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Register a new userRecent advances in the study of the Equivariant Brauer Group
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In this paper we outline a recent construction of a Chern-Weil isomorphism for the equivariant Brauer group of $mathbb R^n$ actions on a principal torus bundle, where the target for this isomorphism is a dimensionally reduced vCech cohomology group. Using this latter group, we demonstrate how to extend the induced algebra construction to algebras with a non-trivial bundle as their spectrum.
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In this paper we construct a Chern-Weil isomorphism for the equivariant Brauer group of R^n-actions on a principal torus bundle, where the target for this isomorphism is a dimensionally reduced Cech cohomology group. From this point of view, the usual forgetful functor takes the form of a connecting homomorphism in a long exact sequence in dimensionally reduced cohomology.
Let $(H, R)$ be a finite dimensional quasitriangular Hopf algebra over a field $k$, and $_Hmathcal{M}$ the representation category of $H$. In this paper, we study the braided autoequivalences of the Drinfeld center $^H_Hmathcal{YD}$ trivializable on $_Hmathcal{M}$. We establish a group isomorphism between the group of those autoequivalences and the group of quantum commutative bi-Galois objects of the transmutation braided Hopf algebra $_RH$. We then apply this isomorphism to obtain a categorical interpretation of the exact sequence of the equivariant Brauer group $mathrm{BM}(k, H,R)$ in [18]. To this aim, we have to develop the braided bi-Galois theory initiated by Schauenburg in [14,15], which generalizes the Hopf bi-Galois theory over usual Hopf algebras to the one over braided Hopf algebras in a braided monoidal category.
We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras $mathcal{O}_infty$ and $mathcal{O}_2$. This generalizes results known for finite groups and poly-$mathbb{Z}$ groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a precursor to a general Kirchberg-Phillips-type classification theory: If two outer actions of an amenable group on a unital Kirchberg algebra are equivariantly homotopy equivalent, then they are conjugate. This marks the first C*-dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups.
Classifying elements of the Brauer group of a variety X over a p-adic field according to the p-adic accuracy needed to evaluate them gives a filtration on Br X. We show that, on the p-torsion, this filtration coincides with a modified version of that defined by Katos Swan conductor, and that the refined Swan conductor controls how the evaluation maps vary on p-adic discs, giving a geometric characterisation of the refined Swan conductor. We give applications to the study of rational points on varieties over number fields.
Let $k$ be a field and $X/k$ be a smooth quasiprojective orbifold. Let $Xto underline{X}$ be its coarse moduli space. In this paper we study the Brauer group of $X$ and compare it with the Brauer group of the smooth locus of $underline{X}$.
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