On Azumaya Graphs

We study Azumaya multiplicative graphs over a suitable base category, generalizing in this way the theory of Azumaya algebras over a ring, with or without unit, and the theory of enriched Azumaya categories. We exhibit the links with the corresponding notions of centrality, separability, Brauer group and Brauer–Taylor group.     

Derived Azumaya algebras and generators for twisted derived categories

We introduce a notion of derived Azumaya algebras over ring and schemes generalizing the notion of Azumaya algebras of Grothendieck (Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses. Dix Exposés sur la Cohomologie des Schémas, pp. 46–66, North-Holland, Amsterdam, 1968). We prove that any such algebra B on a scheme X provides a class ϕ(B) in . We prove that for X a quasi-compact and quasi-separated scheme ϕ defines a bijective correspondence, and in particular that any class in , torsion or not, can be represented by a derived Azumaya algebra on X. Our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories, with coefficients in any local system of reasonable dg-categories, generalizing the well known existence of compact generators in derived categories of quasi-coherent sheaves of Bondal and Van Den Bergh (Mosc. Math. J. 3(1):1–36, 2003). A huge part of this paper concerns the treatment of twisted derived categories, as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator. We present explicit examples of derived Azumaya algebras that are not represented by classical Azumaya algebras, as well as applications of our main result to the localization for twisted algebraic K-theory and to the stability of saturated dg-categories by direct push-forwards along smooth and proper maps.     

Brauer groups for commutative S-algebras

We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a $K\left(n\right)$-local variant of the notion of Brauer groups.     

GENERALIZED BRAUER ALGEBRAS

ABSTRACT

Using some ideas of Brauer, we introduce what we call generalized Brauer algebras and, as a special case, Brauer orders. We show that many well-known classes of so-called crossed product algebras, and in particular, the well-known crossed product orders, can be obtained as special instances of our construction. We prove several results showing when Brauer orders are Azumaya, maximal, hereditary or Gorenstein.     

The Brauer Group of Azumaya Corings and the Second Cohomology Group

Let R be a commutative ring. An Azumaya coring consists of a couple , with S a faithfully flat commutative R-algebra, and an S-coring satisfying certain properties. If S is faithfully projective, then the dual of is an Azumaya algebra. Equivalence classes of Azumaya corings form an abelian group, called the Brauer group of Azumaya corings. This group is canonically isomorphic to the second flat cohomology group. We also give algebraic interpretations of the second Amitsur cohomology group and the first Villamayor–Zelinsky cohomology group in terms of corings.     

Enveloping ringoids

We present notions of module over a universal algebra, and linear representation of a universal algebra, which have gained currency with categorical algebraists, we give several intriguing examples of these objects, showing that important aspects of universal algebras can be studied in this context, and we describe the theory ofenveloping ringoids, which have categories of left modules equivalent to corresponding categories of modules and linear representations. Algebras in many of the familiar varieties of algebras, which have underlying groups, turn out to have enveloping ringoids that are equivalent to familiar rings. Nonempty algebras in an abelian varietyV have an enveloping ringoid equivalent toR(V), the so-calledring of the variety V.Dedicated to the memory of Alan DayPresented by J. Sichler.     

Effectivity of Brauer–Manin obstructions on surfaces

We study Brauer–Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields. A technique for constructing Azumaya algebra representatives of Brauer group elements is given, and this is applied to the computation of obstructions.     

The Brauer-Clifford Group for (S,H)-Azumaya Algebras over a Commutative Ring

The Brauer-Clifford group BrClif(Z,G) corresponding to a finite group G and a finite-dimensional semisimple G-algebra Z was recently introduced by Alexandre Turull in the course of his work on character correspondence conjectures in group representation theory. This Brauer-Clifford group is a group of equivalence classes of Azumaya algebras over Z whose G-algebra structure agrees on restriction to the fixed (and usually nontrivial) G-algebra structure of Z. In this paper we extend the notion of the Brauer-Clifford group to the case of (S,H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is a commutative H-module algebra. These Brauer-Clifford groups turn out to be an example of the Brauer group of the symmetric monoidal category of S # H-modules, a perspective which allows one to construct a dual Brauer-Clifford group for the category of S-modules with compatible right H-comodule structure.     

