# Group cohomology build on rings other than $\mathbb{Z}$

The usual group cohomology $H^n(G, M) = \text{Ext}_{\mathbb{Z}[G]}^n(\mathbb{Z}, M)$ and can be computed via canonical chain complex $$\cdots \rightarrow \mathbb{Z}[G^{n+1}] \rightarrow \cdots \rightarrow \mathbb{Z} \rightarrow 0$$

Is there similar cohomology theory where we replace $\mathbb{Z}$ by a different ring such as $\mathbb{Z}_p$?

• Just replace the integers by your preferred ring of coefficients! – Mariano Suárez-Álvarez Sep 12 '16 at 23:36
• @MarianoSuárez-Álvarez I am aware that the general theory of derived functor still works when we replace $Z$ by an arbitrary ring. What I am worry about is whether the classical results such as Tate's 2-periodicity, Herbrand quotient, cohomological triviality still applies. The proofs does not give off the impression to be dependent on Z; but I want to be sure so a reference material would be great. – An Hoa Sep 13 '16 at 3:45
• I don't think there is a reference which deals with that, really. – Mariano Suárez-Álvarez Sep 13 '16 at 4:03
• @MarianoSuárez-Álvarez I think I just found it. According to en.wikipedia.org/wiki/Ext_functor, the ring does not matter. This is also Exercise 6.1.2, page 161-162 of Weibel's book An introduction to homological algebra. – An Hoa Sep 13 '16 at 4:31
• That exercise only says that you can use the complex with $Z$ replaced by a commutative ring $k$ to compute cohomology in the category of $kG$-modules, not that everything works —periodicity and so on. The exercise is a just a long way of saying "replace the integers by your preferred ring of coefficients". – Mariano Suárez-Álvarez Sep 13 '16 at 4:43