# 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

$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) \to Br(S)$$ which sends the form $$f \, dg$$ to the Brauer class of the Azumaya algebra $$\mathcal O_{S} \langle x,y\rangle/ ([x,y]-1, x^p-f, y^p-g).$$ The existence of such a map for non-affine $$S$$ requires checking the existence of certain canonical Morita equivalences, which ensure that the required map exists.

(In the original paper, $$k = \mathbb F_p$$, but if $$k$$ is perfect, then this map descends to the differentials of $$S/k$$. More generally, one may introduce Frobenius twists.)

Now suppose that I have a vector bundle $$E$$ over $$S$$. Then I can consider the algebra $$A(E, a, a^\vee) = \mathcal O_S \langle E, E^\vee \rangle / ([e, e^\vee] = e^\vee(e), e^p = a^\vee(e), (e^\vee)^p = a(e^\vee) \mid e \in E, e^\vee \in E^\vee)$$ where $$a^\vee: E \to Fr^*\mathcal O_S$$ and $$a: E^\vee \to Fr^*\mathcal O_S$$. This is Azumaya, as it splits over a cover where the $$p$$th roots of $$a$$ and $$a^\vee$$ are extracted. Is there a nice description of the Brauer class of $$A(E,a,a^\vee)$$?

My instinct is that $$[A(E,a,a^\vee)] =$$"$$\alpha(a^\vee(da))$$", except that $$da$$ does not make sense unless $$E$$ is equipped with a connection.

• If $S:=Spec(A)$ and $E$ is a finite rank projective $A$-module, there is always a connection $\nabla:E \rightarrow E\otimes \Omega^1_S$ on $E$, but $\nabla$ is not unique. I suspect you want a "unique" connection $\nabla$. For more general $S, E$ there is not always such a connection $\nabla$
– user122276
Feb 17 '21 at 12:40
• @hm2020 Indeed in the affine case there is a connection, but I am interested in the non-affine case. Feb 17 '21 at 15:50