Let $X$ be a smooth projective variety over a field $K$ of positive characteristic. Let $\mathcal{F}$ be a sheaf of $K$algebras. Is there any criterion when the natural map from $H^i(\otimes^n\mathcal{F})$ to $H^i(\wedge^n \mathcal{F})$ for some $n>0$ is an isomorphism for all $i>0$?
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