Derived Algebraic Geometry XII: Proper Morphisms, Completions, and the Grothendieck Existence Theorem November 8, 2011 Contents 1 Generalities on Spectral Algebraic Spaces 5 1.1 Scallop Decompositions in the Separated Case . 5 1.2 Quasi-Finite Morphisms . 10 1.3 Quasi-Separatedness . 14 1.4 Points of Spectral Algebraic Spaces . 20 1.5 Quasi-Coherent Stacks and Local Compact Generation . 26 2 Noetherian Approximation 34 2.1 Truncated Category Objects . 35 2.2 Finitely n-Presented Morphisms . 37 2.3 Approximation of Spectral Deligne-Mumford Stacks . 40 2.4 Approximation of Quasi-Coherent Sheaves . 45 2.5 Descent of Properties along Filtered Colimits . 47 3 Properness 51 3.1 Strongly Proper Morphisms . 52 3.2 The Direct Image Theorem . 55 3.3 Proper Linear 1-Categories . 61 3.4 Valuative Criteria . 67 4 Completions of Modules 71 4.1 I-Nilpotent and I-Local Modules . 71 4.2 Completion of Modules . 76 4.3 Completion in the Noetherian Case . 81 5 Completions of Spectral Deligne-Mumford Stacks 87 5.1 Formal Completions . 88 ^ 5.2 Truncations in QCoh(XK )...................................... 95 5.3 The Grothendieck Existence Theorem . 103 5.4 Algebraizability of Formal Stacks . 105 6 Relationship with Formal Moduli Problems 111 6.1 Deformation Theory of Formal Thickenings . 112 6.2 Formal Spectra as Formal Moduli Problems . 117 6.3 Schlessinger's Criterion in Spectral Algebraic Geometry . 121 1 A Stone Duality 130 A.1 Stone Spaces . 130 A.2 Upper Semilattices . 136 A.3 Lattices and Boolean Algebras . 140 2 Introduction Let R be a Noetherian ring which is complete with respect to an ideal I, let X be a proper R-scheme, let X0 denote the closed subscheme X ×Spec R Spec R=I of X, and let X denote the formal scheme obtained by completing X along X0. A classical result of Grothendieck asserts that every coherent sheaf on X extends uniquely to a coherent sheaf of X. More precisely, the Grothendieck existence theorem (Corollary 5.1.6 of [8]) implies that the restriction functor Coh(X) ! Coh(X) is an equivalence of categories. Our primary objective in this paper is to prove a version of Grothendieck's existence theorem in the setting of spectral algebraic geometry. For this, we first need to develop a good theory of proper morphisms between spectral algebraic spaces. The main thing we need is the following version of the proper direct image theorem, which we prove in x3 (Theorem 3.2.2): (∗) If f : X ! Y is a proper morphism of spectral algebraic spaces which is locally almost of finite presentation and F 2 QCoh(X) is almost perfect, then the pushforward f∗ F again almost perfect. As a first step towards proving (∗), we might ask when the pushforward functor f∗ carries quasi-coherent sheaves on X to quasi-coherent sheaves on Y. This requires only very mild hypotheses: namely, that f is quasi-compact and quasi-separated. This follows from very general considerations about spectral Deligne- Mumford stacks, and will be proven in x1. One novel feature of assertion (∗) is that, unlike its classical counterpart, it does not require any Noethe- rian hypotheses on X or Y. Nevertheless, our proof of (∗) will proceed by reduction to the Noetherian case, where it can be deduced from the classical coherence theorem for (higher) direct image sheaves. To carry out the reduction, we need to develop the technique of Noetherian approximation in the setting of spectral algebraic geometry. This is the subject of x2. The second half of this paper is devoted to a study of formal completions in spectral algebraic geometry. To every spectral Deligne-Mumford stack X, we can associate an underlying topological space j X j, which ^ we study in x1.4. In x5, we will associate to every closed subset K ⊆ j X j a formal completion XK , which cn we regard as a functor from the 1-category CAlg of connective E1-rings. In particular, we have an ^ ^ 1-category of quasi-coherent sheaves QCoh(XK ) and a restriction functor QCoh(X) ! QCoh(XK ). Our main result asserts that, if X is proper and locally almost of finite presentation over a Noetherian E1-ring R which is complete with respect to an ideal I ⊆ π0R, and K ⊆ j X j is the closed subset determined by I, then the restriction functor ^ QCoh(X) ! QCoh(XK ) restricts to an equivalence of 1-categories on almost perfect objects (Theorem 5.3.2). The proof relies on (∗) together with some facts about completions of modules over Noetherian E1-rings, which we study in x4. In the final section of this paper (x6), we study the relationship between the formal completions studied in this paper and the formal