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 whether there's a way to describe which $k$-linear categories arise this way. Thus I'd like to know if there's a way to define the Brauer group of a field $k$ as classifying certain kinds of $k$-linear categories. I'd also like to know if there's a good description of the sum of elements of the Brauer group in terms of categories (is it some sort of tensor product of categories?).
The one condition I can come up with is that it should be a semisimple abelian category over $k$ for which the endomorphism algebra of the unit object is $k$.
An even more bold hope is to express the invariant map $\mathrm{Br}(\mathbb{Q}_p) \to \mathbb{Q}/\mathbb{Z}$ in terms of this category.
 A: $k$-linear cocomplete categories admit a "tensor product over $\text{Mod}(k)$" (thinking of them as module categories over $\text{Mod}(k)$) and the only thing you need to know about it to answer this question is that the tensor product of $\text{Mod}(A)$ and $\text{Mod}(B)$ is $\text{Mod}(A \otimes_k B)$. Azumaya algebras then give rise to invertible categories with respect to this tensor product and I believe every invertible such category has this form although I don't know how to prove it. More details about the tensor product in this blog post.
A more explicit characterization isn't hard but also isn't particularly enlightening - over a field, Azumaya algebras are central simple algebras, so you are looking for $k$-linear cocomplete categories where

*

*there's exactly one isomorphism class of simple object

*every object is a direct sum of copies of this simple object, and

*the (categorical) center (endomorphisms of the identity functor) is $k$.

But the invertibility characterization should hold for $k$ any commutative ring, and is in my opinion very conceptually clean; it tells us that the Brauer group is a kind of "higher Picard group" classifying a categorified version of line bundles. Some difficulties here in general with the difference between the Brauer group and the cohomological Brauer group.
