Given a flat module $F$ and an exact sequence $A_\bullet$, a tensor product of chain complexes
$$F\otimes A_\bullet$$
is again exact, where $\otimes$ is defined as in Tensor product of chain complexes
and $F$ regarded as a chain complex concentrated in degree $0$.
My question is, is it true that for any chain complex $F_\bullet$ whose components are all flat and an exact sequence $A_\bullet$, a tensor product of chain complexes
$$F_\bullet\otimes A_\bullet$$
is again exact?
Moreover, how about the inner-hom $[F_\bullet, A_\bullet]$?
Best Answer
If there are no boundedness conditions on the complexes then this need not be true. For example, let $F_\bullet=A_\bullet$ be the unbounded complex $$\cdots\xrightarrow{\times 2}\mathbb{Z}/4\mathbb{Z}\xrightarrow{\times 2}\mathbb{Z}/4\mathbb{Z}\xrightarrow{\times 2}\mathbb{Z}/4\mathbb{Z}\xrightarrow{\times 2}\cdots$$ of $\mathbb{Z}/4\mathbb{Z}$-modules (so it is both exact and a complex of flat $\mathbb{Z}/4\mathbb{Z}$-modules).
Then each component of $F_\bullet\otimes_{\mathbb{Z}/4\mathbb{Z}}A_\bullet$ is isomorphic to $$\bigoplus_{i\in\mathbb{Z}}\mathbb{Z}/4\mathbb{Z},$$ the kernel of the differential is $$\bigoplus_{i\in\mathbb{Z}}2\mathbb{Z}/4\mathbb{Z},$$ and the image of the differential is $$\left\{(a_i)_{i\in\mathbb{Z}}\in\bigoplus_{i\in\mathbb{Z}}2\mathbb{Z}/4\mathbb{Z} \,\middle\vert\,\sum_{i\in\mathbb{Z}}a_i=0\right\}.$$ So $F_\bullet\otimes_{\mathbb{Z}/4\mathbb{Z}}A_\bullet$ has homology isomorphic to $2\mathbb{Z}/4\mathbb{Z}$ in each degree, and in particular is not exact.
However, with suitable boundedness conditions, it is true.
If $F_\bullet$ is bounded, then an induction argument on the length of $F_\bullet$ shows that $F_\bullet\otimes A_\bullet$ is exact. From this one can prove that $F_\bullet\otimes A_\bullet$ is exact if
(a) either $F_\bullet$ or $A_\bullet$ is bounded, or
(b) both $F_\bullet$ and $A_\bullet$ are bounded above, or
(c) both $F_\bullet$ and $A_\bullet$ are bounded below.
[Alternatively, spectral sequence arguments can be used.]
Similar results hold for the internal Hom, but conditions (b) and (c) are replaced by the condition that one of $F_\bullet$ and $A_\bullet$ is bounded below, and the other is bounded above.