Computing Certain Tor Modules On page 2 of Bernhard Keller's note, ``Derived categories and their uses'', one reads: please see the picture
My question simply is: Why is the algebra $A$ assumed to be flat over (the commutative ring) $k$ in the highlighted line? After all, given a projective/flat resolution $P$ of the right $A$-module $M$, one can recover $\mathrm{Tor}^A_n (M,N \otimes_k X)$ by tensoring $P$ with the $A$-module $N \otimes_k X$ and then taking homology. In this process, flatness of $A$ over $k$ does not seem relevant to me.
 A: I think the point he's making here is that the object $M\otimes^{\bf L}_AN$ of the derived category of $k$-modules carries more information than its homology modules $\text{Tor}^A_i(M,N)$.
In general it is not possible to recover $\text{Tor}^A_n(M,N\otimes_kX)$ from the $k$-modules $\text{Tor}^A_i(M,N)$ and the $k$-module $X$, even if $A$ is a flat $k$-algebra.
If $A$ is flat over $k$, then 
$$M\otimes^{\bf L}_A(N\otimes_kX)\cong (M\otimes^{\bf L}_AN)\otimes^{\bf L}_kX$$
(using the fact that if $P$ and $Q$ are $A$-projective resolutions of $M$ and $N$, then $P\otimes_AQ$ is a $k$-flat resolution of $M\otimes^{\bf L}_AN$), and so $\text{Tor}^A_n(M,N\otimes_kX)$ can be recovered from the object $M\otimes^{\bf L}_AN$ of the derived category of $k$-modules and the $k$-module $X$.
But if $A$ is not a flat $k$-algebra, then in general it is not even possible to do this. Then $M\otimes^{\bf L}_A(N\otimes_kX)$ may not be isomorphic to $(M\otimes^{\bf L}_AN)\otimes^{\bf L}_kX$.
It is possible to recover $M\otimes^{\bf L}_A(N\otimes_kX)$ from the complex $F\otimes_AN$ for any flat resolution $F$ of $M$, simply by tensoring with $X$, and for different $F$ these complexes $F\otimes_AN$ are quasi-isomorphic. However, there are other complexes quasi-isomorphic to $P\otimes_AN$ that don't come from a flat resolution of $M$, and tensoring these with $X$ doesn't necessarily give $M\otimes^{\bf L}_A(N\otimes_kX)$ if $A$ is not flat over $k$. So in this case it is not enough to know $M\otimes^{\bf L}_AN$ up to isomorphism in the derived category. 
