We provide new equivalent conditions for an algebra $\Lambda$ to be $g$-finite, analogous to those established by L. Demonet, O. Iyama, and G. Jasso, but within the category of projective presentations $\mathcal{ K}^{ [-1,0]}(\text{ proj} \Lambda)$. We show that an algebra has finitely many isomorphism classes of basic $2$-term silting objects if and only if all cotorsion pairs in $\mathcal{ K}^{ [-1,0]}(\text{ proj} \Lambda)$ are complete. Furthermore, we establish that this criterion is also equivalent to all thick subcategories in $\mathcal{ K}^{ [-1,0]}(\text{ proj} \Lambda)$ having enough injective and projective objects.
Categories: math.RT