DOCSLIB.ORG

The Logarithmic Cotangent Complex

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Logarithmic Cotangent Complex The logarithmic cotangent complex Martin C. Olsson Received: date / Revised version: date – c Springer-Verlag 2005 Abstract. We define the cotangent complex of a morphism of fine log schemes, prove that it is functorial, and construct under certain restrictions a transitivity triangle. We also discuss its relationship with deformation theory. 1. Introduction 1.1. The goal of this paper is to develop a logarithmic version of the the- ory of cotangent complexes for morphisms of schemes ([6]). Our interest in the development of such a theory comes from two sources. First, we want to generalize the log smooth deformation theory due to K. Kato ([9], 3.14, see also [8]) in order to understand more complicated deformation theory problems in the logarithmic category (see section 5). Such deformation theory problems arise for example in the study of log Gromov–Witten the- ory ([12], [19]). Secondly, we are hopeful that the development of a theory of log cotangent complexes can be useful in the study of intersection the- ory on degenerations of varieties. In fact, the beginnings of a theory of log cotangent complexes can be found in the recent work of K. Kato and T. Saito ([10]) on logarithmic localized intersection product, where it is used to study a proper scheme over a complete discrete valuation ring whose reduced closed fiber is a divisor with normal crossings. In this paper, we associate to every morphism of fine log schemes f : X → Y ([9]) a projective system ≥−n−1 ≥−n ≥0 LX/Y = (· · · −→ LX/Y −→ LX/Y −→ · · · −→ LX/Y ), ≥−n [−n,0] where each LX/Y is an essentially constant ind-object in D (OX ) (the derived category of sheaves of OX -modules with support in [−n, 0]). We call the system LX/Y the cotangent complex of f, and we show that it has the following properties (1.1 (i))–(1.1 (v)): M. Olsson School of Mathematics, Institute for Advanced Study , Einstein Drive, Princeton, NJ 08540 USA 2 Martin C. Olsson ≥−n−1 ≥−n (1.1 (i)). For any n ≥ 0, the natural map τ≥−nLX/Y → LX/Y is an isomorphism. ∗ (1.1 (ii)). If f is strict (i.e. the map f MY → MX is an isomorphism), ◦ ◦ then (τ≥−nL ◦ ◦) represents LX/Y , where X (resp. Y ) denotes the un- X/Y derlying scheme of X (resp. Y ). 1 (1.1 (iii)). If f is log smooth, then the sheaf of log differentials ΩX/Y represents LX/Y . (1.1 (iv)). If X0 −−−−→a X 0 (1.1.1) f y yf Y 0 −−−−→b Y is a commutative diagram of fine log schemes, then there is a natural map ∗ La LX/Y −→ LX0/Y 0 . (1.1.2) If (1.1.1) is cartesian and if f is log flat, then (1.1.2) is an isomorphism, and if in addition the composite X0 → Y 0 → Y satisfies condition (T ) below (1.4), then the map 0∗ ∗ Lf LY 0/Y ⊕ La LX/Y −→ LX0/Y (1.1.3) is an isomorphism. (1.1 (v)). Given a composite f g X −−−−→ Y −−−−→ Z (1.1.4) satisfying condition (T ) below (1.4), there is a natural map ∗ LX/Y −→ Lf LY/Z [1]. (1.1.5) such that the resulting triangle ∗ ∗ Lf LY/Z −→ LX/Z −→ LX/Y −→ Lf LY/Z [1] (1.1.6) is distinguished (in the sense of (1.8 (iii)). 