Tilting theory for extended module categories
Yu Zhou
TL;DR
This work generalizes Happel–Reiten–Smalø tilting to the $m$-extended heart of a bounded $t$-structure and develops an $s$-torsion theory in $m ext{-mod }A$. It establishes a bijection between $(m+1)$-term silting complexes, functorially finite $s$-torsion pairs, and $ au_{[m]}$-tilting pairs, linking silting theory with higher Auslander–Reiten structures in extended module categories. The authors construct Auslander–Reiten triangles in $m ext{-mod }A$, define the translations $ au_{[m]}$, $ au_{[m]}^-$, and develop a full $ au_{[m]}$-tilting theory, including canonical truncations and canonical triangles, thereby broadening higher tilting, silting, and AR frameworks to truncated subcategories of derived categories. The results provide a coherent triangle/extriangulated-category viewpoint, with explicit descriptions of torsion components and deep connections between tilting, torsion theory, and AR phenomena in extended module settings.
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 $τ_{[m]}$-tilting pairs and show bijections between $τ_{[m]}$-tilting pairs, $(m+1)$-term silting complexes, and functorially finite $s$-torsion pairs.