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

TL;DR

This paper characterizes $g$-finiteness of an algebra via properties of cotorsion pairs and thick subcategories within the category of projective presentations, extending previous conditions to a new categorical context.

Contribution

It introduces new equivalent conditions for $g$-finiteness in the category of projective presentations, linking it to cotorsion pairs and thick subcategories.

Findings

Finiteness of 2-term silting objects corresponds to complete cotorsion pairs.

All thick subcategories have enough injective and projective objects under $g$-finiteness.

Provides new categorical criteria for $g$-finiteness in 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.