1.2. As an application of this log cotangent complex, we explain how to compute crystalline cohomology of “log complete intersections” in terms of the log cotangent complex. This application is modelled on the ap- proach of Illusie in ([7], VIII). The logarithmic cotangent complex 3 1.3. As shown by an argument of W. Bauer (section 7), there does not exist a reasonable theory of cotangent complexes for log schemes for which one has a distinguished triangle (1.1.6) for every composite as in (1.1.4). In this paper, we construct the distinguished triangle (1.1.6) when (1.1.4) satisfies the following condition (which holds for example if g is log flat (3.14) or if f is an integral morphism (3.15)): 1.4 (Condition (T)). There exists a family of commutative diagrams π Xi XX ×Z Zi X ×Y Yi Xi f ? ? ? fi + πYi YY ×Z Zi Yi (1.4.1) g ? ? g π i Zi + ZZi such that ◦ ◦ ◦ (1.4 (i)). The schemes Xi, Y i, and Zi are all affine. (1.4 (ii)). The maps πXi , πYi , and πZi are all strict, and their underlying morphisms of schemes are flat and locally of finite presentation. ` ◦ ◦ (1.4 (iii)). The induced map i Xi → X is surjective. (1.4 (iv)). There exists charts βXi : QXi −→ MXi , βYi : QYi −→ MYi , βZi : QZi −→ MZi and injective maps θ1 : QZi −→ QYi , θ2 : QYi −→ QXi such that the diagrams βXi βYi QXi −−−−→ MXi QYi −−−−→ MY x x x x θ f b θ gb 2 i 1 i β β Yi ∗ Zi ∗ QYi −−−−→ fi MYi QZi −−−−→ gi MZi commute and j Tor (OZ ⊗ [Q ] [QX ], OY [G]) = 0 for all j > 0. OZ ⊗ [Q ] [QY ] i Z Zi Z i i i Z Zi Z i (1.4.2) 4 Martin C. Olsson gp gp Here G := Coker(Q → Q ) and OY [G] is viewed as a OZ ⊗ [QY ] Zi Yi i i Z[QZi ]Z i algebra via the map ∗ OZ ⊗ [QY ] −→ OY [G], t ⊗ e(q) 7−→ g (t)βQ (q) · gq¯, (1.4.3) i Z[QZi ] Z i i Yi where gq¯ denotes the image of q in G. Remark 1.5. As explained to us by Gabber, the vanishing condition (1.4.2) is equivalent to j Tor (OZ ⊗ [Q ] [QX ], OY ) = 0 for all j > 0. (1.5.1) OZ ⊗ [Q ] [QY ] i Z Zi Z i i i Z Zi Z i A proof of this equivalence is given at the end of section 4. 1.6. On the other hand, Gabber has shown that if one does not require 1 LX/S ' ΩX/S for X → S log smooth, then there does exist a theory of log cotangent complexes for which one has a distinguished triangle for all composites X → Y → Z. We explain Gabber’s approach in section 8. 1.7. The paper is organized as follows. In section 2 we construct certain algebraic stacks LogΓ classifying commutative diagrams of log structures of a fixed form (see (2.1) for Γ the precise definition of Log ). These stacks generalize the stacks LogS studied in ([14]), and are the main technical tool used in the construction of the functoriality morphism (1.1.2) and the transitivity triangle (1.1.6). We define LX/Y in section 3 and verify properties (1.1 (i))–(1.1 (v)), assuming a key proposition (3.10) concerning condition (T ). This propo- sition is proven in section 4. In section 5 we discuss the relationship between the logarithmic cotan- gent complex and deformation theory of log schemes generalizing ([6], III.1.2.3, III.2.1.7, III.2.2.4) to log schemes. This section relies heavily on the results of ([13]). In section 6 we apply the cotangent complex to study log versions of results in ([7], VIII) as mentioned above. In section 7 we discuss an argument of W. Bauer which shows that there does not exist a reasonable theory of cotangent complexes for log schemes for which one has a distinguished triangle (1.1.6) for every com- posite as in (1.1.4). Thus some condition, such as condition (T ), is needed in order to have (1.1 (v)). Finally we discuss in section 8 an alternate approach to defining a G log cotangent complex LX/S for a morphism of log schemes X → S due to Gabber. This approach has the advantage that one always has a dis- G tinguished triangle. Furthermore, there is a natural map LX/S → LX/S whose cone is cohomologically concentrated in degrees ≤ −3. This implies The logarithmic cotangent complex 5 G that for the standard deformation theory problems one can use LX/S in- stead of LX/S. We have also included an appendix which contains some basic re- sults about simplicial monoids which are needed in the study of Gabber’s cotangent complex. 