TITLE: Is the tensor product of two projective sheaves projective?
QUESTION [13 upvotes]: The tensor product of two abelian sheaves $\mathcal{F}$ and $\mathcal{G}$ is defined to be the sheafification of $U \mapsto \mathcal{F}(U) \otimes \mathcal{G}(U)$. In the category of abelian groups, the tensor product of projectives is projective, so it seems logical to hope a similar thing with sheaves.
I was trying to look at the presheaf $U \mapsto \mathcal{F} (U) \otimes \mathcal{G}(U)$. It is projective at every section and I was trying to use this to assemble a natural transformation to complete the necessary diagram that would make $\mathcal{F} \otimes \mathcal{G}$ projective, but I don't see a way of doing this that will make the necessary diagrams commutative to assemble a natural transformation.
Another attempt was using the adjunction with the internal sheaf hom because we have that
$$Hom(\mathcal{F}, \mathscr{H}\kern -.5pt om (\mathcal{G}, -)) \cong Hom (\mathcal{F} \otimes \mathcal{G}, -)$$
and $\mathcal{F} \otimes \mathcal{G}$ is projective if and only if $Hom(\mathcal{F} \otimes \mathcal{G}, -)$ is exact. The usual argument with groups is composing exact functors, since the internal hom coincides with the usual hom. However, we don't know here if $\mathscr{H}\kern -.5pt om(\mathcal{G},-)$ is exact when $\mathcal{G}$ is projective (and my attempts at proving this have been unsuccessful as well).
I would also be interested to know if this statement is true in a general closed monoidal category, but the specific application I was looking for is in the category of sheaves.
EDIT: this question is more specifically referring to condensed abelian groups (sheafs of abelian groups on the site of profinite sets), but I thought it might be true in a more general sheaf setting.

REPLY [10 votes]: This fails in general. In fact, it fails for condensed abelian groups, which is a mayor annoyance.
Projective condensed abelian groups $M$ are in fact exactly the retracts of $\mathbb Z[S]$ where $S$ is an extremally disconnected profinite set, or even the Stone-Cech compactification of a discrete set. (A curious, slightly nontrivial fact is that in that case also $M\oplus \mathbb Z[S]\cong \mathbb Z[S]$.) In any case, the question becomes whether $\mathbb Z[S]\otimes \mathbb Z[T]\cong \mathbb Z[S\times T]$ is projective. This can be shown never to be the case when $S$ and $T$ are infinite, for example via Proposition 4.8 in the Analytic Geometry lectures.
I'd be extremely interested in the answer to the following question:

If $S$ and $T$ are extremally disconnected profinite sets, does $\mathbb Z[S\times T]$ have finite projective dimension?

I've spent quite a bit of time in this, but couldn't make progress in either direction. In some ways, it seems unlikely, but a positive answer would be really helpful.