On $g$-finiteness in the category of projective presentations

Abstract

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.

27 pages, 1 figure. V2 : Added Proposition 2.13