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

TL;DR

This work extends the established 2-term silting–torsion–cotorsion correspondence to arbitrary $d\ge 2$ by formulating torsion theory in extriangulated categories and leveraging truncation functors. Central to the approach is the truncation $\tau_{\ge -d+2}$, which links $\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ with $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$, enabling a triad of poset isomorphisms among $d$-term silting objects, complete hereditary cotorsion classes, and positive/functorially finite torsion classes. The paper proves that these correspondences are compatible via explicit maps $\Phi$, $\psi$, and $\psi'$, forming a commutative diagram of bijections, and shows that both the torsion and cotorsion posets are lattices with natural sublattices arising from positivity and completeness properties. Additionally, the results connect to $d$-cluster tilting theory, providing a bridge to $d$-torsion classes and their relation to $d$-term silting through known injections and dualities. Overall, the work provides a unified higher-dimensional framework for silting, torsion, and cotorsion theories with lattice structures and categorical equivalences that generalize the classical $2$-term case.

Abstract

For a finite-dimensional algebra $Λ$ over an algebraically closed field $K$, it is known that the poset of $2$-term silting objects in $\mathrm{K}^b(\operatorname{proj}Λ)$ is isomorphic to the poset of functorially finite torsion classes in $\operatorname{mod}Λ$, and to that of complete cotorsion classes in $\mathrm{K}^{[-1,0]}(\operatorname{proj}Λ)$. 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 $\mathrm{K}^b(\operatorname{proj}Λ)$ is isomorphic to the poset of complete and hereditary cotorsion classes in $\mathrm{K}^{[-d+1,0]}(\operatorname{proj}Λ)$, and to that of positive and functorially finite torsion classes in $D^{[-d+2,0]}(\operatorname{mod}Λ)$, an extension-closed subcategory of $D^b(\operatorname{mod}Λ)$. We further show that the posets $\operatorname{cotors}\mathrm{K}^{[-d+1,0]}(\operatorname{proj}Λ)$ and $\operatorname{tors} D^{[-d+2,0]}(\operatorname{mod}Λ)$ are lattices, and that the truncation functor $τ_{\geq -d+2}$ gives an isomorphism between the two.

Figures (1)

• Figure 1: Figure taken from G1

Theorems & Definitions (73)

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