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Derived Algebraic Geometry XIV: Representability Theorems

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Derived Algebraic Geometry XIV: Representability Theorems Derived Algebraic Geometry XIV: Representability Theorems March 14, 2012 Contents 1 The Cotangent Complex 6 1.1 The Cotangent Complex of a Spectrally Ringed 1-Topos . 7 1.2 The Cotangent Complex of a Spectral Deligne-Mumford Stack . 10 1.3 The Cotangent Complex of a Functor . 14 2 Properties of Moduli Functors 22 2.1 Nilcomplete, Cohesive, and Integrable Functors . 22 2.2 Relativized Properties of Functors . 33 2.3 Finiteness Conditions on Functors . 37 2.4 Moduli of Spectral Deligne-Mumford Stacks . 47 3 Representability Theorems 55 3.1 From Classical Algebraic Geometry to Spectral Algebraic Geometry . 55 3.2 Artin's Representability Theorem . 61 3.3 Application: Existence of Weil Restrictions . 68 3.4 Example: The Picard Functor . 76 4 Tangent Complexes and Dualizing Modules 86 4.1 The Tangent Complex . 87 4.2 Dualizing Modules . 93 4.3 Existence of Dualizing Modules . 97 4.4 A Linear Representability Criterion . 104 4.5 Existence of the Cotangent Complexes . 107 1 Introduction Let R be a commutative ring and let X be an R-scheme. Suppose that we want to present X explicitly. We might do this by choosing a covering of X by open subschemes fUαgα2I , where each Uα is an affine scheme given by the spectrum of a commutative R-algebra Aα. Let us suppose for simplicity that each intersection −1 Uα \Uβ is itself an affine scheme, given as the spectrum of Aα[xα,β] for some element xα,β 2 Aα. To describe X, we need to specify the following data: (a) For each α 2 I, a commutative R-algebra Aα. Such an algebra might be given by generators and relations as a quotient R[x1; : : : ; xn]=(f1(x1; : : : ; xn); : : : ; fm(x1; : : : ; xm)) for some collection of polynomials fi. (b) For every pair of indices α; β 2 I, a pair of elements xα,β 2 Aα and xβ,α 2 Aβ, together with an −1 −1 R-algebra isomorphism φα,β : Aα[xα,β] ' Aβ[xβ,α]. Moreover, the isomorphisms φα,β should be the identity when α = β, and satisfy the following cocycle condition: −1 −1 −1 (c) Given α; β; γ 2 I, the commutative ring Aγ [xγ,β; φβ,γ (xβ,α) ] should be a localization of Aγ [xγ,α]. Moreover, the composite map −1 φα,γ −1 −1 −1 Aα ! Aα[xα,γ ] ! Aγ [xγ,α] ! Aγ [xγ,β; φβ,γ (xβ,α) ] should be obtained by composing (localizations of) φα,β and φβ,γ . In [52], we introduced the notion of a spectral scheme. The definition of a spectral scheme is entirely analogous the classical notion of a scheme. However, the analogues of (a), (b), and (c) are much more complicated in the spectral setting. For example, giving an affine spectral scheme over a commutative ring R is equivalent to giving an E1-algebra over R. These are generally quite difficult to describe using generators and relations. For example, the polynomial ring R[x] generally does not have a finite presentation as an E1-algebra over R, unless we assume that R has characteristic zero. These complications are amplified when we pass to the non-affine situation. In the spectral setting, (b) requires us to construct equivalences between E1-algebras, which are often difficult to describe. Moreover, since E1-rings form an 1-category rather than an ordinary category, the analogue of the cocycle condition described in (c) is not a condition −1 −1 but an additional datum (namely, a homotopy between two E1-algebra maps Aα ! Aγ [xγ,β; φβ,γ (xβ,α) ] for every triple α; β; γ 2 I), which must be supplemented by additional coherence data involving four-fold intersections and beyond. For these reasons, it is generally very difficult to provide \hands-on" constructions in the setting of spectral algebraic geometry. Fortunately, there is another approach to describing a scheme X. Rather than trying to explicitly con- struct the commutative rings associated to some affine open covering of X, one can instead consider the functor FX represented by X, given by the formula FX (R) = Hom(Spec R; X). The scheme X is determined by the functor FX up to canonical isomorphism. The situation for spectral schemes is entirely analogous: cn every spectral scheme X determines a functor FX : CAlg ! S, and the construction X 7! FX deter- mines a fully faithful embedding from the 1-category of spectral schemes to the 1-category Fun(CAlgcn; S) (Theorem V.2.4.1). Generally speaking, it is much easier to describe a spectral scheme (or spectral Deligne- Mumford stack) X = (X; OX) by specifying the functor FX than it is to specify the structure sheaf OX explicitly. This motivates the following general question: Question 0.0.1. Given a functor F : CAlgcn ! S, under what circumstances is F representable by a spectral Deligne-Mumford stack? 