Tilting theory for extended module categories

TL;DR

This paper develops a tilting theory framework for extended module categories, establishing new correspondences between silting complexes, torsion pairs, and tilting pairs, thereby advancing the understanding of their structure and relationships.

Contribution

It introduces $ au_{[m]}$-tilting pairs and characterizes torsion pairs induced by $(m+1)$-term silting complexes in extended module categories.

Findings

Established a Happel-Reiten-Smalo tilting theorem for extended hearts.

Characterized torsion pairs induced by $(m+1)$-term silting complexes.

Bijections between $ au_{[m]}$-tilting pairs, silting complexes, and $s$-torsion pairs.

Abstract

In extended hearts of bounded $t$-structures on a triangulated category, we provide a Happel-Reiten-Smalo tilting theorem and a characterization for $s$-torsion pairs. Applying these to $m$-extended module categories, we characterize torsion pairs induced by $(m+1)$-term silting complexes. After establishing Auslander-Reiten theory in extended module categories, we introduce $\tau_{[m]}$-tilting pairs and show bijections between $\tau_{[m]}$-tilting pairs, $(m+1)$-term silting complexes, and functorially finite $s$-torsion pairs.