1.8. Notation and prerequisites (i). Logarithmic geometry: We assume that the reader is familiar with logarithmic geometry at the level of the first two sections of ([9]) and with the paper ([14]). Throughout the paper, a log structure on a scheme X means a log structure on the ´etalesite Xet. If α : M → OX is a pre-log structure on a scheme X, then we denote by (Ma, α) (or just Ma) the associated log structure, and if (M, α) is a ∗ log structure on X then we denote by M the sheaf M/OX . If X is a log scheme, we usually denote its log structure by MX and its ◦ underlying scheme by X. If f : X → Y is a morphism of log schemes, we ◦ ◦ ◦ denote the underlying morphism of schemes by f : X → Y . For a fine log ◦ scheme X, we denote by LogX the algebraic stack over X whose objects are fine X-log schemes and whose morphisms (T1 → X) → (T2 → X) are b ∗ morphisms of X-log schemes g : T1 → T2 for which g : g MT2 → MT1 is an isomorphism ([14]). If f : T → X is a morphism of fine log schemes, ◦ then we denote by Lf : T → LogX the associated 1-morphism. Recall that a chart for a fine log structure M on a scheme X is a map P → M from the constant sheaf associated to a fine (i.e.
Load more

Recommended publications
  • 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]
  • THE DERIVED CATEGORY of a GIT QUOTIENT Contents 1. Introduction 871 1.1. Author's Note 874 1.2. Notation 874 2. the Main Theor
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 3, July 2015, Pages 871–912 S 0894-0347(2014)00815-8 Article electronically published on October 31, 2014 THE DERIVED CATEGORY OF A GIT QUOTIENT DANIEL HALPERN-LEISTNER Dedicated to Ernst Halpern, who inspired my scientific pursuits Contents 1. Introduction 871 1.1. Author’s note 874 1.2. Notation 874 2. The main theorem 875 2.1. Equivariant stratifications in GIT 875 2.2. Statement and proof of the main theorem 878 2.3. Explicit constructions of the splitting and integral kernels 882 3. Homological structures on the unstable strata 883 3.1. Quasicoherent sheaves on S 884 3.2. The cotangent complex and Property (L+) 891 3.3. Koszul systems and cohomology with supports 893 3.4. Quasicoherent sheaves with support on S, and the quantization theorem 894 b 3.5. Alternative characterizations of D (X)<w 896 3.6. Semiorthogonal decomposition of Db(X) 898 4. Derived equivalences and variation of GIT 901 4.1. General variation of the GIT quotient 903 5. Applications to complete intersections: Matrix factorizations and hyperk¨ahler reductions 906 5.1. A criterion for Property (L+) and nonabelian hyperk¨ahler reduction 906 5.2. Applications to derived categories of singularities and abelian hyperk¨ahler reductions, via Morita theory 909 References 911 1. Introduction We describe a relationship between the derived category of equivariant coherent sheaves on a smooth projective-over-affine variety, X, with a linearizable action of a reductive group, G, and the derived category of coherent sheaves on a GIT quotient, X//G, of that action.
