DERIVED ALGEBRAIC GEOMETRY 1. Introduction 1.1. Bezout's Theorem
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Cohomological Descent on the Overconvergent Site
COHOMOLOGICAL DESCENT ON THE OVERCONVERGENT SITE DAVID ZUREICK-BROWN Abstract. We prove that cohomological descent holds for finitely presented crystals on the overconvergent site with respect to proper or fppf hypercovers. 1. Introduction Cohomological descent is a robust computational and theoretical tool, central to p-adic cohomology and its applications. On one hand, it facilitates explicit calculations (analo- gous to the computation of coherent cohomology in scheme theory via Cechˇ cohomology); on another, it allows one to deduce results about singular schemes (e.g., finiteness of the cohomology of overconvergent isocrystals on singular schemes [Ked06]) from results about smooth schemes, and, in a pinch, sometimes allows one to bootstrap global definitions from local ones (for example, for a scheme X which fails to embed into a formal scheme smooth near X, one actually defines rigid cohomology via cohomological descent; see [lS07, comment after Proposition 8.2.17]). The main result of the series of papers [CT03,Tsu03,Tsu04] is that cohomological descent for the rigid cohomology of overconvergent isocrystals holds with respect to both flat and proper hypercovers. The barrage of choices in the definition of rigid cohomology is burden- some and makes their proofs of cohomological descent very difficult, totaling to over 200 pages. Even after the main cohomological descent theorems [CT03, Theorems 7.3.1 and 7.4.1] are proved one still has to work a bit to get a spectral sequence [CT03, Theorem 11.7.1]. Actually, even to state what one means by cohomological descent (without a site) is subtle. The situation is now more favorable. -
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 . -
A Surface of Maximal Canonical Degree 11
A SURFACE OF MAXIMAL CANONICAL DEGREE SAI-KEE YEUNG Abstract. It is known since the 70's from a paper of Beauville that the degree of the rational canonical map of a smooth projective algebraic surface of general type is at most 36. Though it has been conjectured that a surface with optimal canonical degree 36 exists, the highest canonical degree known earlier for a min- imal surface of general type was 16 by Persson. The purpose of this paper is to give an affirmative answer to the conjecture by providing an explicit surface. 1. Introduction 1.1. Let M be a minimal surface of general type. Assume that the space of canonical 0 0 sections H (M; KM ) of M is non-trivial. Let N = dimCH (M; KM ) = pg, where pg is the geometric genus. The canonical map is defined to be in general a rational N−1 map ΦKM : M 99K P . For a surface, this is the most natural rational mapping to study if it is non-trivial. Assume that ΦKM is generically finite with degree d. It is well-known from the work of Beauville [B], that d := deg ΦKM 6 36: We call such degree the canonical degree of the surface, and regard it 0 if the canonical mapping does not exist or is not generically finite. The following open problem is an immediate consequence of the work of [B] and is implicitly hinted there. Problem What is the optimal canonical degree of a minimal surface of general type? Is there a minimal surface of general type with canonical degree 36? Though the problem is natural and well-known, the answer remains elusive since the 70's. -
Hirschdx.Pdf
130 5. Degrees, Intersection Numbers, and the Euler Characteristic. 2. Intersection Numbers and the Euler Characteristic M c: Jr+1 Exercises 8. Let be a compad lHlimensionaJ submanifold without boundary. Two points x, y E R"+' - M are separated by M if and only if lkl(x,Y}.MI * O. lSee 1. A complex polynomial of degree n defines a map of the Riemann sphere to itself Exercise 7.) of degree n. What is the degree of the map defined by a rational function p(z)!q(z)? 9. The Hopfinvariant ofa map f:5' ~ 5' is defined to be the linking number Hlf) ~ 2. (a) Let M, N, P be compact connected oriented n·manifolds without boundaries Lk(g-l(a),g-l(b)) (see Exercise 7) where 9 is a c~ map homotopic 10 f and a, b are and M .!, N 1. P continuous maps. Then deg(fg) ~ (deg g)(deg f). The same holds distinct regular values of g. The linking number is computed in mod 2 if M, N, P are not oriented. (b) The degree of a homeomorphism or homotopy equivalence is ± 1. f(c) * a, b. *3. Let IDl. be the category whose objects are compact connected n-manifolds and whose (a) H(f) is a well-defined homotopy invariant off which vanishes iff is nuD homo- topic. morphisms are homotopy classes (f] of maps f:M ~ N. For an object M let 7t"(M) (b) If g:5' ~ 5' has degree p then H(fg) ~ pH(f). be the set of homotopy classes M ~ 5". -
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]. -
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. -
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. -
ON the CONSTRUCTION of NEW TOPOLOGICAL SPACES from EXISTING ONES Consider a Function F
ON THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM EXISTING ONES EMILY RIEHL Abstract. In this note, we introduce a guiding principle to define topologies for a wide variety of spaces built from existing topological spaces. The topolo- gies so-constructed will have a universal property taking one of two forms. If the topology is the coarsest so that a certain condition holds, we will give an elementary characterization of all continuous functions taking values in this new space. Alternatively, if the topology is the finest so that a certain condi- tion holds, we will characterize all continuous functions whose domain is the new space. Consider a function f : X ! Y between a pair of sets. If Y is a topological space, we could define a topology on X by asking that it is the coarsest topology so that f is continuous. (The finest topology making f continuous is the discrete topology.) Explicitly, a subbasis of open sets of X is given by the preimages of open sets of Y . With this definition, a function W ! X, where W is some other space, is continuous if and only if the composite function W ! Y is continuous. On the other hand, if X is assumed to be a topological space, we could define a topology on Y by asking that it is the finest topology so that f is continuous. (The coarsest topology making f continuous is the indiscrete topology.) Explicitly, a subset of Y is open if and only if its preimage in X is open. With this definition, a function Y ! Z, where Z is some other space, is continuous if and only if the composite function X ! Z is continuous. -
