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Let $U$ be a normal subgroup of a group $G$ of finite index. On cohomology, somewhat dual to the functorially defined restriction map, $\text{res}^G_U\colon H^n(G, A) \to H^n(U, A)$, the finite index allows us to define a $0$-dimensional map that induces a correstriction $\text{cor}^G_H\colon H^n(U, A) \to H^n(G, A)$. This map has interesting properties relating it to $\text{res}^G_H$, such as the index formula and the double coset formula.

Has the same been done for the inflation map $\text{inf}^G_{G/U} \colon H^n(G/U, A^U) \to H^n(G, A)$? A way to define a $0$-dimensonal map that would induce a coinflation $$\text{coinf}^G_{G/U}\colon H^n(G, A) \to H^n(G/U, A^U)$$ on cohomology with properties analogous to the correstriction?

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  • $\begingroup$ @ChrisGerig I haven't been so lucky, the first 5 pages of a search for "coinflation on cohomology" only shows coinflation defined at homology, not cohomology groups. Same goes for "$\text{coinf}$" references on Weibel's book, all three appearances refer to maps defined on group homology rather than cohomology. $\endgroup$
    – HASouza
    Commented Nov 22, 2017 at 3:10
  • $\begingroup$ You’re right! And inflation does not exist on homology. I think this boils down to an issue with the map on group rings $\mathbb{Z}[G]\to\mathbb{Z}[G/H]$. All the other maps use functoriality of (co)homology (in both variables: groups and modules) and Shapiro’s lemma associated to (co)extension of scalars coming from inclusion of H into G. $\endgroup$
    – Chris Gerig
    Commented Nov 22, 2017 at 4:19
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    $\begingroup$ One candidate that could be named coinf is the map $H^{n-i}(G,A)\to H^i(G/H,D_n(H,A))^{\vee}$ where $\vee$ is the Pontryagin dual and $D_n(H,A)$ is the inductive limit of $H^n(U,A)^{\vee}$ as $U$ runs through subgroups of $G$ containing $H$. The dual of this map appears in Tate's spectral sequence together with the dual of the corestriction map, just like inf and res do in the Hochschild-Serre spectracl sequence. (Cohomology of number fields II.5.4) $\endgroup$
    – Chris Wuthrich
    Commented Nov 22, 2017 at 9:37

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