# How cohomology group varies with tensor product [closed]

Let $\mathfrak{F}$ be a sheaf of abelian groups on a smooth scheme $X$. Suppose for some $i>1$, there exists a surjective morphism $H^i(\mathfrak{F}^{\otimes i-1}) \to H^i(\mathfrak{F}^{\otimes i})$. Is it true that this means $h^i(\mathfrak{F}^{\otimes i-1})=h^i(\mathfrak{F}^{\otimes i})=0$?

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