Differential Graded Lie Algebras and Deformation Theory
Differential graded Lie algebras and Deformation theory Marco Manetti 1 Differential graded Lie algebras Let L be a lie algebra over a fixed field K of characteristic 0. Definition 1. A K-linear map d : L ! L is called a derivation if it satisfies the Leibniz rule d[a; b] = [da; b]+[a; db]. n d P dn Lemma. If d is nilpotent (i.e. d = 0 for n sufficiently large) then e = n≥0 n! : L ! L is an isomorphism of Lie algebras. Proof. Exercise. Definition 2. L is called nilpotent if the descending central series L ⊃ [L; L] ⊃ [L; [L; L]] ⊃ · · · stabilizes at 0. (I.e., if we write [L]2 = [L; L], [L]3 = [L; [L; L]], etc., then [L]n = 0 for n sufficiently large.) Note that in the finite-dimensional case, this is equivalent to the other common definition that for every x 2 L, adx is nilpotent. We have the Baker-Campbell-Hausdorff formula (BCH). Theorem. For every nilpotent Lie algebra L there is an associative product • : L × L ! L satisfying 1. functoriality in L; i.e. if f : L ! M is a morphism of nilpotent Lie algebras then f(a • b) = f(a) • f(b). a P an 2. If I ⊂ R is a nilpotent ideal of the associative unitary K-algebra R and for a 2 I we define e = n≥0 n! 2 R, then ea•b = ea • eb. Heuristically, we can write a • b = log(ea • eb). Proof. Exercise (use the computation of www.mat.uniroma1.it/people/manetti/dispense/BCHfjords.pdf and the existence of free Lie algebras). -
Etale Homotopy Theory of Non-Archimedean Analytic Spaces
Etale homotopy theory of non-archimedean analytic spaces Joe Berner August 15, 2017 Abstract We review the shape theory of ∞-topoi, and relate it with the usual cohomology of locally constant sheaves. Additionally, a new localization of profinite spaces is defined which allows us to extend the ´etale realization functor of Isaksen. We apply these ideas to define an ´etale homotopy type functor ´et(X ) for Berkovich’s non-archimedean analytic spaces X over a complete non-archimedean field K and prove some properties of the construction. We compare the ´etale homotopy types coming from Tate’s rigid spaces, Huber’s adic spaces, and rigid models when they are all defined. 0 Introduction The ´etale homotopy type of an algebraic variety is one of many tools developed in the last century to use topological machinery to attack algebraic questions. For example sheaves and their cohomology, topoi, spectral sequences, Galois categories, algebraic K-theory, and Chow groups. The ´etale homotopy type enriches ´etale cohomology and torsor data from a collection of groups into a pro-space. This allows us to almost directly apply results from algebraic topol- ogy, with the book-keeping primarily coming from the difference between spaces and pro-spaces. In this paper, we apply this in the context of non-archimedean geometry. The major results are two comparison theorems for the ´etale homo- topy type of non-archimedean analytic spaces, and a theorem indentifying the homotopy fiber of a smooth proper map with the ´etale homotopy type of any geometric fiber. The first comparison theorem concerns analytifications of va- rieties, and states that after profinite completion away from the characteristic arXiv:1708.03657v1 [math.AT] 11 Aug 2017 of the field that the ´etale homotopy type of a variety agrees with that of its analytification. -
New Ideas in Algebraic Topology (K-Theory and Its Applications)
NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems. -
Derived Differential Geometry
The definition of schemes Different kinds of spaces in algebraic geometry Some basics of category theory What is derived geometry? Moduli spaces and moduli functors Algebraic spaces and (higher) stacks Derived Differential Geometry Lecture 1 of 14: Background material in algebraic geometry and category theory Dominic Joyce, Oxford University Summer 2015 These slides, and references, etc., available at http://people.maths.ox.ac.uk/∼joyce/DDG2015 1 / 48 Dominic Joyce, Oxford University Lecture 1: Algebraic Geometry and Category Theory The definition of schemes Different kinds of spaces in algebraic geometry Some basics of category theory What is derived geometry? Moduli spaces and moduli functors Algebraic spaces and (higher) stacks Plan of talk: 1 Different kinds of spaces in algebraic geometry 1.1 The definition of schemes 1.2 Some basics of category theory 1.3 Moduli spaces and moduli functors 1.4 Algebraic spaces and (higher) stacks 2 / 48 Dominic Joyce, Oxford University Lecture 1: Algebraic Geometry and Category Theory The definition of schemes Different kinds of spaces in algebraic geometry Some basics of category theory What is derived geometry? Moduli spaces and moduli functors Algebraic spaces and (higher) stacks 1. Different kinds of spaces in algebraic geometry Algebraic geometry studies spaces built using algebras of functions. Here are the main classes of spaces studied by algebraic geometers, in order of complexity, and difficulty of definition: Smooth varieties (e.g. Riemann surfaces, or algebraic complex n manifolds such as CP . Smooth means nonsingular.) n n Varieties (at their most basic, algebraic subsets of C or CP . 2 Can have singularities, e.g.