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Let $k$ be a field, $R,S$ commutative $k$-algebras. If $\mathsf{Mod}_{fp}$ denotes the category of finitely presented modules, the external tensor product $\mathsf{Mod}_{fp}(R) \otimes_k \mathsf{Mod}_{fp}(S) \to \mathsf{Mod}_{fp}(R \otimes_k S)$, $(A,B) \mapsto A \otimes_k B$ is fully faithful (does someone have a reference for this basic fact?), which means that $$\hom_R(A,A') \otimes_k \hom_S(B,B') \cong \hom_{R \otimes_k S}(A \otimes_k B,A' \otimes_k B')$$ for $R$-modules $A,A'$ and $S$-modules $B,B'$ of finite presentation.

Question. How can we characterize those morphisms in $\mathsf{Mod}_{fp}(R) \otimes_k \mathsf{Mod}_{fp}(S)$ which represent an epimorphism in $\mathsf{Mod}_{fp}(R \otimes_k S)$?

In other words, given $\sum_i \alpha_i \otimes \beta_i \in \hom_R(A,A') \otimes_k \hom_S(B,B')$, when is the corresponding homomorphism $\sum_i \alpha_i \otimes \beta_i : A \otimes_k B \to A' \otimes_k B'$ an epimorphism?

A necessary condition is that $(\alpha_i) : \oplus_i A \to A'$ and $(\beta_i) : \oplus_i B \to B'$ are epimorphisms. It is also sufficient for pure tensors: If $\alpha : A \to A'$ and $\beta : B \to B'$ and wlog $A',B' \neq 0$, then $\alpha \otimes \beta$ is an epimorphism iff $\alpha$ and $\beta$ are epimorphisms.

I would like to have a condition which takes place in $\mathsf{Mod}(R)$ and $\mathsf{Mod}(S)$ separately.

Idea. Define recursively good epimorphisms $A \otimes_k B \to A' \otimes_k B'$ by the following clauses: a) Diagonal epimorphisms are good. b) Compositions of good epimorphisms are good. c) If there there is a morphism $C \otimes_k D \to A \otimes_k B$ such that the composition $C \otimes_k D \to A' \otimes_k B'$ is a good epimorphism, then $A \otimes_k B \to A' \otimes_k B'$ is a good epimorphism. Technically, we call an epimorphism good if it is obtained from a) by applying a finite number of clauses a),b),c).

Question. Is every epimorphism good?

Background. Actually I'm interested in a more general situation where $R,S$ are replaced by sufficiently nice $k$-schemes $X,Y$ and $A,A'$ (resp. $B,B'$) are quasi-coherent sheaves on $X$ (resp. $Y$) of finite presentation. I want to show (in order to prove this) that if $F : \mathsf{Qcoh}(X) \to \mathcal{C}$ and $G : \mathsf{Qcoh}(Y) \to \mathcal{C}$ are cocontinuous $k$-linear tensor functors, then for every epimorphism $A \boxtimes_k B \to A' \boxtimes_k B'$ also the induced morphism $F(A) \otimes G(B) \to F(A') \otimes G(B')$ is an epimorphism. I know this (quite indirectly) when $X$ is quasiprojective. For the general case I need a "separated" characterization of epimorphisms.

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What happens for vector spaces, i.e. the case $R=k=S$? –  HenrikRüping Oct 2 '13 at 19:03
    
Well in this case epis = split epis are easy to characterize (because we have fully-faithfulness). –  Martin Brandenburg Oct 2 '13 at 22:48
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