    [Show full text]
  • [Math.AG] 5 Feb 2006 Omlvra Aiyfrapoe Ii Nltcspace
    DEFORMATION THEORY OF RIGID-ANALYTIC SPACES ISAMU IWANARI Abstract. In this paper, we study a deformation theory of rigid an- alytic spaces. We develop a theory of cotangent complexes for rigid geometry which fits in with our deformations. We then use the com- plexes to give a cohomological description of infinitesimal deformation of rigid analytic spaces. Moreover, we will prove an existence of a formal versal family for a proper rigid analytic space. Introduction The purpose of the present paper is to formulate and develop a deforma- tion theory of rigid analytic spaces. We will give a cohomological description of infinitesimal deformations. Futhermore we will prove an existence of a formal versal family for a proper rigid analytic space. The original idea of deformations goes back to Kodaira and Spencer. They developed the theory of deformations of complex manifolds. Their deformation theory of complex manifolds is of great importance and becomes a standard tool for the study of complex analytic spaces. Our study of deformations of rigid analytic spaces is motivated by analogy to the case of complex analytic spaces. More precisely, our interest in the development of such a theory comes from two sources. First, we want to construct an analytic moduli theory via rigid analytic stacks by generalizing the classical deformation theory due to Kodaira-Spencer, Kuranishi and Grauert to the non-Archimedean theory. This viewpoint will be discussed in Section 5. Secondly, we may hope that our theory is useful in arithmetic geometry no less than the complex-analytic deformation theory is very useful in the study of complex-analytic spaces.
    [Show full text]
  • Arxiv:Math/0609151V1 [Math.AC] 5 Sep 2006 Stesbeto Hs Oe.Te R Osbttt O Eith for Substitute No Are They Notes
    Contemporary Mathematics Andr´e-Quillen homology of commutative algebras Srikanth Iyengar Abstract. These notes are an introduction to basic properties of Andr´e- Quillen homology for commutative algebras. They are an expanded version of my lectures at the summer school: Interactions between homotopy theory and algebra, University of Chicago, 26th July - 6th August, 2004. The aim is to give fairly complete proofs of characterizations of smooth homomorphisms and of locally complete intersection homomorphisms in terms of vanishing of Andr´e-Quillen homology. The choice of the material, and the point of view, are guided by these goals. Contents 1. Introduction 1 2. K¨ahler differentials 3 3. Simplicial algebras 6 4. Simplicial resolutions 9 5. The cotangent complex 15 6. Basic properties 19 7. Andr´e-Quillen homology and the Tor functor 22 8. Locally complete intersection homomorphisms 23 9. Regular homomorphisms 28 References 31 arXiv:math/0609151v1 [math.AC] 5 Sep 2006 1. Introduction In the late 60’s Andr´eand Quillen introduced a (co)-homology theory for com- mutative algebras that now goes by the name of Andr´e-Quillen (co)-homology. This is the subject of these notes. They are no substitute for either the panoramic view that [22] provides, or the detailed exposition in [23] and [2]. My objective is to provide complete proofs of characterizations of two impor- tant classes of homomorphisms of noetherian rings: regular homomorphisms and locally complete intersection homomorphisms, in terms of Andr´e-Quillen homology. However, I have chosen to treat only the case when the homomorphism is essentially Supported by the National Science Foundation, under grant DMS 0442242.
    [Show full text]
  • A GUIDE to the LITERATURE 03B0 Contents 1. Short Introductory Articles 1 2. Classic References 1 3. Books and Online Notes 2 4
    A GUIDE TO THE LITERATURE 03B0 Contents 1. Short introductory articles 1 2. Classic references 1 3. Books and online notes 2 4. Related references on foundations of stacks 3 5. Papers in the literature 3 5.1. Deformation theory and algebraic stacks 4 5.2. Coarse moduli spaces 5 5.3. Intersection theory 6 5.4. Quotient stacks 7 5.5. Cohomology 9 5.6. Existence of finite covers by schemes 10 5.7. Rigidification 11 5.8. Stacky curves 11 5.9. Hilbert, Quot, Hom and branchvariety stacks 12 5.10. Toric stacks 13 5.11. Theorem on formal functions and Grothendieck’s Existence Theorem 14 5.12. Group actions on stacks 14 5.13. Taking roots of line bundles 14 5.14. Other papers 14 6. Stacks in other fields 15 7. Higher stacks 15 8. Other chapters 15 References 16 1. Short introductory articles 03B1 • Barbara Fantechi: Stacks for Everybody [Fan01] • Dan Edidin: What is a stack? [Edi03] • Dan Edidin: Notes on the construction of the moduli space of curves [Edi00] • Angelo Vistoli: Intersection theory on algebraic stacks and on their moduli spaces, and especially the appendix. [Vis89] 2. Classic references 03B2 • Mumford: Picard groups of moduli problems [Mum65] This is a chapter of the Stacks Project, version fac02ecd, compiled on Sep 14, 2021. 1 A GUIDE TO THE LITERATURE 2 Mumford never uses the term “stack” here but the concept is implicit in the paper; he computes the picard group of the moduli stack of elliptic curves. • Deligne, Mumford: The irreducibility of the space of curves of given genus [DM69] This influential paper introduces “algebraic stacks” in the sense which are now universally called Deligne-Mumford stacks (stacks with representable diagonal which admit étale presentations by schemes).
