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Suppose $X$ is a scheme, and $A, B$ two étale sheaves of locally compact topological abelian groups. Assume there is a map of étale sheaves such that for any $f: U\to X$ étale, $\Gamma(U_{\rm ét}, A)\to \Gamma(U_{\rm ét}, B)$ is continuous with discrete image.

Can the higher cohomology groups be topologized with locally compact topologies such that $H^j(X_{\rm ét}, A)\to H^j(X_{\rm ét}, B)$ is continuous with discrete image?

In other words, if this is possible for $j=0$ on all objects of the small étale site of $X$, is it possible for all $j$?

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    $\begingroup$ You can put the discrete topology on $H^j(X_{\text{ét}},-)$. I somehow imagine this is not what you want... $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 9, 2018 at 4:28

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