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Derived Algebraic Geometry VII: Spectral Schemes

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Derived Algebraic Geometry VII: Spectral Schemes November 5, 2011 Contents 1 Sheaves of Spectra 4 2 Spectral Schemes 11 3 Coherent 1-Topoi 26 4 Deligne's Completeness Theorem 33 5 Flat Descent 42 6 Flat Descent for Modules 52 7 Digression: Henselian Rings 63 8 Spectral Deligne-Mumford Stacks 68 9 Comparison Results 85 1 Introduction Our goal in this paper is to introduce a variant of algebraic geometry, which we will refer to as spectral algebraic geometry. We will take as our starting point Grothendieck's theory of schemes. Recall that a scheme is a pair (X; OX ), where X is a topological space, OX is a sheaf of commutative rings on X, and the pair (X; OX ) is locally (with respect to the topology of X) isomorphic to the Zariski spectrum of a commutative ring. We can regard a commutative ring R as a set equipped with addition and multiplication maps a : R × R ! R m : R × R ! R which are required to satisfy certain identities. For certain applications (particularly in algebraic topology), it is useful to consider a variation, where R is equipped with a topology. Roughly speaking, a (connective) E1-ring is a space X equipped with continuous addition and multiplication maps a : X × X ! X m : X × X ! X which are required to satisfy the same identities up to (coherent) homotopy. The theory of E1-rings is a robust generalization of commutative algebra: in particular, the basic formal constructions needed to set up the theory of schemes (such as localization) make sense in the setting of E1-rings. We will use this observation to introduce the notion of a spectral scheme: a mathematical object which is obtained by \gluing together" a collection of (connective) E1-rings, just as a scheme is obtained by \gluing together" a collection of commutative rings. The collection of commutative rings can be organized into a category Ring. That is, to every pair of 0 0 0 commutative rings R and R , we can associate a set HomRing(R; R ) of ring homomorphisms from R to R . 0 The analogous statement for E1-rings is more complicated: to every pair of E1-rings R and R , we can associate a space Map(R; R0) of morphisms from R to R0. Moreover, these mapping spaces are equipped with a composition products Map(R; R0) × Map(R0;R00) ! Map(R; R00), which are associative (and unital) up to coherent homotopy. To adequately describe this type of structure, it is convenient to use the language of 1-categories developed in [40]. The collection of all E1-rings is naturally organized into an 1-category which we will denote by CAlg, which contains (the nerve of) the category Ring as a full subcategory. Let us now outline the contents of this paper. Recall first that the category of schemes can be realized as a subcategory of the category of ringed spaces, whose objects are pairs (X; OX ) where X is a topological space and OX is a sheaf of commutative rings on X. Our first goal will be to introduce a suitable 1-categorical version of this category. In x1, we will introduce the 1-category RingTop of spectrally ringed 1-topoi. The objects of RingTop are given by pairs (X; OX), where X is an 1-topos and OX is a sheaf of E1-rings on X. In x2, we will introduce the notion of a spectral scheme. The collection of spectral schemes is organized into an 1-category, which we regard as a subcategory of the 1-category RingTop of spectrally ringed 1- topoi. This subcategory admits a number of characterizations (see Definitions 2.2, 2.7, and 2.27) which we will show to be equivalent. We will also explain the relationship between our theory of spectral schemes and the classical theory of schemes (Proposition 2.37). Let (X; OX ) be a scheme. Recall that X is said to be quasi-compact if every open covering of X has a finite subcovering, and quasi-separated if the collection of quasi-compact open subsets of X is closed under pairwise intersections. In x3, we will generalize these conditions to our 1-categorical setting by introducing the notion of a coherent 1-topos. If X is the 1-topos of sheaves on a topological space X, then X is coherent if and only if X is quasi-compact and quasi-separated. For every spectral scheme (X; OX), the 1-topos X is locally coherent (because coherence is automatic in the affine case), and the coherence of X is an important hypothesis for almost any nontrivial application. Our theory of coherent 1-topoi is an adaptation of the classical theory of coherent topoi (see, for example, [28]). A theorem of Deligne asserts that every coherent topos X has enough points: that is, that there exists a collection of geometric morphisms ffα : Set ! Xg such that a morphism φ in X is invertible if and ∗ only if each pullback fα(φ) is a bijection of sets. In x4, we will prove an 1-categorical analogue of this statement (Theorem 4.1). As an application, we prove a connectivity result for the geometric realization of a hypercovering (Theorem 4.20) which is useful in x5. 