I'm working through some results in Almost split sequences in subcategories by M. Auslander and S. Smalø, specifically the results 5.6 - 5.10.
The specific result I'm trying to understand is proposition 5.6 $(b) \sim (c)$
$\underline{\operatorname{Hom}}_{\Lambda}(\operatorname{Tr}DM, N') = 0$ for all submodules $N'$ of $N$ if and only if $\operatorname{Hom}_{\Lambda}(\operatorname{Tr}DM, N) = 0$
In the proof, they choose a non-zero morphism $f : \operatorname{Tr}DM \rightarrow N$ with $\operatorname{Im}(f) = N'$. Then they say that $\operatorname{Tr}DM$ has no non-trivial projective summands.
So, $\operatorname{Tr}D : \overline{\operatorname{mod}}\Lambda \rightarrow \underline{\operatorname{mod}}\Lambda$ is the functor above. If $\operatorname{Tr}DM$ did have a non-trivial projective summand, say $P$ such that $P \oplus Q \cong \operatorname{Tr}DM$, then there would exist a morphism $\underline{\operatorname{Hom}}_{\Lambda}(P, \operatorname{Tr}DM)$ which would have to be non-zero, thus breaking the assumption that we have a stable category. Is this the reason?
It’s not that modules in the stable category are not allowed to have nonzero projective summands. But by definition of the transpose $\text{Tr}$, the transpose $\text{Tr}X$ of a module $X$ will never have a nonzero projective summand.
Recall that $\text{Tr}X$ is constructed by taking a minimal projective presentation $P_1\to P_0\to X\to0$, then taking $P_0^*\to P_1^*$, where $P^*=\text{Hom}_\Lambda(P,\Lambda)$, and taking the cokernel. But $\text{Hom}_\Lambda(-,\Lambda)$ is a contravariant equivalence from finitely generated projective right modules to finitely generated projective left modules, so if $P_0^*\to P_1^*$ has a nonzero projective summand $Q^*$ in its cokernel, so $0\to Q^*$ is a summand of $P_0^*\to P_1^*$, then $Q\to 0$ is a summand of $P_1\to P_0$, contradicting minimality.