moduli problems of [46]. In particular, we show that if a local Noetherian E1-ring A is complete with respect to the maximal ideal m ⊆ π0A, then A is completely determined by an associated formal moduli problems (defined on local Artinian E1-rings with residue field k = π0A=m). Moreover, we characterize those formal moduli problems which arise via this construction (Theorem 6.2.2). We have included in this paper an appendix, where we review some facts about Stone duality which we will use in this series of papers. Remark 0.0.1. Many of the notions introduced in this paper are straightforward adaptations of the corre- sponding notions in classical algebraic geometry. For example, a map of spectral algebraic spaces (X; OX) ! (Y; OY) is proper if and only if the underlying map of ordinary algebraic spaces (X; π0 OX) ! (Y; π0 OY) is proper. Consequently, many of the results of this paper follow immediately from their classical counterparts. However, it seems worthwhile to give an independent exposition, since some of our definitions differ from those given in [31] and elsewhere in the literature (in particular, we do not require the diagonal of a spectral algebraic space to be schematic, though this follows from Theorem 1.2.1 in the quasi-separated case). 3 Notation and Terminology We will use the language of 1-categories freely throughout this paper. We refer the reader to [40] for a general introduction to the theory, and to [41] for a development of the theory of structured ring spectra from the 1-categorical point of view. We will also assume that the reader is familiar with the formalism of spectral algebraic geometry developed in the earlier papers in this series. For convenience, we will adopt the following reference conventions: (T ) We will indicate references to [40] using the letter T. (A) We will indicate references to [41] using the letter A. (V ) We will indicate references to [42] using the Roman numeral V. (VII) We will indicate references to [43] using the Roman numeral VII. (VIII) We will indicate references to [44] using the Roman numeral VIII. (IX) We will indicate references to [45] using the Roman numeral IX. (X) We will indicate references to [46] using the Roman numeral X. (XI) We will indicate references to [47] using the Roman numeral XI. For example, Theorem T.6.1.0.6 refers to Theorem 6.1.0.6 of [40]. If C is an 1-category, we let C' denote the largest Kan complex contained in C: that is, the 1- category obtained from C by discarding all non-invertible morphisms. We will say that a map of simplicial sets f : S ! T is left cofinal if, for every right fibration X ! T , the induced map of simplicial sets FunT (T;X) ! FunT (S; X) is a homotopy equivalence of Kan complexes (in [40], we referred to a map with this property as cofinal). We will say that f is right cofinal if the induced map Sop ! T op is left cofinal: that is, if f induces a homotopy equivalence FunT (T;X) ! FunT (S; X) for every left fibration X ! T . If S and T are 1-categories, then f is left cofinal if and only if for every object t 2 T , the fiber product S ×T Tt= is weakly contractible (Theorem T.4.1.3.1). cn Throughout this paper, we let CAlg denote the 1-category of E1-rings and CAlg the full subcategory spanned by the connective E1-rings. We will need to study several different notions of the \spectrum" of a ring: (i) Given a commutative ring R, we can consider its Zariski spectrum: that is, the topological space whose points are prime ideals of R. We will denote this topological space by SpecZ R. More generally, if A is Z an E1-ring, we let Spec A denote the Zariski spectrum of the commutative ring π0A. (ii) Given a commutative ring, we can consider the affine scheme given by the spectrum of R. We will denote this affine scheme simply by Spec R (so that the underlying topological space of Spec R is given by SpecZ R). ´et (iii) Given a connective E1-ring A, we let Spec A denote the affine spectral Deligne-Mumford stack ´et ´et associated to A. That is, we have Spec A = (X; O), where X is the 1-topos of ShvA of sheaves on the ´et 1-category CAlgA of ´etale A-algebras, and O is the sheaf of E1-rings given by the forgetful functor ´et CAlgA ! CAlg. (iv) If A is a connective E1-ring equipped with an ideal I ⊆ π0A, then we can consider the formal spectrum Spf A of A (with respect to I). We will regard this formal spectrum as a functor CAlgcn ! S, which assigns to a connective E1-ring R the summand of MapCAlg(A; R) consisting of maps from A into R which carry each element of I to a nilpotent element of π0R.