2 To every connective E1-ring R, one can associate a spectral scheme SpecZ(R) 2 SpSch. Consequently, cn every spectral scheme X = (X; OX) represents a functor X on the 1-category CAlg of connective E1-rings, given by the formula X(R) = MapSpSch(SpecZ(R); X). In x5, we will show that if X is 0-localic (meaning that the underlying 1-topos of X can be realized as the category of sheaves on a topological space), then X is a sheaf with respect to the flat topology (Theorem 5.15). The proof makes use of the fact that the flat topology is subcanonical on the 1-category of E1-rings: that is, that every corepresentable functor is a sheaf with respect to the flat topology. We will deduce this subcanonicality from a more general result concerning descent for modules over E1-rings, which is proven in x6. In classical algebraic geometry, the category of schemes can be regarded as a full subcategory of a larger 2-category of Deligne-Mumford stacks. In x8, we will introduce the notion of a spectral Deligne-Mumford stack. Our definition involves the notion of a strictly Henselian sheaf of E1-rings, which is generalization of the classical theory of strictly Henselian rings; we include a brief review of the classical theory in x7. As with spectral schemes, we can think of spectral Deligne-Mumford stacks as mathematical objects obtained by \gluing together" connective E1-rings. The difference lies in the nature of the gluing: in the setting of spectral Deligne-Mumford stacks, we replace the Zariski topology by the (far more flexible) ´etaletopology on E1-rings. The collection of all spectral Deligne-Mumford stacks is organized into an 1-category Stk, and there is an evident functor SpSch ! Stk. In x9, we will show that this functor is fully faithful when restricted to the 1-category of 0-localic spectral schemes. Remark 0.1. Our theory of spectral algebraic geometry is closely related to the theory of homotopical algebraic geometry introduced by To¨enand Vezzosi, and there is substantial overlap between their work (see [68], [69], [70], and [71]) and the ideas treated in this paper. Perhaps the primary difference in our presentation is that we stick closely to the classical view of scheme as a kind of ringed space, while To¨enand Vezzosi make use of the \functor of points" philosophy which identifies an algebro-geometric object X with the underlying functor R 7! Hom(Spec R; X). Notation and Terminology This paper will make extensive use of the theory of 1-categories, as developed in [40]. We will also need the theory of structured ring spectra, which is presented from an 1-categorical point of view in [41]. Finally, we will make use of the theory of geometries developed in [42], and earlier paper 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. For example, Theorem T.6.1.0.6 refers to Theorem 6.1.0.6 of [40]. Let R be a commutative ring. We let SpecZ R denote the collection of all prime ideals in R. We will refer to SpecZ R as the Zariski spectrum of R. We regard SpecZ R as endowed with the Zariski topology: a set Z Z U ⊆ Spec R is open if and only if there exists an ideal I ⊆ R such that U = fp 2 Spec R : I * pg. This Z topology has a basis of open sets given by Ux = fp 2 Spec R : x2 = pg, where x ranges over the collection of elements of R. Z Z If R is an E1-ring, we let Spec R denote the Zariski spectrum Spec (π0R) of the commutative ring π0R. We will occasionally need the following result from commutative algebra: Proposition 0.2. Let f : R ! R0 be an ´etalemap of commutative rings. Then f induces an open map of topological spaces SpecZ(R0) ! SpecZ(R). 3 Let X be an 1-topos and let OX be a sheaf on X with values in an 1-category C (that is, a functor op X ! C which preserves small limits).
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