Kunneth formula for cohomology

Statement

Suppose $X$ and $Y$ are topological spaces. We then have the following relation betwen the cohomology groups of $X$, $Y$, and the product space $X\times Y$.

For any $n\ge 0$ and any module $M$ over a principal ideal domain $R$, we have:

$H^n(X\times Y;M)\cong ({\sum }_{i+j=n}{H}^i(X;M)\otimes H^j(Y;M))\oplus ({\sum }_{p+q=n+1}\mathrm{T}\mathrm{o}\mathrm{r}(H^p(X;M),H^q(Y;M)))$

Here, $\mathrm{T}\mathrm{o}\mathrm{r}$ is torsion of modules over the ring $R$.

Particular cases

Case of free modules

If all the cohomology groups $H^i(X;M)$ are free (or more generally, torsion-free) modules over $R$, and/or all the cohomology groups $H^j(Y;M)$ are free (or more generally, torsion-free) modules over $R$, then all the torsion part vanishes and we simply get:

$H^n(X\times Y;M)\cong ({\sum }_{i+j=n}{H}^i(X;M)\otimes H^j(Y;M))$