On $d$-term silting objects, torsion classes, and cotorsion classes

TL;DR

This paper generalizes the correspondence between silting objects, torsion classes, and cotorsion classes from 2-term to d-term cases in derived categories of finite-dimensional algebras, introducing new notions for extriangulated categories.

Contribution

It extends known bijections to d-term silting objects for any d ≥ 2, involving new torsion class concepts in extriangulated categories and establishing lattice structures.

Findings

Isomorphism between d-term silting objects and torsion/cotorsion classes.

Lattice structures of cotorsion and torsion class posets.

Truncation functor induces isomorphism between these posets.

Abstract

For a finite-dimensional algebra $\Lambda$ over an algebraically closed field $K$, it is known that the poset of $2$-term silting objects in $\mathrm{K}^b(\operatorname{proj}\Lambda)$ is isomorphic to the poset of functorially finite torsion classes in $\operatorname{mod}\Lambda$, and to that of complete cotorsion classes in $\mathrm{K}^{[-1,0]}(\operatorname{proj}\Lambda)$. In this work, we generalise this result to the case of $d$-term silting objects for arbitrary $d\geq 2$ by introducing the notion of torsion classes for extriangulated categories. In particular, we show that the poset of $d$-term silting objects in $\mathrm{K}^b(\operatorname{proj}\Lambda)$ is isomorphic to the poset of complete and hereditary cotorsion classes in $\mathrm{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$, and to that of positive and functorially finite torsion classes in $D^{[-d+2,0]}(\operatorname{mod}\Lambda)$, an extension-closed subcategory of $D^b(\operatorname{mod}\Lambda)$. We further show that the posets $\operatorname{cotors}\mathrm{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ and $\operatorname{tors} D^{[-d+2,0]}(\operatorname{mod}\Lambda)$ are lattices, and that the truncation functor $\tau_{\geq -d+2}$ gives an isomorphism between the two.