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A Derived Stack? Gabriele Vezzosi

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A Derived Stack? Gabriele Vezzosi WHAT IS... ? a Derived Stack? Gabriele Vezzosi Derived stacks are the “spaces” studied in de- with their isomorphisms instead of just objects rived algebraic geometry, a relatively new theory modulo isomorphisms. These functors are called in which algebraic geometry meets homotopy stacks, and aficionados of the WHAT IS column theory—or higher category theory, depending on already have met this notion (“What is a stack?” one’s taste. Just as a scheme is locally modeled by Dan Edidin, Notices, April 2003). More recently, on commutative rings, derived schemes or stacks higher stacks came into play; they arise naturally are modeled on some kind of derived commutative when one is interested in classifying geometric rings, a homotopy version of commutative rings. objects (say, over a given scheme) for which the In order to define derived stacks more precisely, natural notion of equivalence is broader than just it will be useful to reexamine briefly the functorial isomorphisms. Example: perfect complexes over point of view in (underived) algebraic geometry. a given scheme with equivalences given by quasi- Let k be a base commutative ring. In algebraic isomorphisms, i.e., maps inducing isomorphisms geometry, a k-scheme may be given at least two on cohomology. In such cases it is natural to equivalent definitions. The first one is as a special enlarge the target category for the correspond- kind of pair (X, OX ), where X is a topological space ing moduli functors to the category of simplicial and OX is a sheaf of commutative k-algebras on X sets or, equivalently, to the category of topological (this is the so-called ringed space approach). The spaces. A stack may be viewed as a higher stack via second one is as a special kind of functor from the the nerve construction: the nerve of a groupoid is category CommAlgk of commutative k-algebras to the simplicial set whose n-th level is the set of n the category of sets (this is the functor of points composable morphisms in the groupoid. This sim- approach). For example, the n-dimensional pro- plicial set has homotopy groups only in degrees Pn jective space k over k may be identified with the ≤ 1, with π1 roughly corresponding to automor- functor sending A ∈ CommAlgk to the set of sur- n+1 phisms of a given object. General simplicial sets or jective maps of A-modules A → A, modulo the topological spaces are needed in order to accom- equivalence relation generated by multiplication modate “higher autoequivalences” of the objects by units in A. In the following, we will concentrate being classified. on the functor-of-points description. Another example of a higher stack is given by Prompted by the study of moduli problems iterating the so-called classifying stack construc- (e.g., classifying families of elliptic curves or vec- tion. For a k-group scheme G, there is a stack tor bundles on a given algebraic variety), algebraic BG ≡ K(G, 1) classifying principal G-bundles; by geometers have long been led to enlarge the target taking the nerve, we may view K(G, 1) as a functor category of the functor of points from sets to to simplicial sets. If G is abelian, this functor is groupoids (i.e., categories whose morphisms are equivalenttoafunctortosimplicialabeliangroups, all invertible) in order to classify objects together and the classifying stack construction may then Gabriele Vezzosi is professor of geometry at the Univer- be applied to any simplicial level again to get sità di Firenze, Italy. His email address is gabriele. BBG ≡ K(G, 2). And so on. For any n ≥ 2, K(G, n) [email protected]. is not a stack but a higher stack (classifying August 2011 Notices of the AMS 955 “higher” principal G-bundles); it is the algebro- What about the general case? By definition, geometric analog of the Eilenberg-Mac Lane space in order to compute such multiplicities, i.e., the in topology, from which we borrowed the notation. Tor-groups, one has to resolve OZ,p (or OT ,p) via a It is useful to draw a diagram summarizing the complex of projective or flat OX,p-modules, tensor still underived situation we have just discussed. this resolution with OT ,p (or OZ,p), and compute the cohomology of the resulting complex. This schemes / CommAlg / Sets is very much homological algebra, but where I Tk I TTT has the geometry gone? We started with three II TTTstacks II TT i0 II TTTT varieties and a point on each of them, and we II TTT* II ended up computing the cohomology of a rather II Grpds higher stacks II geometrically obscure complex. Is there a way to II II reconcile the general case with the flat intersection II Nerve II$ case in a possibly wider geometrical picture? A SimplSets possible answer is the following: we can still keep the two peculiar geometric features of the flat Here i0 is the functor identifying a set with intersection case mentioned above, provided we the groupoid having that set of objects and only are willing to contemplate a notion of commutative identities as arrows. rings that is more general than the usual one. More The main point of derived algebraic geome- precisely, one can first observe that it is possible try is to enlarge (also) the source category, i.e., to choose the resolution of, say, OZ,p (as an OX,p- to replace commutative algebras with a more module) in such a way that it has the structure flexible notion of commutative rings serving as of a nonpositively commutative differential graded new or derived rings. Why? I could list here, OX,p-algebra with the differential increasing the among some of the actual historical motivations, degree (a cdga, for short) or equivalently of a the Kontsevich hidden smoothness philosophy and simplicial commutative OX,p-algebra. This gives a geometrical definition of universal elliptic co- us an extension of the first feature of the flat homology (aka topological modular forms). For intersection case. Then we can force the second expository reasons, I will concentrate instead on feature by insisting that the tensor product of this two more down-to-earth and classical instances cdga resolution with OT ,p does give the “scheme that naturally lead to building a geometry based structure” of Z ∩ T (locally at the point p). Of on these derived rings rather than on the usual course this is not a usual scheme structure, but commutative rings. rather a new kind of scheme-like structure that we will call a derived structure, the name coming Derived Intersections from the fact that what we are computing is In algebraic geometry, the so-called intersection L the derived tensor product OZ,p ⊗OX,p OT ,p (whose multiplicities are given by Serre’s formula. Here is cohomology groups are the Tor-groups appearing oneformofit.Let X be anambientcomplex smooth in Serre’s intersection formula). But we may, and projective variety, and let Z, T be possibly singular L therefore we do, view OZ,p ⊗OX,p OT ,p as a derived subvarieties of X whose dimensions sum up to commutative ring, i.e., a cdga or a simplicial dim X and which intersect on a 0-dimensional commutative algebra. locus. If p ∈ Z ∩ T is a point, its “weight” in the intersection, i.e., its intersection multiplicity, is Deformation Theory given by Another more or less classical topic in algebraic OX,p µp(X; Z,T) = X dimC Tori (OZ,p, OT ,p), geometry that also leads to considering these i≥0 kinds of derived rings is the theory of the cotan- where OY ,p denotes the local ring of a variety gent complex. This object dates back to Quillen, Y at p ∈ Y and the Tors are computed in the Grothendieck, and Illusie (see L. Illusie, Complexe category of OX,p-modules. One can easily prove cotangent et déformations. I, Lecture Notes in Math. that the sum is finite and much less easily that 239, Springer-Verlag, 1971) and is too technical to it is nonnegative. But here we are interested in be carefully defined here. Let me just say that in another aspect of this formula. In the lucky case the affine case X = Spec A, for A a commutative of a flat intersection, i.e., when either OZ,p or OT ,p k-algebra, it is defined as the classical module of are flat OX,p-modules, this formula tells us that the Kähler differentials, but only after having replaced multiplicity is given by the dimension of the tensor A itself by a resolution that is a simplicial com- product OZ,p ⊗OX,p OT ,p, which, for our purposes, mutative k-algebra that is free at each simplicial has two peculiar features: it is a commutative ring, level (in characteristic 0, one might as well resolve and it is the local ring of the scheme-theoretic using cdga’s). Recall that a simplicial commutative intersection Z ∩ T at p. In other words, it carries algebrais asimplicialsetthat isateachlevel acom- a nice geometrical interpretation. mutative algebra and whose structural maps (face 956 Notices of the AMS Volume 58, Number 7 and degeneracies) are required to be morphisms rise to a homotopy theory (technically speaking, of algebras. Even with such a sketchy definition, it a Quillen model category structure); the same is will not perhaps be too surprising that most topics true, with weak homotopy equivalences as equiv- in the deformation theory of schemes, and more alences, for the target category SimplSets. It is a generally of moduli stacks, are handled through basic rule of derived algebraic geometry that all this cotangent complex.
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