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 Lie Algebra in prime characteristic](https://arxiv.org/abs/math/0205144), we have the following sentence:



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> The sheaf $D\_X$ of crystalline differential operators on a smooth variety $X$ over $k$ has a non-trivial center, canonically identified with the sheaf of functions on the Frobenius twist $T^∗X^{(1)}$ of the cotangent bundle. Moreover $D\_X$ is an Azumaya algebra over $T^∗X^{(1)}$.
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Instead of going through the general proof given, I only want to understand, in as simple a manner as possible, the situation when $X$ is the affine $n$-space over $k$. In this case, $D\_X$ is simply the Weyl algebra and the Azumaya property, if I understand correctly, means that the quotient of the Weyl algebra by its centre is isomorphic to some matrix algebra over $k$. Is there a way to construct such a matrix algebra and a corresponding isomorphism to the quotient explicitly? Any help, even in the case of the affine line, would be highly appreciated.


**P.S.** If my understanding is incorrect, could you please point out the flaw(s) and how the question could be turned into something reasonable?


**Major Edit** It has been pointed out that my understanding of the Azumaya property is incorrect. But my question remains the same: is there a direct way to prove the claim of the paper in the case when $X$ is the affine $n$-space (or even the affine line) over $k$.



Weyl algebra as an Azumaya algebra over its centre

> 
> But my question remains the same: is there a direct way to prove the
> claim of the paper in the case when is the affine -space (or even
> the affine line) over .
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> 
> 


Yes, read the proof of Proposition 1 in "[The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture](https://arxiv.org/abs/math/0512171)" by Alexei Belov-Kanel and Maxim Kontsevich.


If you are only interested in the $n=1$ case $A=k \langle X,Y \mathrel| XY-YX=1 \rangle$, then give $X$ degree $1$ and $Y$ degree $-1$, so that $A$ becomes a $\mathbb{Z}$-graded ring. It is even strongly graded, meaning $A\_1\cdot A\_{-1} = A\_0$, so all graded info is determined by the zero-part $A\_0=k[XY]$. In particular, if you divide out a graded maximal ideal you get a strongly graded ring with part of degree $0$ a field and finite over its center, and these must be central simple algebras (in this case, just $p \times p$ matrices) proving that $A$ is what is called a graded Azumaya algebra. Now, for the fun part, as there exist homogeneous central identities, a graded Azumaya algebra is a genuine Azumaya algebra. Done.


