The finite generation conjecture for cohomology of finite tensor categories is equivalent to finite generation of Hochschild cohomology for the endomorphism algebras of projective generators.
Witherspoon,Varieties for modules of finite dimensional Hopf algebras, inGeometric and topological aspects of the representation theory of finite groups, 481–495, Springer Proc
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On the cohomology of finite tensor categories
The finite generation conjecture for cohomology of finite tensor categories is equivalent to finite generation of Hochschild cohomology for the endomorphism algebras of projective generators.