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.

  • $\begingroup$ 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 $\nabla$ $\endgroup$
    – user122276
    Feb 17, 2021 at 12:40
  • $\begingroup$ @hm2020 Indeed in the affine case there is a connection, but I am interested in the non-affine case. $\endgroup$ Feb 17, 2021 at 15:50


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