    [Show full text]
  • Lecture 5 the Cotangent Complex and Nisnevich Descent Last Lecture We
    Lecture 5 The cotangent complex and Nisnevich descent Last lecture we saw the proof of Zariski descent in algebraic K-theory. Once we define the notion of Nisnevich square, the proof for Nisnevich descent will be exactly the same. To this end, we will begin this lecture by introducing the notion of the cotangent complex and ´etale morphisms in derived algebraic geometry. Since we will need to work with the cotangent complex later in the course, we will take this opportunity to cover it in some detail. 1. Derived derivations. Recall that the module of (algebraic) K¨ahlerdifferentials admits a universal property in terms of derivations. The cotangent complex can be viewed as a derived version of K¨ahlerdifferentials, and admits a universal property in terms of a derived version of derivations. 1.1. Recall that for an ordinary commutative ring R and an ordinary R-module M, one can define a new commutative ring R ⊕ M, the trivial square-zero extension of R by M, where the multiplication is defined by the formula (r; m) · (r0; m0) = (rr0; r0m + rm0): cn Now let R 2 SCRing and M 2 ModR be a connective R-module. One can then construct a new simplicial commutative ring whose underlying R-module is the direct sum R ⊕ M, and such that the ring structure on π0(R) ⊕ π0(M) is given by the formula above. Remark 1.2. There are various possible ways to actually construct R ⊕ M 2 SCRing. Of course if we choose strict models for the 1-category SCRing then this is trivial.
    [Show full text]
  • Derived Algebraic Geometry
    DERIVED ALGEBRAIC GEOMETRY BERTRAND TOEN¨ Abstract. This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization. Contents Introduction1 1. Selected pieces of history6 2. The notion of a derived scheme 17 2.1. Elements of the language of 1-categories 20 2.2. Derived schemes 30 3. Derived schemes, derived moduli, and derived stacks 35 3.1. Some characteristic properties of derived schemes 35 3.2. Derived moduli problems and derived schemes 40 3.3. Derived moduli problems and derived Artin stacks 43 3.4. Derived geometry in other contexts 48 4. The formal geometry of derived stacks 50 4.1. Cotangent complexes and obstruction theory 50 4.2. The idea of formal descent 52 4.3. Tangent dg-lie algebras 54 4.4. Derived loop spaces and algebraic de Rham theory 55 5. Symplectic, Poisson and Lagrangian structures in the derived setting 59 5.1. Forms and closed forms on derived stacks 59 5.2. Symplectic and Lagrangian structures 63 5.3. Existence results 67 5.4. Polyvectors and shifted Poisson structures 71 References 76 arXiv:1401.1044v2 [math.AG] 12 Sep 2014 Introduction Derived algebraic geometry is an extension of algebraic geometry whose main purpose is to propose a setting to treat geometrically special situations (typically bad intersections, quotients by bad actions,. ), as opposed to generic situations (transversal intersections, quotients by free and proper actions,. ). In order to present the main flavor of the subject we will start this introduction by focussing on an emblematic situation in the context of algebraic geometry, or in geometry in Bertrand To¨en,Universit´ede Montpellier 2, Case courrier 051, B^at9, Place Eug`eneBataillon, Montpellier Cedex 5, France.
