hartshorne's book Chapter I and II plus some background on flat/etale morphisms). Some category theory (such as Vakil's Notes on Algebraic Geometry, Chapter 1). References: (1) Olsson, Algebraic Spaces and Stacks (2) Stacks Project (3) Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (available online) (4) Halpern-Leistner, The Moduli space, https://themoduli.space (5) Guide to the literature by Jarod Alper (6) Behrend, Introduction to algebraic stacks, lecture notes. The main reference of the coarse is Olsson and will be used throughout, in particular for many of the exercises. However, we will diverge in some details from Olsson's presentation, in particular on quasi-coherent sheaves on algebraic stacks where we follow the stacks project. The notes of Vistoli are also highly recommended for the first part which covers aspects of Grothendieck topologies, fibered categories and descent theory. In the second part we will use the notes of Halpern-Leistner. Exercises and student presentations: Friday's session will be used sometimes for informal discussion of the material covered in the problem sets. From time to time there will also be student presentations, see: http://www.math.uni-bonn.de/~georgo/stacks/presentations.pdf Schedule1 Oct 28: Motivation mostly.
    [Show full text]