I need a reference for the following fact:
Let $R$ be (for simplicity) an algebra over the field $k$, let $A, B$ be $R$-modules. Let $E = Ext^1_R(B, A)$. There is a natural isomorphism between $Hom_k(E, E)$ and $Ext^1_R(E\otimes_k B, A)$. Let $\eta: 0 \to A \to N \to E \otimes_k B \to 0$ be the extension which corresponds to $1_E$ under this isomorphism.
Then $\eta$ is the universal extension in the sense that the map $Hom_R(B, E\otimes_k B) \to E$ induced by applying $Hom_R(B, -)$ to $\eta$ sends $e \otimes 1_B \mapsto e$.
This is written down somewhere, but I can't find it. Thanks for your help.
PS $R$ is associative but not commutative, and contains $k$; it can be assumed noetherian if that helps. $A$, $B$ might be infinite-dimensional over $k$, but it's sufficiently general if we assume they're noetherian (as $R$-modules).
