7
$\begingroup$

Let $R,S,T$ be commutative rings and assume that $R,S$ are $T$-algebras.

In an answer to this question, Pierre-Yves Gaillard gives an example of an $R \otimes_T S$-module that cannot be written as the tensor product of an $R$-module and an $S$-module (there, $T=k$, $R=S=k^2$ where $k$ is a field).

I'm interested in the relation between the module categories Mod-$R$, Mod-$S$ and Mod-$R \otimes_T S$. Is there some kind of general operation (a "tensor product") on abelian categories that takes Mod-$R$, Mod-$S$ and Mod-$T$ (or $T$ itself) and produces Mod-$R \otimes_T S$?

  • 0
    @Sebastian: Well, by the general theory of Morita equivalence, $\textrm{Mod-}R$ is equivalent to $\textrm{Mod-}S$ if and only if $R \cong S$ as rings, so in theory, yes, it should be possible if you are willing to restrict your attention to commutative rings. But the general case looks more doubtful, as the passage from $R$ to $\textrm{Mod-}R$ loses information.2012-01-31

1 Answers 1

3

Yes, Deligne's tensor product, Kelly's tensor product, as all as the coproduct of cocomplete $\otimes$-tensor categories. They all have the property $\mathsf{Mod}(R) \otimes_k \mathsf{Mod}(S) \simeq \mathsf{Mod}(R \otimes_k S)$. See Schäppi's work for generalizations to schemes and quasi-coherent sheaves.

  • 0
    Thank you for digging up the old question! I had come across the Deligne tensor product in the meantime, but the other two references are new to me.2014-01-07