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Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich topology). I came across the following implicit claim in what I am currently reading :

Claim: Automorphisms of $X$ which acts trivially on the residue field $k(x)$ of $x$, act trivially on $H^n_x(X,M)$,

where $H^n_x(X,M)$:= cohomology with supports along $x$.

I think the above claim should be false. I am looking for a counter example (or proof!).

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    $\begingroup$ How do automorphisms of $X$ act on $H^n_x(X, M)$? $\endgroup$ Apr 30, 2015 at 17:32
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    $\begingroup$ (An $f:X\to X$ induces maps $H^n_x(X, M)\to H^n_x(X, f^* M)$ and $H^n(X, f_* M)\to H^n(X, M)$ but there is no canonical way of identifying $f^* M$ and $f_* M$ with $M$...) $\endgroup$ Apr 30, 2015 at 17:41
  • $\begingroup$ You are right. actually in the situation I am looking at M is a sheaf on the category of smooth varieties over k. I have edited the question accordingly. $\endgroup$
    – Amit H
    May 1, 2015 at 5:17
  • $\begingroup$ Another comment: If $f:X\to X$ is an isomorphism, then $H^n_x(X, f^*(-))$ form a universal $\delta$-functor. The system of maps $f^* : H^n_x(X, -)\to H^n_x(X, f^*(-))$ forms a map between two universal $\delta$-functors, so it suffices to check whether $f^*:H^0_x(X, -)\to H^0_x(X, f^*(-))$ is an isomorphism. $\endgroup$ May 1, 2015 at 6:12

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