DOCSLIB.ORG

Algebraic Stacks TCC Course, Autumn 2014

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebraic Stacks TCC Course, Autumn 2014 Algebraic Stacks TCC course, Autumn 2014 Michael Groechenig with contributions by: Alexander Betts Andrea Petracci Claudio Onorati Craig Smith December 15, 2014 Contents 1 Introduction 3 1.1 What is a stack? . .3 1.2 What is algebraic geometry? A functorial approach. .7 2 Faithfully flat descent 13 2.1 Tensor products and flatness . 14 2.1.1 A reminder on tensor products and base change . 14 2.1.2 Base change invariant properties . 17 2.1.3 Flatness . 18 2.2 Faithfully flat descent . 20 2.2.1 Basic properties . 20 2.2.2 Descending modules . 22 2.2.3 Descent for ring homomorphisms . 24 3 Sheaves and stacks 24 3.1 Sheaves . 25 3.1.1 Topological spaces . 25 3.1.2 Grothendieck topologies and sheaves . 26 3.1.3 Three examples . 28 3.2 Stacks . 29 3.2.1 Groupoids as generalised sets . 29 3.2.2 Stacks as groupoid-valued sheaves . 30 3.2.3 Examples . 33 3.2.4 Fibre products in 2-categories . 35 1 4 Algebraicity 36 4.1 Schemes . 37 4.1.1 Affine schemes . 37 4.1.2 Open subfunctors of affine schemes . 39 4.1.3 Schemes . 40 4.2 Algebraic spaces and stacks . 45 4.2.1 Finite presentation . 45 4.2.2 Smooth and ´etalemorphisms . 47 4.2.3 Various Grothendieck topologies . 48 4.2.4 Algebraic spaces . 49 4.2.5 Algebraic stacks . 49 4.2.6 Presentations . 50 4.2.7 Base change . 51 5 Torsors and quotient stacks 52 5.1 Group objects . 52 5.2 Torsors . 54 5.2.1 Definition and basic properties . 54 5.2.2 Faithfully flat descent revisited . 57 5.2.3 The tautological trivialisation . 58 5.3 Algebraicity of quotient stacks . 59 5.3.1 Algebraicity of the stack of torsors . 59 5.3.2 Quotient stacks . 59 5.4 Philosophy: quotient stacks and equivariant objects . 60 6 The stack of G-bundles on a curve 61 6.1 Preliminaries . 61 6.1.1 Curves and vector bundles on curves . 61 6.1.2 Recollection of cohomology of coherent sheaves . 61 6.1.3 Stacks which are locally of finite presentation . 62 6.1.4 A criterion for surjectivity . 62 6.1.5 Quot schemes . 62 6.2 The stack of vector bundles on a curve . 63 6.2.1 Families of vector bundles . 63 6.2.2 The atlas . 63 6.3 The stack of G-bundles on a curve . 65 6.3.1 Stacks of G-bundles as mapping stacks . 65 6.3.2 Presentability statements . 66 7 Smooth algebraic stacks 69 7.1 Smooth morphisms . 69 7.2 The cotangent complex . 73 7.2.1 Dual numbers and the tangent space . 73 7.2.2 K¨ahlerdifferentials and basic deformation theory . 76 7.2.3 The cotangent sheaf for smooth Deligne-Mumford stacks . 77 7.2.4 Towards the cotangent complex for smooth algebraic stacks . 78 2 8 Quasi-coherent sheaves 79 8.1 Quasi-coherent sheaves on prestacks . 79 8.1.1 A stacky definition of quasi-coherent sheaves . 80 8.1.2 The category of quasi-coherent sheaves as a limit . 82 8.1.3 Stackification and quasi-coherent sheaves . 83 8.2 BG and G-representations . 84 8.2.1 The Barr-Beck Theorem . 84 8.2.2 Quasi-coherent sheaves on BG: the proof . 87 1 Introduction This introduction (just like the whole course) aims to explain two things: the meaning of the word stack, and its connection with algebraic geometry. We begin by discussing three typical examples of stacks, two of which take place in the category of topological spaces, while the third is of arithmetic nature. 1.1 What is a stack? Glueing of vector bundles The archetypical example of a stack is well-known to every mathematician. In order to describe the data of a rank n vector bundle E on a topological space X one often resorts to a so-called cocycle description. This amounts to choosing an open covering fUigi2I of X, and for each pair (i; j) of indices, a vector bundle automorphism φij of the trivial vector bundle n (Ui \ Uj) × R on the open subset Ui \ Uj. One then hopes to find a vector bundle E on X, together with vector bundle isomorphisms (called local trivializations) n ∼= φi : Ui × R −! EjUi ; such that the identities −1 φij = φi ◦ φj (1) are satisfied on Ui \ Uj. A necessary condition for the existence of (E; (φi)i2I ) is that the cocycle condition φij ◦ φjk = φik (2) is satisfied for every triple (i; j; k). Note that this identity implies for i = j = k that φii = id, and −1 hence for i = k that φji = φij . It is essential