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 computational 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 local geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, theétale local triviality of Azumaya algebras over connective derived schemes, and a local to global principle for the algebraicity of stacks of stable categories. Mathematics Subject Classification 2010Primary: 14F22, 18G55. Secondary: 14D20, 18E30.ization) follow from the untwisted case as in [Toë12], using the fact that our categories of α-twisted sheaves Mod α X ≃ Mod A admit global generators with endomorphism algebra A. SummaryWe now give a detailed summary of the paper. By definition, an R-algebra A is Azumaya if it is a compact generator of the ∞-category of R-modules and if the multiplication actionan equivalence. This definition is due to Auslander-Goldman [AG60] in the case of discrete commutative rings, and it has been studied in the settings of schemes by Grothendieck [Gro68a], E ∞ -ring spectra by Baker-Richter-Szymik [BRS10], and derived algebraic geometry over simplicial commutative rings by Toën [Toë12]. In a slightly different direction, it has also been studied in the setting of higher categories by Borceux-Vitale [BV02] and Johnson [Joh10]. All of these variations ultimately rely on the idea of an Azumaya algebra as an algebra whose module category is invertible with respect to a certain "Morita" symmetric monoidal structure.Although we restrict to Azumaya algebras over commutative ring spectra, we note that the notion of Azumaya algebra makes sense over any E 3 -ring spectrum. The reason for this is that if R is an E 3 -ring, then Mod R is naturally a E 2 -monoidal ∞-category, and so its ∞-category of modules is naturally E 1 -monoidal. The theory of Azumaya algebras is closely related to the notions of smoothness and properness in non-commutative geometry, which have been studied extensively starting from Kontsevich [Kap98]. These and related ideas have been used to great success to prove theorems in algebraic geometry. For instance, van den Bergh [VdB04] uses non-commutative algebras to give a proof of the Bondal-Orlov conjecture, showing that birational smooth projective 3-folds are derived equivalent.One of main points of the paper is to establish the following theorem, which says that all Azumaya algebras over the sphere spectrum are Morita equivalent.Theorem 1.1. The Brauer group of the sphere spectrum is zero.This theorem follows from several other important results, which we now outline. In Theorem 3.15, we show that a compactly generated R-linear category C (a stable presentable ∞-category enriched in R-mo...

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of dimension d , we give upper bounds on the index depending only on d and the order of the class. By the Oka principle, this also solves the period-index problem for the analytic Brauer group of any Stein space that has the homotopy type of a finite CW-complex. Our methods use twisted topological K -theory, which was first introduced by Donovan and Karoubi. We also study the cohomology of the projective unitary groups to give cohomological obstructions to a class being represented by an Azumaya algebra of degree n. Applying this to the finite skeleta of the Eilenberg-MacLane space K(Z/ℓ, 2), where ℓ is a prime, we construct a sequence of spaces with an order ℓ class in the Brauer group, but whose indices tend to infinity.19L50, 16K50, 57T10, 55Q10, 55S35 IntroductionThis paper gives a solution to a period-index problem for twisted topological K -theory. The solution should be viewed as an existence theorem for twisted vector bundles.Let X be a connected CW-complex. An Azumaya algebra A of degree n on X is a noncommutative algebra over the sheaf C of complex-valued functions on X such that A is a vector bundle of rank n 2 and the stalks are finite dimensional complex matrix algebras M n (C). Examples of Azumaya algebras include the sheaves of endomorphisms of complex vector bundles and the complex Clifford bundles Cl(E) of oriented evendimensional real vector bundles E . The Brauer group Br(X) classifies topological Azumaya algebras on X up to the usual Brauer equivalence: A 0 and A 1 are Brauer equivalent if there exist vector bundles E 0 and E 1 and an isomorphismof sheaves of C -algebras. Define Br(X) to be the free abelian group on isomorphism classes of Azumaya algebras modulo Brauer equivalence.The period-index problem for twisted topological K -theory 13 Proposition 2.5 If A is an Azumaya algebra of degree n with class α, then A ∼ = End(E) for some α-twisted vector bundle E of rank n. The α-twisted sheaf is unique up to tensoring with untwisted line bundles. Proof See [16, Theorem 1.3.5] or [35, Section 3]. Proposition 2.6 Tensoring with E * , the dual of E , induces an equivalence of categories Vect α → Vect A . Proof See [16, Theorem 1.3.7] or [35, Section 3]. Proposition 2.7 If A and B are Brauer-equivalent Azumaya algebras over X , then Vect A and Vect B are equivalent categories.Proof This follows from the previous two propositions.If X is a finite CW-complex, the twisted topological K -group KU 0 (X) α may be identified with the Grothendieck group of left A-modules.Proposition 2.8 If X is compact and Hausdorff, and if A is an Azumaya algebra on X with class α (so that α is torsion), then K A 0 (X) ∼ = KU 0 (X) α . This isomorphism is uniquely defined up to the natural action of H 2 (X, Z) on the left.Proof This follows from [9, Section 3.1]. In fact, they show that KU ...

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period-index problem in full for finite CW complexes of dimension less than 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period-index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.

Schlichting conjectured that the negative K-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree −1. The main results of this paper are that K−1(E) vanishes when E is a small stable ∞-category with a bounded t-structure and that K−n(E) vanishes for all n 1 when additionally the heart of E is noetherian. It follows that Barwick's theorem of the heart holds for nonconnective K-theory spectra when the heart is noetherian. We give several applications, to non-existence results for bounded t-structures and stability conditions, to possible K-theoretic obstructions to the existence of the motivic t-structure, and to vanishing results for the negative K-groups of a large class of dg algebras and ring spectra.

We use the Beilinson t-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme X with graded pieces given by the Hodge-completion of the derived de Rham cohomology of X. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt-Morrow-Scholze for p-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.