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Derived Algebraic Geometry V: Structured Spaces

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Derived Algebraic Geometry V: Structured Spaces Derived Algebraic Geometry V: Structured Spaces February 22, 2011 Contents 1 Structure Sheaves 7 1.1 C-Valued Sheaves . 9 1.2 Geometries . 12 1.3 The Factorization System on StrG(X)................................ 20 1.4 Classifying 1-Topoi . 26 1.5 1-Categories of Structure Sheaves . 30 2 Scheme Theory 35 2.1 Construction of Spectra: Relative Version . 36 2.2 Construction of Spectra: Absolute Version . 43 2.3 G-Schemes . 50 2.4 The Functor of Points . 60 2.5 Algebraic Geometry (Zariski topology) . 69 2.6 Algebraic Geometry (Etale´ topology) . 75 3 Smoothness 82 3.1 Pregeometries . 83 3.2 Transformations and Morita Equivalence . 87 3.3 1-Categories of T-Structures . 93 3.4 Geometric Envelopes . 97 3.5 Smooth Affine Schemes . 105 4 Examples of Pregeometries 109 4.1 Simplicial Commutative Rings . 109 4.2 Derived Algebraic Geometry (Zariski topology) . 115 4.3 Derived Algebraic Geometry (Etale´ topology) . 122 4.4 Derived Differential Topology . 133 1 Introduction: Bezout's Theorem Let C; C0 ⊆ P2 be two smooth algebraic curves of degrees m and n in the complex projective plane P2. If C and C0 meet transversely, then the classical theorem of Bezout (see for example [4]) asserts that C \ C0 has precisely mn points. We may reformulate the above statement using the language of cohomology. The curves C and C0 have fundamental classes [C]; [C0] 2 H2(P2; Z). If C and C0 meet transversely, then we have the formula [C] [ [C0] = [C \ C0]; where the fundamental class [C \ C0] 2 H4(P2; Z) ' Z of the intersection C \ C0 simply counts the number of points where C and C0 meet. Of course, this should not be surprising: the cup product on cohomology classes is defined so as to encode the operation of intersection. However, it would be a mistake to regard the equation [C] [ [C0] = [C \ C0] as obvious, because it is not always true. For example, if the curves C and C0 meet nontransversely (but still in a finite number of points), then we always have a strict inequality [C] [ [C0] > [C \ C0] if the right hand side is again interpreted as counting the number of points in the set-theoretic intersection of C and C0. If we want a formula which is valid for non-transverse intersections, then we must alter the definition of [C \ C0] so that it reflects the appropriate intersection multiplicities. Determination of these intersection multiplicities requires knowledge of the intersection C \ C0 as a scheme, rather than simply as a set. This is one of the classic arguments that nonreduced scheme structures carry useful information: the intersection number [C][[C0] 2 Z, which is defined a priori by perturbing the curves so that they meet transversally, can also be computed directly (without perturbation) if one is willing to contemplate a potentially non-reduced scheme structure on the intersection. In more complicated situations, the appropriate intersection multiplicities cannot always be determined from the scheme-theoretic intersection alone. Suppose that C and C0 are smooth subvarieties of Pn of complementary dimension, having a zero-dimensional intersection. In this case, the appropriate intersection number associated to a point p 2 C \ C0 is not always given by the complex dimension of the local ring 0 0 OC\C ;p = OC;p ⊗OPn;p OC ;p : The reason for this is easy to understand from the point of view of homological algebra. Since the tensor product functor ⊗OPn;p is not exact, it does not have good properties when considered alone. According to Serre's intersection formula, the correct intersection multiplicity is instead the Euler characteristic X i OPn;p (−1) dim Tori (OC;p; OC0;p): This Euler characteristic contains the dimension of the local ring of the scheme-theoretic intersection as its leading term, but also higher-order corrections. We refer the reader to [15] for further discussion of this formula for the intersection multiplicity. If we would like the equation [C] [ [C0] = [C \ C0] to remain valid in the more complicated situations described above, then we will need to interpret the intersection C \ C0 in some way which remembers not 0 only the tensor product OC;p ⊗OPn;p OC ;p, but the higher Tor groups. Moreover, we should not interpret L these Tor-groups separately, but rather should think of the total derived functor O ⊗ O 0 as a kind C;p OPn;p C ;p of generalized ring. These considerations lead us naturally to the subject of derived algebraic geometry. Using an appropriate notion of \generalized ring", we will mimic the constructions of classical scheme theory to obtain a theory of