Reduction of $\tau$-tilting modules and torsion pairs

Abstract

The class of support $\tau$-tilting modules was introduced recently by Adachi, Iyama and Reiten. These modules complete the class of tilting modules from the point of view of mutations. Given a finite dimensional algebra $A$, we study all basic support $\tau$-tilting $A$-modules which have given basic $\tau$-rigid $A$-module as a direct summand. We show that there exist an algebra $C$ such that there exists an order-preserving bijection between these modules and all basic support $\tau$-tilting $C$-modules; we call this process $\tau$-tilting reduction. An important step in this process is the formation of $\tau$-perpendicular categories which are analogs of ordinary perpendicular categories. Finally, we show that $\tau$-tilting reduction is compatible with silting reduction and 2-Calabi-Yau reduction in appropiate triangulated categories.

32 pages. Shortened abstract, corrected typos in references [1] and [9]

References in corpus (1)

• Intermediate co-t-structures, two-term silting objects, tau-tilting modules, and torsion classes

Cited by in corpus (26)

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• Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$
• Tilting modules over Auslander-Gorenstein Algebras
• Torsion pairs and quasi-abelian categories
• Universal localisations via silting
• On support $\tau$-tilting graphs of gentle algebras
• ICE-closed subcategories and wide $\tau$-tilting modules
• Tropical $F$-polynomials and General Presentations
• $\tau$-Tilting Theory and $\tau$-Slices
• Pairwise Compatibility for 2-Simple Minded Collections
• $\tau$-perpendicular wide subcategories
• Triangulated categories with cluster-tilting subcategories
• Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank
• $\mathcal{G}$-systems
• The category of a partitioned fan
• Counting the number of $\tau$-exceptional sequences over Nakayama algebras
• Support $\tau$-tilting subcategories in exact categories
• A characterisation of higher torsion classes
• A uniqueness property of τ exceptional sequences
• Infinitesimal semi-invariant pictures and co-amalgamation
• $\tau$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras
• Relative left Bongartz completions and their compatibility with mutations