A silting theorem

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 $\mathop{\rm mod}\nolimits A$.