Künneth theorem for profinite cohomology.

Question

Let $G,H$ be groups and $k$ a field, there is a well known formula for the group cohomology of the product:

$$H^\ast(G\times H,k)\cong H^\ast(G,k)\otimes H^\ast(G,k)$$ I was wondering whether this is also true for $G,H$ profinite groups. My thoughts were that you should argue that the formula commutes with limits and therefore reduce the statement to the case of finite groups. Howevery this seems to gloss over many details, so I would love to see a reference.

Answer 1

A reference can be found in the book Cohomology of Number Fields Theorem 2.4.6, which states:

Let $G$ and $H$ be profinite groups and let $A$ be a discrete $H$-module, regarded as a $(G \times H)$-module via trivial action of the group $G$. Then the Hochschild-Serre spectral sequence $$ E_{2}^{p q}=H^{p}\left(G, H^{q}(H, A)\right) \Rightarrow H^{n}(G \times H, A) $$ degenerates at $E_{2}$. Furthermore, it splits in the sense that there is a decomposition $$ H^{n}(G \times H, A) \cong \bigoplus_{p+q=n} H^{p}\left(G, H^{q}(H, A)\right), $$

from which one can follow the Künneth theorem for fields.