Pinching Azumaya algebras Fischer, Johannes
Journal of pure and applied algebra,
November 2023, 2023-11-00, Letnik:
227, Številka:
11
Journal Article
Recenzirano
We show, that for a morphism of schemes from X to Y, that is a finite modification in finitely many closed points, a cohomological Brauer class on Y is represented by an Azumaya algebra if its ...pullback to X is represented by an Azumaya algebra. Part of the proof uses an extension of a result by Ferrand, on pinching of finite locally free sheaves, to Azumaya algebras.
Purity for flat cohomology Česnavičius, Kęstutis; Scholze, Peter
Annals of mathematics,
01/2024, Letnik:
199, Številka:
1
Journal Article
We establish the flat cohomology version of the Gabber-Thomason purity for \'etale cohomology: for a complete intersection Noetherian local ring $(R, \mathfrak{m})$ and a commutative, finite, flat ...$R$-group $G$, the flat cohomology $H^i_{\mathfrak{m}}(R, G)$ vanishes for $i < \mathrm{dim}(R)$. For small $i$, this settles conjectures of Gabber that extend the Grothendieck-Lefschetz theorem and give purity for the Brauer group for schemes with complete intersection singularities. For the proof, we reduce to a flat purity statement for perfectoid rings, establish $p$-complete arc descent for flat cohomology of perfectoids, and then relate to coherent cohomology of $\mathbb{A}_{\mathrm{inf}}$ via prismatic Dieudonn\'e theory. We also present an algebraic version of tilting for \'etale cohomology, use it reprove the Gabber-Thomason purity, and exhibit general properties of fppf cohomology of (animated) rings with finite, locally free group scheme coefficients, such as excision, agreement with fpqc cohomology, and $p$-adic continuity.
The Bogomolov multiplier B0(G) of a finite group G is the subgroup of the Schur multiplier H2(G,Q/Z) consisting of the cohomology classes which vanish after restricting to every abelian subgroup of ...G. We give a new proof of a Hopf-type formula for B0(G) and derive an exact sequence for the cohomological version of the Bogomolov multiplier. Using this exact sequence we provide necessary and sufficient conditions for the corresponding inflation homomorphism to be an epimorphism and to be the zero map. Finally, we give a complete list of groups of order p6, for odd prime p, having trivial Bogomolov multiplier, so completing the 2020 investigation of Chen and Ma.
An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and ...isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.
We provide explicit generators of the torsion of the second cohomology of bielliptic surfaces, and we use this to study the pullback map between the Brauer group of a bielliptic surface and that of ...its canonical cover.
We develop an equivariant Dixmier-Douady theory for locally trivial bundles of C⁎-algebras with fibre D⊗K equipped with a fibrewise T-action, where T denotes the circle group and D=End(V)⊗∞ for a ...T-representation V. In particular, we show that the group of T-equivariant ⁎-automorphisms AutT(D⊗K) is an infinite loop space giving rise to a cohomology theory ED,T⁎(X). Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to ED,T1(X)≅X,BAutT(D⊗K). We compute this group for tori and compare the case D=C to the equivariant Brauer group for trivial actions on the base space.
We give formulas for calculating the unramified Brauer group of a homogeneous space X of a semisimple simply connected group G with finite geometric stabilizer \bar F over a wide family of fields of ...characteristic 0. When k is a number field, we use these formulas in order to study the Brauer-Manin obstruction to the Hasse principle and weak approximation. We prove in particular that the Brauer-Manin pairing is constant on X(k_v) for every v outside of an explicit finite set of nonarchimedean places of k.
We determine the Brauer group and Picard group of the moduli space
of stable parabolic vector bundles of rank r with determinant L on a real curve Y of arithmetic genus
with at most nodes as ...singularities.
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