to the theory of topological vector bundles that the cocycle condition is also sufficient. Indeed, we can define E by glueing the trivial vector bundles n Ui × R with respect to the isomorphisms φij: let Ee be the disjoint union of the topological spaces n n n Ui × R ; we define an equivalence relation, by declaring (x; v) 2 Ui × R and (y; w) 2 Uj × R as equivalent if x = y and v = φij(w). The topological quotient space will be denoted by E and has a canonical map E / X, and canonical trivializations (φi)i2I , which satisfy the identity (1) above. In modern language we would say that the theory of vector bundles on topological spaces is a stack. Let's be more precise and actually nail down what kind of mathematical object the stack 3 is in this example. It is the following assignment, which associates to an open subset U ⊂ X the groupoid of vector bundles on U. Groupoid is a fancy word for a simple thing. In fact, it simply denotes a category where every morphism is invertible. Hence, the groupoid of vector bundles on U denotes the category of vector bundles on U, where we discard non-invertible morphisms. Remark 1.1. It is possible to formulate a theory of stacks (or rather sheaves of categories), which takes values in honest categories and not just groupoids. However, it will not be necessary for us to pursue that level of generality. As we have seen in the example of vector bundles on topological spaces, we only need vector bundle isomorphisms to describe the glueing data. Hence it is sensible to discard all the other morphisms of vector bundles. The cocycle condition (2) can be elegantly reformulated. This requires the usage of fibre products. Definition 1.2. For continuous maps of topological spaces f : X / Z and g : Y / Z, we denote by X ×Z Y the subspace of X × Y , consisting of tuples (x; y) with f(x) = g(y). We denote by a Y = Ui; i2I and by π : Y / X the canonical map, induced by the inclusions Ui ,! X. Let Ee be the trivial rank n vector bundle on Y . The fibre product Y ×X Y is equivalent to the disjoint union a Ui \ Uj: (i;j)2I2 The cocycle (φij)(i;j)2I2 is captured by an isomorphism of vector bundles on Y ×X Y , namely ∗ ∼= ∗ φ: p2Ee −! p1E:e (3) Here, we use the suggestive notation which denotes the projection to the first component of a fibre product X ×Z Y / X by p1, and similarly for the projection to the second component. Let's pause a second to see that this is indeed equivalent to a collection of automorphisms (φij) of the trivial vector bundles on Ui \ Uj. The trivial vector bundle always pulls back to the trivial vector bundle, no matter which map we considered. Hence, φ is really an automorphism of the trivial vector bundle on Y ×X Y , which is, as remarked above, equivalent to the disjoint union of all the spaces Ui \ Uj. The cocycle condition amounts to the identity of vector bundle isomorphisms on Y ×X Y ×X Y : ∗ ∗ ∗ ∗ ∗ p12φ ◦ p23φ = p13φ: p3Ee / p1E:e (4) The family of local trivializations (φi)i2I amounts to ∗ n π E =∼ Ee = Y × R : (5) We will re-encounter a similar formulation below. 4 Descending vector bundles to quotients One of the first non-trivial vector bundles one encounters in topology is a M¨obiusstrip of infinite width over the circle S1. A rigorous description of this bundle can be given as follows. At first we observe the existence of a homeomorphism 1 ∼ 1 S /µ2 = S ; 1 where µ2 = f1; −1g denotes the group of second order roots of unity, acting on S ⊂ C multi- 1 1 2 plicatively. The homeomorphism is induced by the µ2-invariant map S / S , z 7! z . Although both spaces are homeomorphic, we will distinguish between both sides, and denote the canonical projection by 1 ∼ 1 ∼ 1 π : Y = S / X = S /µ2 = S : 1 1 The M¨obiusbundle E on S can be defined by extending the action of µ2 on S to the total space Ee 1 1 of the trivial rank 1 bundle S × R / R. Namely, we define the action of ζ 2 µ2 on (z; λ) 2 S × R to be ζ · (z; λ) = (ζx; ζλ): 1 ∼ 1 The quotient E/µe 2 will be denoted by E. It defines a rank 1 vector bundle on S /µ2 = X = S . One observes that π∗E =∼ S1 ×R, which should be compared to equation (5). But there are even more similarities with the glueing procedure described in the proceeding subsection. Let's take a look at the fibre products Y ×X Y and Y ×X Y ×X Y .
