Let $A$ be a finite dimensional algebra (you can assume it is selfinjective in case this helps) and $M$ an $A$-module without projective summands. Let $B=\underline{End_A(M)}$, the stable endomorphism ring of $M$.
Question: Can we write down minimal injective copresentations of the indecomposable projective $B$-modules (which should be given by $\underline{Hom_A(M,M_i)}$) in terms of data in mod-A ? In particular, can we easily decide when such an indecomposable projective module has injective dimension at most one?
I just saw this in works of Auslander and Reiten (see https://link.springer.com/chapter/10.1007/BFb0059259 proposition 4.6.) , but there $M$ was the direct sum of all $A$-modules, which makes things easier. Maybe there is a reference for the more general case?
