Let $A$ be a finite dimensional algebra and $\mathcal{T}$ a full subcategory of mod$A$. $\mathcal{T}$ is said to be contravariantly finite in mod$A$ if for every module $M \in mod A$, there is some $X \in \mathcal{T}$ and a morphism $f:X \rightarrow M$ such that for every $X' \in \mathcal{T}$, the sequence $Hom_A(X',X) \overset{f_{\ast}}{\rightarrow} Hom_A(X',M) \rightarrow 0$ is exact. Dually, we can define covariantly finit subcategories. $\mathcal{T}$ is said to be functorially finite in mod$A$ if it is both contravariantly and covariantly finite.
$X \in \mathcal{T}$ is Ext-projective if $Ext_A^1(X,\mathcal{T})=0$. If $\mathcal{T}$ is a torsion class of a torsion pair and functorially finite in mod$A$, then how to get that there are finitely many indecomposable Ext-projective modules in $\mathcal{T}$ up to isomorphism?
Best Answer
This essentially follows from the results of
Auslander, M.; Smalø, Sverre O., Preprojective modules over Artin algebras, J. Algebra 66, 61-122 (1980). ZBL0477.16013.
and
Auslander, M.; Smalø, Sverre O., Almost split sequences in subcategories, J. Algebra 69, 426-454 (1981). ZBL0457.16017.
but an explicit statement and proof can be found in
Smalø, Sverre O., Torsion theories and tilting modules, Bull. Lond. Math. Soc. 16, 518-522 (1984). ZBL0519.16016.
Concretely, if $\mathcal{T}$ is a functorially finite torsion class and $$A\stackrel{\alpha}{\longrightarrow} T_0\longrightarrow T_1\longrightarrow0$$ is an exact sequence with $\alpha$ a minimal left $\mathcal{T}$-approximation, then the indecomposable Ext-projective modules are the indecomposable summands of $T_0\oplus T_1$.