Does projective imply flat?

Question

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care most about the situation where $\mathcal C$ is finite $\mathbb C$-linear in the sense of arXiv:1406.4204 (in which case the monoidal structure is closed iff it is right exact in each variable). Note that, unlike in that paper, I specifically care about monoidal structures which are not rigid. An example of the category I have in mind is the category $\mathrm{Mod}^f_A$ of finite-dimensional modules for any finite-dimensional commutative algebra $A$ (with $\otimes = \otimes_A$).

Recall that an object $P \in \mathcal C$ is projective if $\hom(P,-) : \mathcal C \to \mathrm{AbGp}$ is right exact. (It is already left exact.) Note that this has nothing to do with the monoidal structure.

An object $F \in \mathcal C$ is flat if $F \otimes : \mathcal C \to \mathcal C$ is left exact. (It is already right exact.) Note that this has everything to do with the monoidal structure.

Are projective objects necessarily flat?

Of course, in $\mathrm{Mod}_A^f$ they are. The other examples I usually use of non-rigid monoidal categories are the representation theories of non-Hopf bialgebras, but there every object is flat.

Answer 1

I believe the following is a counterexample. Let $\mathcal{A}$ and $\mathcal{B}$ be closed symmetric monoidal abelian categories such that the unit object $1\in\mathcal{B}$ is projective and let $F:\mathcal{A}\to\mathcal{B}$ be a non-exact strong symmetric monoidal functor which has a right adjoint $G:\mathcal{B}\to\mathcal{A}$. For instance, if $A$ is a commutative ring and $B$ is a commutative $A$-algebra which is not flat over $A$, you could have $\mathcal{A}=\mathrm{Mod}_A$ and $\mathcal{B}=\mathrm{Mod}_B$ and $F(M)=M\otimes_A B$. Let $\mathcal{C}=\mathcal{A}\times\mathcal{B}$, and equip it with the symmetric monoidal structure given by $$(M,V)\otimes (N,W)=(M\otimes N,F(M)\otimes W\oplus V\otimes F(N)\oplus V\otimes W).$$

The unit is $(1,0)$, and associativity follows from $F$ being strong symmetric monoidal. Furthermore, this monoidal structure is closed, with internal hom given by $$\operatorname{hom}((M,V),(N,W))=(\operatorname{hom}(M,N)\oplus G(\operatorname{hom}(V,W)),\operatorname{hom}(F(M),W)\oplus\operatorname{hom}(V,W)).$$

In this category, the object $(0,1)$ is projective by hypothesis, but it is not flat because $(M,0)\otimes (0,1)=(0,F(M))$ and $F$ is not exact.

In the example mentioned above where $\mathcal{A}=\mathrm{Mod}_A$ and $\mathcal{B}=\mathrm{Mod}_B$ and $B$ happens to be a quotient of $A$, this construction has the following intuitive explanation. The monoidal product is defined as if $(M,V)$ were secretly the $A$-module $M\oplus V$ and the tensor product is just the ordinary tensor product of $A$-modules. In particular, since $B=0\oplus B$ is not flat over $A$, the object $(0,B)$ is not flat. However, the category itself doesn't believe that $(M,V)$ is just a single $A$-module $M\oplus V$, and in particular the quotient map $A\to B$ does not exist as a map $(A,0)\to (0,B)$ that would cause $(0,B)$ to fail to be projective.

For a finite $\mathbb{C}$-linear version of this example, you can take $A$ and $B$ to be finite-dimensional $\mathbb{C}$-algebras and restrict to finitely generated modules everywhere.

Answer 2

The paper:

When projective does not imply flat, and other homological anomalies, Theory and Applications of Categories, Vol 5, pp. 202-250, 1999, available here

by Gaunce Lewis shows that this behaviour is quite common in categories of Mackey functors for compact Lie groups. These categories arise very naturally in equivariant stable homotopy category. Here is a quote from the abstract: "These examples were not fabricated to illustrate the abstract possibility of misbehavior. Rather, they are drawn from the literature."