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Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$?

This is true in characteristic zero by Weibel and Cortiñas, but their proof doesn’t generalise to prime characteristic. Is there a characteristic-independent proof, or is there a counter example in positive characteristic?

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  • $\begingroup$ The paper Noncommutative motives of Azumaya algberas by Tabuada and Van den Bergh proves this when the rank of $A$ is invertible in $k$. I do not know a counterexample in the case when the characteristic divides the rank. $\endgroup$ Jun 17, 2020 at 15:08

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