derived schemes in which a version of the formula [C][[C0] = [C \C0] can be shown to hold with (essentially) no hypotheses on C and C0. Here, we must interpret the intersection C \ C0 in the sense of derived schemes, 2 and we must take great care to give the proper definition for the fundamental classes (the so-called virtual fundamental classes of [1]). What sort of objects should our generalized rings be? To answer this question, we begin by considering the simplest case of Bezout's theorem, in which C and C0 are lines in the projective plane P2. In this case, we know that [C] [ [C0] is the cohomology class of a point, and that C intersects C0 transversely in one point so long as C and C0 are distinct. However, when the equality C = C0 holds, the scheme-theoretic intersection C \ C0 does not even have the correct dimension. Let us now try to formulate a theory of intersections which will handle the degenerate situation where C = C0. To simplify the discussion, we will consider only lines in the affine plane A2 ⊆ P2, with coordinate ring C[x; y]. Two distinct lines in A2 may be given (without loss of generality) by the equations x = 0 and y = 0. The scheme-theoretic intersection of these two lines is the spectrum of the ring C[x; y]=(x; y) ' C, obtained from C[x; y] by enforcing the defining equations of both lines. This ring has Krull dimension 0 because C[x; y] has Krull dimension 2 and we have imposed 2 independent conditions. Now suppose that instead of C and C0 being two distinct lines, they are actually two identical lines, both of which are defined by the equation x = 0. In this case, the affine ring of the scheme theoretic intersection is given by C[x; y]=(x; x) ' C[y]. This ring has Krull dimension 1, rather than the expected dimension 0, because the two equations are not independent: setting x equal to zero twice has the same effect as setting x equal to zero once. To obtain the theory we are looking for, we need a notion of generalized ring which remembers not only whether or not x is equal to 0, but how many different ways x is equal to 0. One way to obtain such a formalism is by categorifying the notion of a commutative ring. That is, in place of ordinary commutative rings, we consider categories equipped with addition and multiplication operations (which are encoded by functors, rather than ordinary functions). For purposes of the present discussion, let us call such an object a categorical ring. We will not give a precise axiomatization of this notion; this turns out to be somewhat complicated (see [7], for example). Example 0.0.1. Let Z≥0 denote the semiring of nonnegative integers. We note that Z≥0 arises in nature through the process of decategorification. The nonnegative integers were originally introduced in order to count: in other words, in order to measure the size of finite sets. To make this statement more precise, let us denote by Fin the category whose objects are finite sets, and whose morphisms are isomorphisms of finite sets. Then we can identify Z≥0 with the set of isomorphism classes of objects in Fin. The addition and multiplication operations on Z≥0 are induced by functors Fin × Fin ! Fin, given by the formation of disjoint union and Cartesian product. Moreover, all of the axioms for a commutative semiring have analogues that hold at the categorical level: for example, the distributive law xy+xz = x(y+z) translates into the existence of a canonical isomorphism a a (X × Y ) (X × Z) ' X × (Y Z) for every triple of objects X; Y; Z 2 Fin. (In order to complete the analogy with the above discussion, we should \complete" the category Fin by formally adjoining inverses, to obtain a categorical ring rather than a categorical semiring, but we will ignore this point for the time being.) To simplify the discussion, we will consider only categorical rings which are groupoids: that is, every morphism in the underlying category is an isomorphism. If C is a categorical ring, then the collection of isomorphism classes of objects in C forms an ordinary commutative ring, which we will denote by π0 C. Every commutative ring R arises in this way: for example, we may take C to be a category whose objects are the elements of R and which contains only identity maps for morphisms. The categorical rings which arise in this way are very special: their objects have no nontrivial automorphisms. For a given commutative ring R, there are usually many other ways to realize an isomorphism of R with the collection of isomorphism classes of objects in a categorical ring C. Although C is not uniquely determined by R, there is often a natural choice for C which is dictated by the manner in which R is constructed.
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