Unramified division algebras do not always contain Azumaya maximal orders

We show that, in general, over a regular integral noetherian affine scheme X of dimension at least 6, there exist Brauer classes on X for which the associated division algebras over the generic point have no Azumaya maximal orders over X. Despite the algebraic nature of the result, our proof relies on the topology of classifying spaces of algebraic groups.     

Quadratic functors on pointed categories

We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups Ab, and whose source category is an arbitrary category C with null object such that all objects are colimits of copies of a generating object E which is small and regular projective; this includes all categories of models V of a pointed theory T. More specifically, we are interested in such quadratic functors F from C to Ab which preserve filtered colimits and suitable coequalizers.A functorial equivalence is established between such functors F:C→Ab and certain minimal algebraic data which we call quadratic C-modules: these involve the values on E of the cross-effects of F and certain structure maps generalizing the second Hopf invariant and the Whitehead product.Applying this general result to the case where E is a cogroup these data take a particularly simple form. This application extends results of Baues and Pirashvili obtained for C being the category of groups or of modules over some ring; here quadratic C-modules are equivalent with abelian square groups or quadratic R-modules, respectively.     

Hereditary noetherian categories of positive Euler characteristic

We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function. Dedicated to Otto Kerner on the occasion of his 60th birthday     

Rugged modules: The opposite of flatness

Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M⁺ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.     

On the finiteness of the Brauer group of an arithmetic scheme

The Artin conjecture on the finiteness of the Brauer group is shown to hold for an arithmetic model of a K3 surface over a number field k. The Brauer group of an arithmetic model of an Enriques surface over a sufficiently large number field is shown to be a 2-group. For almost all prime numbers l, the triviality of the l-primary component of the Brauer group of an arithmetic model of a smooth projective simply connected Calabi-Yau variety V over a number field k under the assumption that V (k) ≠ Ø is proved.     

Operads and varieties of algebras defined by polylinear identities

We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of C-linear multioperator algebras; in particular, of algebras, defined by C-polylinear identities (here C is a commutative operad). We introduce and study symmetric C-linear operads. The main result of this article is as follows: A variety of C-linear multioperator algebras is defined by C-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric C-linear operad.     

On Coprime Modules and Comodules

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C* is prime. This definition, however, is not intrinsic—it strongly depends on the base ring being a field. The purpose of the article is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory.

Dualizing classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular, we obtain various characterisations of prime and coprime coalgebras over rings and fields.     

Actions in the Category of Precrossed Modules in Lie Algebras

For any object L in the category of precrossed modules in Lie algebras PXLie, we construct the object Act(L), which we call the actor of this object. From this construction, we derive the notions of action, center, semidirect product, derivation, commutator, and abelian precrossed module in PXLie. We show that the notion of action is equivalent to the one given in semi-abelian categories, and Act(L) is the split extension classifier for L. In the case of a crossed module in Lie algebras we show how to recover its actor in the category of crossed modules from its actor in the category of precrossed modules.     

Scott modules in finite groups with cyclic Sylow p-subgroups

We present a simple method for determining Scott modules of finite groups using just the Brauer tree of the principal block and the values of ordinary characters at non-identity p-elements. Previously, determining Scott modules often involved long calculations. The method is applicable to the case where finite groups have cyclic Sylow p-subgroups. The structure of a Scott module is closely related to the length of the shortest path from the vertex labeled by the trivial ordinary character to the exceptional vertex in the Brauer tree. We classify the structures of the Scott modules and the ordinary characters they afford, by the length of the associated path in the Brauer tree.     

Endopermutation Scott modules,slash indecomposability and saturated fusion systems

This paper introduces the notion of Frobenius-friendly modules which is a generalisation of p-permutation modules and we obtain the slash functors for these modules. Generalisations of Scott modules and Brauer indecomposability, which are called endopermutation Scott modules and slash indecomposability respectively, are given by using the slash functors. We also describe slash indecomposability in terms of saturated fusion systems, which is a generalisation of the case of Brauer indecomposability.