Greg, what do you mean by "preserving Bockstein homomorphisms"? At first, are they defined at all? The definition I know involves tensoring an exact sequence of abelian groups with $C_n$ for every $n$, hoping that this will produce another short exact sequence; however, this requires the $C_n$ to be flat.

Thanks a lot (particularly for the geometric interpretation of df + fd, which has nothing to do with my question but is a real gem). I'll ponder about your quotient category question a bit more.

I can't check your example (it leads to the question whether $3d^2-2c^2-3cd=\pm 3$ is solvable over $\mathbb Z$, what seems to require Pell equation theory) but Example 12 in the article referenced by Pete gives another example of a matrix which is not similar to its transpose over $\mathbb Z$, so I consider my question answered. Thanks anyway.

Many thanks! Example 12 from this PDF (which Google has in its cache - accessible through the preview link even when the site is down) negatively answers my question.

Yes, you did, by your definition of the action on $M^{\mathbb{ad}}$. Anyway, my knowledge of group cohomology is rather... trivial (done a few exercises in Lang), and I'm more interested in the general case of Hochschild cohomology (and Harrison as well).

This is puzzling me. Is the $\delta$ in group cohomology really defined by $\left(\delta f\right) \left(a_1 , ... , a_{n+1}\right) = a_1 f\left(a_2 , ... , a_{n+1}\right)$ $ + \sum\limits_{i=1}^{n} f\left(a_1 , ... , a_{i-1} , a_{i}a_{i+1} , a_{i+2} , ... , a_{n+1}\right) + \left(-1\right)^{n+1} a_{n+1} f\left(a_1 , ... , a_n\right) a_{n+1}^{-1}$ ? This was new to me. Anyway, I don't care too much about the group case; thanks a lot for the help in the case of general algebras.

Sorry, the hom doesn't go to $\mathbb{Z}$, but to the group module $M$ (which is turned into a $\mathbb{Z}G$-bimodule as follows: The left action of $\mathbb{Z}G$ on $M$ is the linearization of the group action that we started with; the right action of $\mathbb{Z}G$ on $M$ is the linearization of $mg=m$ for every $m\in M$ and $g\in G$).

And, if I am not mistaken, these isomorphisms commute with $\delta$.

Okay, I was imprecise as always. What I meant is: In group cohomology (written the inhomogenous way), a n-cochain is an element of $\mathrm{Hom}_{\mathrm{Set}}\left(G^n,\mathbb{Z}\right)$. (If it's already here that I am wrong, sorry.) By the universal property of the free $\mathbb{Z}$-module, $\mathrm{Hom}_{\mathrm{Set}}\left(G^n,\mathbb{Z}\right)\cong\mathrm{Hom}_{\mathbb{Z}}\left(\mathbb{Z}G^n,\mathbb{Z}\right)$ canonically. Finally, $\mathbb{Z}G^n\cong\left(\mathbb{Z}G\right)^{\otimes n}$.