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Brauer Groups and Etale Cohomology in Derived Algebraic Geometry

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Brauer Groups and Etale Cohomology in Derived Algebraic Geometry Brauer groups and etale´ cohomology in derived algebraic geometry Benjamin Antieau∗ and David Gepner October 2, 2012 Abstract In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a com- putational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the lo- cal geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, the ´etale local triviality of Azumaya algebras over connective derived schemes, and a local to global principle for the algebraicity of stacks of stable categories. Key Words Commutative ring spectra, derived algebraic geometry, moduli spaces, Azumaya algebras, and Brauer groups. Mathematics Subject Classification 2010 Primary: 14F22, 18G55. Secondary: 14D20, 18E30. Contents 1 Introduction 2 1.1 Setting......................................... 2 1.2 Summary ....................................... 5 1.3 Acknowledgments ................................. 9 arXiv:1210.0290v1 [math.AG] 1 Oct 2012 2 Ring and module theory 9 2.1 Ringsandmodules ................................. 9 2.2 Compactobjectsandgenerators . ...... 9 2.3 Topologieson affineconnectivederivedschemes . 12 2.4 Tor-amplitude................................... 13 2.5 Vanishingloci................................... 15 3 Module categories and their module categories 16 3.1 R-linearcategories .................................. 16 3.2 Smoothandproperalgebras . .... 20 3.3 Azumayaalgebras................................. 22 ∗This material is based upon work supported in part by the NSF under Grant No. DMS-0901373. 1 4 Sheaves 23 4.1 Stacksofalgebraandmodulecategories . ......... 23 4.2 Thecotangentcomplexandformalsmoothness . ......... 26 4.3 Geometricsheaves ................................ ... 31 4.4 Cotangentcomplexesofsmoothmorphisms . ........ 39 4.5 Etale-local sections of smooth geometric morphisms . ............. 43 5 Moduli of objects in linear ∞-categories 43 5.1 Localmoduli ..................................... 43 5.2 Themodulisheafofobjects. ..... 45 5.3 EtalelocaltrivialityofAzumayaalgebras´ . ........ 49 6 Gluing generators 50 6.1 AzumayaalgebrasandBrauerclassesoversheaves . ............ 51 6.2 TheZariskilocal-globalprinciple . ......... 53 6.3 The´etalelocal-globalprinciple . .......... 57 6.4 Liftingtheorems................................. ... 60 7 Brauer groups 61 7.1 TheBrauerspace .................................. 61 7.2 Picardgroupsofconnectiveringspectra . .......... 62 7.3 TheexactsequenceofPicardandBrauergroups . .......... 64 7.4 TheBrauerspacespectralsequence . ....... 65 7.5 ComputationsofBrauergroupsofringspectra. ........... 66 1 Introduction 1.1 Setting Derived algebraic geometry is a generalization of classical Grothendieck-style algebraic geometry aimed at bringing techniques from geometry to bear on problems in homotopy theory, and used to unify and explain many disparate results about categories of sheaves on schemes. It has been used by Arinkin-Gaitsgory [AG12] to formulate a precise version of the geometric Langlands conjecture, by Ben-Zvi-Francis-Nadler [BZFN10] to study integral transforms and Hochschild homology of co- herent sheaves, by Lurie and others to study topological modular forms, by To¨en-Vaqui´e[TV07] to study moduli spaces of complexes of vector bundles, and by To¨en [To¨e12] to study derived Azumaya algebras. Moreover, the philosophy of derived algebraic geometry is closely related to non-commutative geometry and to the idea of hidden smoothness of Kontsevich. The basic objects in derived algebraic geometry are “derived” versions of commutative rings. There are various things this might mean. For instance, it could mean simply a graded commutative ring, or a commutative differential-graded ring, such as the de Rham complex Ω∗(M) of a mani- fold M. Or, it could mean a commutative ring spectrum, which is to say a spectrum equipped with a coherently homotopy commutative and associative multiplication. The basic exampleof such a com- mutative ring spectrum is the sphere spectrum S, which is the initial commutative ring spectrum, and hence plays the role of the integers Z in derived algebraic geometry. Commutative ring spectra are in a precise sense the universal class of derived commutative rings. We work throughout this paper with connective commutative ring spectra, their module categories, and their associated schemes. 