Reduction of τ-Tilting Modules and Torsion Pairs
By Gustavo Jasso. 2013
169 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bc318613e5c798fbc7ae56465ef3806e3c12cd94
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
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In Mathematical Physics
- Lidia Angeleri Hugel
- Thomas Brustle
- H. Treffinger
- Dong Yang
- David Andrew Smith
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- Thomas McConville
Citations in the corpus
- Semibricks (2016, unknown): “Now we recall a result of Jasso [20] on reductions of support τ -tilting modules.”
- Ordered Exchange Graphs (2013, background): “The compatibility of these reductions are studied in [64, 54, 56].”
- Silting Objects (2018, methodology): “A reduction process for support τ -tilting modules which is compatible with silting reduction is introduced in [61].”
- Wall and chamber structure for finite-dimensional algebras (2018, methodology): “In general, it is shown in [26] that the full subcategory M ∩(τM)∩P of modA is equivalent to the category of modules modC(M,P ) where the algebra C(M,P ) is obtained as endomorphism ring of a module M ′ over A such that (M ⊕ M , P ) is τ -tilting, see Theorem 2.”