    [Show full text]
  • DEFINING the COTANGENT COMPLEX Opening Nonsense I've Recently Learned a Bit About the General Framework of Model Categories. I
    DEFINING THE COTANGENT COMPLEX ARPON RAKSIT original: December 17, 2013 updated: January 17, 2015 Opening nonsense I’ve recently learned a bit about the general framework of model categories. It’s sort of amazing, in the sense that seemingly a lot of the basic homotopy theory of spaces which I’ve seen can be axiomatised in a totally formal and categorical way. This includes something like Whitehead’s theorem (and its converse)—that a map of CW complexes is a homotopy equivalence if and only if it’s a weak homotopy equivalence—which I never would have thought of as a categorical fact. Anyway, this axiomatisation and generalisation is beautiful and all, but of course one hopes that it leads to new homotopy theory, in addition to providing a nice framework for existing homotopy theory to live in. Obviously it did. My goal in writing these notes was to learn about one of the earliest examples of this, the development of a (co)homology theory of commutative algebras by Andr´e[And74] and Quillen [Qui68]. Notation. • Denote by Set the category of sets. • All rings (and algebras) are commutative. • For a ring R denote by Mod(R) the category of R-modules and Alg(R) the category of R-algebras. • For a category C denote by sC the category of simplicial objects in C. 1. Homology and abelianisation In hopes of generalising the notion of homology via the theory of model categories, we first ask the following. 1.1. Question. How can we view singular homology in a homotopyish way? 1.2.
    [Show full text]
  • The Tangent Complex and Hochschild Cohomology Of
    THE TANGENT COMPLEX AND HOCHSCHILD COHOMOLOGY OF En-RINGS JOHN FRANCIS Abstract. In this work, we study the deformation theory of En-rings and the En analogue of the tangent complex, or topological Andr´e-Quillencohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence A[n − 1] ! T ! HH∗ (A)[n], relating A En the En-tangent complex and En-Hochschild cohomology of an En-ring A. We give two proofs: The first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as n−1 × the Lie algebras of a fiber sequence of derived algebraic groups, B A ! AutA ! AutBnA. n Here B A is an enriched (1; n)-category constructed from A, and En-Hochschild cohomology is realized as the infinitesimal automorphisms of BnA. These groups are associated to moduli problems in En+1-geometry, a less commutative form of derived algebraic geometry, in the sense of To¨en-Vezzosi and Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital En+1-algebra structure; in particular, the shifted tangent complex TA[−n] is a nonunital En+1-algebra. The En+1-algebra structure of this sequence extends the previously known E -algebra structure on HH∗ (A), given in the higher Deligne conjecture. In order to n+1 En establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed n-manifolds with coefficients given by En-algebras, constructed as a topological analogue of Beilinson-Drinfeld's chiral homology.
    [Show full text]
  • Mumford Stacks Withperfect Obstruction Theories
    Geometry & Topology 17 (2013) 73–92 msp Deriving Deligne–Mumford stacks with perfect obstruction theories TIMO SCHÜRG We show that every n–connective quasi-coherent obstruction theory on a Deligne– Mumford stack comes from the structure of a connective spectral Deligne–Mumford stack on the underlying topos. Working over a base ring containing the rationals, we obtain the corresponding result for derived Deligne–Mumford stacks. 14A20, 18G55; 55P43 1 Introduction Some moduli spaces playing an important role in enumerative geometry carry an additional structure. Apart from the cotangent complex, which controls deformations and obstructions of the objects parametrized by the moduli space, there sometimes exist another complex doing the same job. If the moduli space in question is very singular, the cotangent complex will have cohomology in arbitrary many degrees. In many cases, the replacement complex has much better finiteness properties, being locally isomorphic to a finite complex of vector bundles. If the replacement complex is perfect and of Tor–amplitude at most 1 it gives rise to the virtual fundamental class of Li and Tian[3], and Behrend and Fantechi[1], which is the key to actually producing numbers. Ever since this phenomenon was observed by Kontsevich[2], it was suspected that the replacement complex is a shadow of a derived structure on the moduli space. In the meantime, the foundations of derived algebraic geometry have been firmly laid out by Toën and Vezzosi[11], and Lurie[4]. Using these theories, in many examples derived moduli spaces having the ‘correct’ cotangent complex have been found (for examples see Toën’s overview[9]).