Load more

Recommended publications
  • Arxiv:1108.5351V3 [Math.AG] 26 Oct 2012 ..Rslso D-Mod( on Results Introduction the to 0.2
    ON SOME FINITENESS QUESTIONS FOR ALGEBRAIC STACKS VLADIMIR DRINFELD AND DENNIS GAITSGORY Abstract. We prove that under a certain mild hypothesis, the DG category of D-modules on a quasi-compact algebraic stack is compactly generated. We also show that under the same hypothesis, the functor of global sections on the DG category of quasi-coherent sheaves is continuous. Contents Introduction 3 0.1. Introduction to the introduction 3 0.2. Results on D-mod(Y) 4 0.3. Results on QCoh(Y) 4 0.4. Ind-coherent sheaves 5 0.5. Contents of the paper 7 0.6. Conventions, notation and terminology 10 0.7. Acknowledgments 14 1. Results on QCoh(Y) 14 1.1. Assumptions on stacks 14 1.2. Quasi-coherent sheaves 15 1.3. Direct images for quasi-coherent sheaves 18 1.4. Statements of the results on QCoh(Y) 21 2. Proof of Theorems 1.4.2 and 1.4.10 23 2.1. Reducing the statement to a key lemma 23 2.2. Easy reduction steps 24 2.3. Devissage 24 2.4. Quotients of schemes by algebraic groups 26 2.5. Proof of Proposition 2.3.4 26 2.6. Proof of Theorem 1.4.10 29 arXiv:1108.5351v3 [math.AG] 26 Oct 2012 3. Implications for ind-coherent sheaves 30 3.1. The “locally almost of finite type” condition 30 3.2. The category IndCoh 32 3.3. The coherent subcategory 39 3.4. Description of compact objects of IndCoh(Y) 39 3.5. The category Coh(Y) generates IndCoh(Y) 42 3.6.
    [Show full text]
  • 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]
  • 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]
  • 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]
  • 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.
    [Show full text]
  • MAPPING STACKS and CATEGORICAL NOTIONS of PROPERNESS Contents 1. Introduction 2 1.1. Introduction to the Introduction 2 1.2
    MAPPING STACKS AND CATEGORICAL NOTIONS OF PROPERNESS DANIEL HALPERN-LEISTNER AND ANATOLY PREYGEL Abstract. One fundamental consequence of a scheme being proper is that there is an algebraic space classifying maps from it to any other finite type scheme, and this result has been extended to proper stacks. We observe, however, that it also holds for many examples where the source is a geometric stack, such as a global quotient. In our investigation, we are lead naturally to certain properties of the derived category of a stack which guarantee that the mapping stack from it to any geometric finite type stack is algebraic. We develop methods for establishing these properties in a large class of examples. Along the way, we introduce a notion of projective morphism of algebraic stacks, and prove strong h-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Contents 1. Introduction 2 1.1. Introduction to the introduction2 1.2. Mapping out of stacks which are \proper enough"3 1.3. Techniques for establishing (GE) and (L)5 1.4. A long list of examples6 1.5. Comparison with previous results7 1.6. Notation and conventions8 1.7. Author's note 9 2. Artin's criteria for mapping stacks 10 2.1. Weil restriction of affine stacks 11 2.2. Deformation theory of the mapping stack 12 2.3. Integrability via the Tannakian formalism 16 2.4. Derived representability from classical representability 19 2.5. Application: (pGE) and the moduli of perfect complexes 21 3. Perfect Grothendieck existence 23 3.1.
    [Show full text]
  • 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.
