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

21 pages; improved the presentation