Tilting theory for extended module categories

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 $\ddot{I}{„}_{[m]}$-tilting pairs and show bijections between $\ddot{I}{„}_{[m]}$-tilting pairs, $(m+1)$-term silting complexes, and functorially finite $s$-torsion pairs.

v2: added Proposition 2.8 to correct an error in the proof of Theorem 4.7, included several corollaries at the end of Section 4,fixed typos, and made other small changes to the exposition. 32 pages, comments welcome