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I encountered the Hochschild-Serre spectral sequence in étale cohomology $$H^i(\text{Gal}(\overline{k}/k), H^j_{et}(X_{\overline{k}}, F))\Rightarrow H^{i+j}_{et}(X_{{k}}, F)$$ How is the filtration ...

Let $X$ be a smooth projective variety over a field $k$. Suppose we have étale abelian sheaves $A, B$ on $X_{\rm ét}$ such that $$H^j(X_{\rm ét}, A),\ H^j(X_{\rm ét}, B)$$ are finitely generated ...