# Hochschild cohomology of an Azumaya algebra

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?

• 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. Jun 17, 2020 at 15:08