A silting theorem

TL;DR

This paper generalizes the classical tilting theorem to 2-term silting complexes, establishing new relationships between endomorphism algebras, torsion pairs, and Auslander-Reiten theory in the context of finite-dimensional algebras.

Contribution

It introduces a generalized silting theorem connecting 2-term silting complexes with algebra endomorphisms and torsion pairs, extending classical tilting theory.

Findings

Existence of a 2-term silting complex with specific properties

Natural equivalences between module categories via Hom and Ext

Description of Auslander-Reiten theory in terms of original algebra

Abstract

We give a generalization of the classical tilting theorem. We show that for a 2-term silting complex $\mathbf{P}$ in the bounded homotopy category $K^b(\mathop{\rm proj}\nolimits A)$ of finitely generated projective modules of a finite dimensional algebra $A$, the algebra $B = \mathop{\rm End}\nolimits_{K^b(\mathop{\rm proj}\nolimits A)}(\mathbf{P})$ admits a 2-term silting complex $\mathbf{Q}$ with the following properties: (i) The endomorphism algebra of $\mathbf{Q}$ in $K^b(\mathop{\rm proj}\nolimits B)$ is a factor algebra of $A$, and (ii) there are induced torsion pairs in $\mathop{\rm mod}\nolimits A$ and $\mathop{\rm mod}\nolimits B$, such that we obtain natural equivalences induced by $\mathop{\rm Hom}\nolimits$- and $\mathop{\rm Ext}\nolimits$-functors. Moreover, we show how the Auslander-Reiten theory of $\mathop{\rm mod}\nolimits B$ can be described in terms of the Auslander-Reiten theory of $\mathop{\rm mod}\nolimits A$.