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Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional.

There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow H^*(\mathbb{Z}, M)$. It would of course be an isomorphism if $M$ were finite-dimensional. Are there natural conditions that guarantee it is an isomorphism in general?

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    $\begingroup$ Is this map inflation or restriction? $\endgroup$
    – Ehud Meir
    Commented Nov 15, 2021 at 16:12
  • $\begingroup$ Given $A \rightarrow B$, I call "inflation" the procedure of regarding a $B$-module as an $A$-module via this map. Maybe other people call it "restriction" but I think there is only one possible map in the given situation. $\endgroup$
    – user125639
    Commented Nov 15, 2021 at 17:08
  • $\begingroup$ Maybe such a condition would be that the module is smooth, namely that every point of $M$ is stabilized by an open subgroup of $\mathbb{Z}_p$. In general the map is not surjective on $H^1$. $\endgroup$
    – Uri Bader
    Commented Nov 20, 2021 at 18:27

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