2 While a substantial portion of the theory we develop in this paper has been studied previously for simplicial commutative rings, it is important for applications to homotopy theory and differential geometry to have results applicable to the much broader class of commutative (or E∞) ring spec- tra, as the vast majority of the rings which arise in these contexts are only of this more general form. Simplicial commutative rings are special cases of commutative differential graded rings, and an E∞-ring spectrum admits an E∞-dg model if and only if it is a commutative algebra over the Eilenberg-MacLane spectrum H Z. To give an idea of how specialized a class this is, note that an arbitrary spectrum M is an H Z-module precisely when all of its k-invariants are trivial, meaning n that it decomposes as a product of spectra Σ H πnM, or that it has no nontrivial extensions in its “composition series” (Postnikov tower). Rather, the basic E∞-ring is the sphere spectrum S, which is the group completion of the symmetric monoidal category of finite sets and automorphisms (as op- posed to only identities, which yields Z) and captures substantial information from differential and F1-geometry and contains the homological complexity of the symmetric groups. Similarly, the alge- braic K-theory spectra, as well as other important spectra such those arising from bordism theories of manifolds, in the study of the mapping class group and the Mumford conjecture, or in topological Hochschild or cyclic homology, tend not to exist in the differential graded world. Nevertheless, an E∞-ring spectrum R should be regarded as a nilpotent thickening of its un- derlying commutative ring π0R, in much the same way as the Grothendieck school successfully incorporated nilpotent elements of ordinary rings into algebraic geometry via scheme theory. Of course, this relies upon the “local” theory of homotopical commutative algebra, which, thanks to the efforts of many mathematicians, is now well established. In particular, there is a good notion of ´etale map of commutative ring spectra, and so the basic geometric objects in our paper will be glued together, in this topology, from commutative ring spectra. We adopt Grothendieck’s “functor of points” perspective; specifically, we fix a base E∞-ring R and consider the category of connective cn cn commutative R-algebras, CAlgR . A sheaf is then a space-valued functor on CAlgR which satisfies descent for the ´etale topology in the appropriate homotopical sense. For instance, if S is a commuta- tive R-algebra, there is the representable sheaf Spec S whose space of T-points is the mapping space map(S, T) in the ∞-category of connective commutative R-algebras. Just as in ordinaryalgebraicgeometry,one is really only interested in a subclass of sheaves which are geometric in some sense. An important feature of derived algebraic geometry is the presence of higher versions of Artin stacks, an idea due to Simpson [Sim96]; roughly, this is the smallest class of sheaves which contain the representables Spec S and is closed under formation of quotients by smooth groupoid actions. By restricting attention to these sheaves, it is possible to prove many EGA- style statements. The situation is entirely analogous to the classes of schemes or algebraic spaces in ordinary algebraic geometry, which can be similarly expressed as the closure of the affines under formation of Zariski or ´etale quotients, respectively. The difference is that we allow our sheaves to take values in spaces, a model for the theory of higher groupoids, and that we require the larger class which is closed under smooth actions, so that it contains objects such as the deloopings BnA of a smooth abelian group scheme A. These are familiar objects: the Artin 1-stack BA is the moduli space of A-torsors, and the Artin 2-stack B2A is the moduli space of gerbes with band A. One of the main goals of this paper is to study Azumaya algebras over these derived geomet- ric objects. Historically, the notion of Azumaya algebra, due to Auslander-Goldman [AG60], arose froman attemptto generalizethe Brauergroupof afield. Itwas then globalized by Grothendieck[Gro68a], who defined an Azumaya algebra A over a scheme X as a sheaf of coherent OX-algebras that is ´etale locally a matrix algebra. In other words, there is a surjective ´etale map p : U → X such that ∗ p A Mn(OU). The Brauer group of a scheme classifies Azumaya algebras up to Morita equiv- alence, that is, up to equivalence of their module categories. The original examples of Azumaya algebras are central simple algebras over a field k; by Wedderburn’s theorem, these are precisely the 3 algebras Mn(D), where D is a division algebra of finite dimension over its center k. The algebra of quaternion numbers over R is thus an example of an Azumaya R-algebra, and represents the generator of Br(R) Z/2. In more geometric
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