Questions tagged [azumaya-algebras]
A generalization of central simple algebras over arbitrary commutative rings, or over a scheme
7
questions
3
votes
1answer
183 views
A local-to global principle for splitting of Azumaya algebras
Let $S$ be a finitely generated domain with the field of fractions $F.$ Let X be a smooth,
geometrically connected affine variety over $S.$ Let $A$ be an Azumaya algebra over $X.$
Assume that for all ...
6
votes
1answer
434 views
An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich
Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$.
In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map
$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
7
votes
1answer
218 views
Weyl algebra as an Azumaya algebra over its centre
Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple ...
7
votes
1answer
403 views
Category of modules over an Azumaya algebra and the Brauer group
Let $k$ be a field, and let $\alpha \in \mathrm{Br}(k)$. Let $A$ be an Azumaya algebra representing $\alpha$. Then the category $A$–$\mathrm{mod}$ depends only on $\alpha$.
I would like to know ...
5
votes
0answers
104 views
Hochschild cohomology of an Azumaya algebra
Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$?
This is ...
2
votes
1answer
493 views
Are differential forms related to Azumaya algebras?
While studying vector bundle valued differential forms, $\Omega^{\bullet}(M, E)$, or $\Omega^{\bullet}(M, \mathrm{End}(E))$ if that helps this discussion, I've come across some work in Azumaya ...
5
votes
1answer
298 views
An example of an Azumaya algebra that isn't free over its centre
Azumaya originally defined an Azumaya algebra (which he called a proper maximally central algebra) to be an algebra A which is a free module of finite rank over its centre Z such that the natural map
$...