    [Show full text]
  • Noncommutative Stacks
    Noncommutative Stacks Introduction One of the purposes of this work is to introduce a noncommutative analogue of Artin’s and Deligne-Mumford algebraic stacks in the most natural and sufficiently general way. We start with quasi-coherent modules on fibered categories, then define stacks and prestacks. We define formally smooth, formally unramified, and formally ´etale cartesian functors. This provides us with enough tools to extend to stacks the glueing formalism we developed in [KR3] for presheaves and sheaves of sets. Quasi-coherent presheaves and sheaves on a fibered category. Quasi-coherent sheaves on geometric (i.e. locally ringed topological) spaces were in- troduced in fifties. The notion of quasi-coherent modules was extended in an obvious way to ringed sites and toposes at the moment the latter appeared (in SGA), but it was not used much in this generality. Recently, the subject was revisited by D. Orlov in his work on quasi-coherent sheaves in commutative an noncommutative geometry [Or] and by G. Laumon an L. Moret-Bailly in their book on algebraic stacks [LM-B]. Slightly generalizing [R4], we associate with any functor F (regarded as a category over a category) the category of ’quasi-coherent presheaves’ on F (otherwise called ’quasi- coherent presheaves of modules’ or simply ’quasi-coherent modules’) and study some basic properties of this correspondence in the case when the functor defines a fibered category. Imitating [Gir], we define the quasi-topology of 1-descent (or simply ’descent’) and the quasi-topology of 2-descent (or ’effective descent’) on the base of a fibered category (i.e.
    [Show full text]
  • Algebra & Number Theory Vol. 10 (2016)
    Algebra & Number Theory Volume 10 2016 No. 4 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Anand Pillay University of Notre Dame, USA Brian D. Conrad Stanford University, USA Victor Reiner University of Minnesota, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Sergey Fomin University of Michigan, USA Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Vasudevan Srinivas Tata Inst. of Fund. Research, India Andrew Granville Université de Montréal, Canada J. Toby Stafford University of Michigan, USA Joseph Gubeladze San Francisco State University, USA Ravi Vakil Stanford University, USA Roger Heath-Brown Oxford University, UK Michel van den Bergh Hasselt University, Belgium Craig Huneke University of Virginia, USA Marie-France Vignéras Université Paris VII, France Kiran S. Kedlaya Univ. of California, San Diego, USA Kei-Ichi Watanabe Nihon University, Japan János Kollár Princeton University, USA Efim Zelmanov University of California, San Diego, USA Yuri Manin Northwestern University, USA Shou-Wu Zhang Princeton University, USA Philippe Michel École Polytechnique Fédérale de Lausanne PRODUCTION [email protected] Silvio Levy, Scientific Editor See inside back cover or msp.org/ant for submission instructions.
    [Show full text]
  • The Galois Group of a Stable Homotopy Theory
    The Galois group of a stable homotopy theory Submitted by Akhil Mathew in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Harvard University March 24, 2014 Advisor: Michael J. Hopkins Contents 1. Introduction 3 2. Axiomatic stable homotopy theory 4 3. Descent theory 14 4. Nilpotence and Quillen stratification 27 5. Axiomatic Galois theory 32 6. The Galois group and first computations 46 7. Local systems, cochain algebras, and stacks 59 8. Invariance properties 66 9. Stable module 1-categories 72 10. Chromatic homotopy theory 82 11. Conclusion 88 References 89 Email: [email protected]. 1 1. Introduction Let X be a connected scheme. One of the basic arithmetic invariants that one can extract from X is the ´etale fundamental group π1(X; x) relative to a \basepoint" x ! X (where x is the spectrum of a separably closed field). The fundamental group was defined by Grothendieck [Gro03] in terms of the category of finite, ´etalecovers of X. It provides an analog of the usual fundamental group of a topological space (or rather, its profinite completion), and plays an important role in algebraic geometry and number theory, as a precursor to the theory of ´etalecohomology. From a categorical point of view, it unifies the classical Galois theory of fields and covering space theory via a single framework. In this paper, we will define an analog of the ´etalefundamental group, and construct a form of the Galois correspondence, in stable homotopy theory. For example, while the classical theory of [Gro03] enables one to define the fundamental (or Galois) group of a commutative ring, we will define the fundamental group of the homotopy-theoretic analog: an E1-ring spectrum.
    [Show full text]