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For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is bigger than in char=0), so gives a class $[\mathcal{D}_X]\in Br(T^*X^{(1)})$ which is always non-zero, because differential operators form a non-trivial Azumaya algebra and is functorial under morphisms of cotangent bundles coming from etale morphisms of the base.

Are there any other non-trivial functorial classes in Brauer group?

Namely, can we choose a class $c_X\in Br(X)$ for every $X$ in some reasonable subcategory $\mathcal{C}\subset \mathrm{Var}_k$ such that for all morphisms $f:X\to Y$ we have $c_X=f^*c_Y$ and not all $c_X$ are zero?

Probably, when considering the whole category, the answer is no. Can something be said for smaller subcategories?

In char=0 I don't even know any examples with reasonably large subcategory.

(Edited thanks to David Ben-Zvi)

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    $\begingroup$ Here is another example: $c_X = 2[\mathcal{D}_X]$. Just kidding :) $\endgroup$
    – Jason Starr
    Commented Sep 22, 2016 at 1:19
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    $\begingroup$ Would you elaborate on your example? I guess you're saying that $D_X$ is Azumaya over the Frobenius-twisted cotangent of $X$, and that that has same $Br$ as $X$, and that the resulting classes are functorial just under pullback on the base? $\endgroup$
    – David Ben-Zvi
    Commented Sep 22, 2016 at 1:21
  • $\begingroup$ The classes are functorial for etale morphisms. $\endgroup$
    – Jason Starr
    Commented Sep 22, 2016 at 1:36
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    $\begingroup$ $[D_X]$ can't be functorial for etale morphisms as you could kill the class and the OP says it is never zero. So the element in Br of twisted cotangent does not come from X (I'm guessing it is ramified at $\infty$). So I think we have no examples whatsoever right now. $\endgroup$
    – Count Dracula
    Commented Sep 22, 2016 at 12:54
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    $\begingroup$ Sorry for ignorance, just would like some more details to try to understand the example. I would have thought $D_X$ can't be Azumaya over $X$ since it's not finite rank over $O_X$ and is not etale locally on $X$ a matrix algebra. It has a quotient that is (mod out by p-curvatures) but that quotient I think is Morita trivial (it's descent data for the Frobenius twist of $X$). So it seems you'd need to transfer the Brauer class from the twisted cotangent to the base, and then I would like to understand why that's functorial under etale morphisms of the base rather than of twisted cotangent. $\endgroup$
    – David Ben-Zvi
    Commented Sep 22, 2016 at 12:56

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