    [Show full text]
  • THE COTANGENT STACK 1. Introduction 1.1. Let
    THE COTANGENT STACK SAM RASKIN 1. Introduction 1.1. Let us fix our notations and conventions. 1.2. We refer to [LMB] for the basic background on stacks. Let us quickly recall: a stack over a scheme S is a sheaf of groupoids on the faithfully flat topology of S-schemes, where the presheaf requirement that the composition of the restriction maps is the restriction map of the composition is understood in the weak sense. Stacks form a 2-category with fiber products and there is a clear embedding of the category of schemes as a full subcategory. A morphism X −! Y is representable if for any map to Y from a scheme the pull-back of X along this map is an algebraic space. For representable morphisms, it makes sense to say it has a local property like flatness or smoothness, and we will call such morphisms flat or smooth withholding the word representable where convenient. An Artin (or algebraic) stack is a stack X for which there exists a smooth pre- sentation, i.e., a smooth, surjective map X −! X from a scheme X, and such that the diagonal map X −! X × X is representable and quasicompact. Note that to say the diagonal map is representable is equivalent to requiring that any morphism from a scheme to X is representable. By this condition, for any point x : S −! X from a scheme to a stack, one can make sense of the automorphisms of this point Aut(x) = S ×sX S, and if this stack is an Artin stack then this is naturally given the structure of a group scheme over S.
    [Show full text]
  • Derived Algebraic Geometry IV: Deformation Theory
    Derived Algebraic Geometry IV: Deformation Theory October 8, 2009 Contents 1 The Cotangent Complex: General Theory 4 1.1 Stable Envelopes and Tangent Bundles . 6 1.2 Relative Adjunctions . 10 1.3 The Tangent Correspondence . 16 1.4 The Relative Cotangent Complex . 21 1.5 The Tangent Bundle of the 1-Category of E1-Rings . 30 2 Cotangent Complexes of E1-Rings 39 2.1 Connectivity Estimates . 39 2.2 Finiteness Properties of the Cotangent Complex . 46 2.3 Etale´ Algebras . 50 3 Deformation Theory 52 3.1 Square-Zero Extensions . 54 3.2 Small Extensions . 56 3.3 Deformation Theory of E1-Rings . 61 3.4 Classification of Etale´ Algebras . 68 1 Introduction Let X = Spec A be a smooth affine algebraic variety, defined over the field C of complex numbers. The ∗ holomorphic cotangent bundle TX has the structure of an algebraic vector bundle over X, associated to a projective A-module ΩA=C of finite rank. The A-module ΩA=C is called the module of K¨ahlerdifferentials of A (relative to C). In order to describe it, we first recall a bit of terminology. Let M be an arbitrary A-module. A C-linear derivation from A into M is a map of complex vector spaces d : A ! M which satisfies the Leibniz rule d(ab) = a(db) + b(da). Let DerC(A; M) denote the set of all C-linear derivations of A into M. The module ΩA=C is the receptacle for the universal derivation; in other words, there exists a derivation d : A ! ΩA=C such that, for every A-module M, composition with d induces an isomorphism HomA(ΩA=C;M) ! DerC(A; M): More concretely, ΩA=C is generated (as an A-module) by symbols fdfgf2A, which are subject to the usual rules of calculus: d(f + g) = df + dg d(fg) = f(dg) + g(df) dλ = 0; λ 2 C: The module of K¨ahlerdifferentials ΩA=C is a fundamental invariant of the algebraic variety X, and plays an important role in the study of deformations of X.
    [Show full text]