A new characterization of silting subcategories in the stable category of a Frobenius extriangulated category

TL;DR

This paper introduces a new way to characterize silting subcategories in the stable category of Frobenius extriangulated categories, extending previous results and establishing a bijective correspondence with covariantly finite subcategories.

Contribution

It generalizes existing characterizations of silting subcategories, providing a bijective correspondence with covariantly finite subcategories in Frobenius extriangulated categories.

Findings

Established a bijective correspondence between silting subcategories and covariantly finite subcategories.

Extended the characterization to stable categories of Frobenius exact categories.

Applied the results to homotopy, derived, and Gorenstein projective module categories.

Abstract

We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type correspondence for silting subcategories over triangulated categories. More specifically, for any Frobenius extriangulated category $\mathcal{C}$, we establish a bijective correspondence between silting subcategories of the stable category $\underline{\mathcal{C}}$ and certain covariantly finite subcategories of $\mathcal{C}$. As a consequence, a characterization of silting subcategories in the stable category of a Frobenius exact category is given. This result is applied to homotopy categories over abelian categories with enough projectives, derived categories over Grothendieck categories with enough projectives as well as to the stable category of Gorenstein projective modules over a ring $R$.