In the past months I have been (slowly) studying homological algebra by myself. I have undestood that many arguments and conceps are "dual" to each other.
Here is my naive question.
Let $H$ be a subgroup of a group $G$, and let $M$ be a finite $G$-module. Is it true that $res:H^\bullet(G,M)\to H^\bullet(H,M)$ is dual of $cor:H_\bullet(H,M)\to H_\bullet(G,M)$ with respect to $Hom_{\mathbb Z}(\cdot,\mathbb{Q/Z})$?
I can't prove this even in the case of G acting trivially on M. You may assume this for concreteness.
It is not clear to me even if $res$ is null then $cor$ is null.
Thanks in advance.
PS. The book I'm reading is Rotman's Homological Algebra. I want to learn about cohomology of groups.