2 In the setting of classical algebraic geometry, the analogous question is addressed by the following theorem of Artin: Theorem 1 (Artin Representability Theorem). Let R be a Grothendieck ring (see Definition 0.0.4) and let F be functor of commutative R-algebras to the category of sets. Then F is representable by an algebraic space which is locally of finite presentation over R if the following conditions are satisfied: (1) The functor F commutes with filtered colimits. (2) The functor F is a sheaf for the ´etaletopology. (3) If B is a complete local Noetherian R-algebra with maximal ideal m, then the natural map F(B) ! lim F(B=mn) is bijective. − (4) The functor F admits an obstruction theory and a deformation theory, and satisfies Schlessinger's criteria for formal representability. (5) The diagonal map F ! F ×Spec R F is representable by algebraic spaces (which must be quasi-compact schemes, if we wish to require that F is quasi-separated). This result is of both philosophical and practical interest. Since conditions (1) through (5) are reasonable expectations for any functor F of a reasonably geometric nature, Theorem 1 provides evidence that the theory of algebraic spaces is natural and robust (in other words, that it exactly captures some intuitive notion of \geometricity"). On the other hand, if we are given a functor F, it is usually reasonably easy to check whether or not Artin's criteria are satisfied. Consequently, Theorem 1 can be used to construct a great number of moduli spaces. Remark 0.0.2. We refer the reader to [1] for the original proof of Theorem 1. Note that in its original formulation, condition (3) was replaced by the weaker requirement that the map F(B) ! lim F(B=mn) has − dense image (with respect to the inverse limit topology). Moreover, Artin's theorem proof required a stronger assumption on the commutative ring R. For a careful discussion of the removal of this hypothesis, we refer the reader to [8]. Our goal in this paper is to prove an analogue of Theorem 1 in the setting of spectral algebraic geometry. Let R be an Noetherian E1-ring such that π0R is a Grothendieck ring, and suppose we are given a functor cn F : CAlgR ! S. Our main result (Theorem 2) supplies necessary and sufficient conditions for F to be representable by a spectral Deligne-Mumford n-stack which is locally almost of finite presentation over R. For the most part, these conditions are natural analogues of the hypotheses of Theorem 1. The main difference is in the formulation of condition (4). In the setting of Artin's original theorem, a deformation and obstruction theory are auxiliary constructs which are not uniquely determined by the functor F. The meaning of these conditions are clarified by working in the spectral setting: they are related to the problem of extending the functor F to E1-rings which are nondiscrete. In this setting, the analogue of condition (4) is that the functor F should be infinitesimally cohesive (Definition 2.1.9) and admits a cotangent complex (Definition 1.3.13). This assumption is more conceptually satisfying: in the setting of spectral algebraic geometry, the cotangent complex of a functor F is uniquely determined by F. Let us now outline the contents of the this paper. We begin in x1 with a general discussion of the cotangent complex formalism. Recall that if A is an E1-ring, then the cotangent complex LA is an A- module spectrum which is universal among those A-modules for which the projection map A ⊕ LA ! A admits a section (which we can think of as a derivation of A into LA; see xA.8.3.4). This definition can be globalized: if X is an arbitrary spectral Deligne-Mumford stack, then there is a quasi-coherent sheaf LX with ∗ the following property: for every ´etalemap u : Spec A ! X, the pullback u LX is the quasi-coherent sheaf on Spec A associated to the A-module LA. Moreover, we will explain how to describe the quasi-coherent sheaf LX directly in terms of the functor represented by X, and use this description to define the cotangent complex for a large class of functors F : CAlgcn ! S. 3 cn Let F : CAlg ! S be a functor which admits a cotangent complex LF. By definition, LF controls the deformation theory of F along trivial square-zero extensions. That is, if A is a connective E1-ring and M is a connective A-module, then the space F(A ⊕ M) is determined by the space F(A) and cotangent complex LF. However, for many applications of deformation theory, this is not enough: we would like also to describe the spaces F(Ae), where Ae is a nontrivial square-zero extension of A by M.
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