A silting theorem

2015/03/20 by Aslak Bakke Buan, Yu Zhou, Buan, Aslak Bakke +1 · 3 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1503.06129

openalex publication_date 2015/03/20 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

We give a generalization of the classical tilting theorem. We show that for a 2-term silting complex P in the bounded homotopy category K^b(\mathop\rm proj\nolimits A) of finitely generated projective modules of a finite dimensional algebra A, the algebra B = \mathop\rm End\nolimits_K^b(\mathop\rm proj\nolimits A)(P) admits a 2-term silting complex Q with the following properties: (i) The endomorphism algebra of Q in K^b(\mathop\rm proj\nolimits B) is a factor algebra of A, and (ii) there are induced torsion pairs in \mathop\rm mod\nolimits A and \mathop\rm mod\nolimits B, such that we obtain natural equivalences induced by \mathop\rm Hom\nolimits- and \mathop\rm Ext\nolimits-functors. Moreover, we show how the Auslander-Reiten theory of \mathop\rm mod\nolimits B can be described in terms of the Auslander-Reiten theory of \mathop\rm mod\nolimits A.

Cited by

• Functor-induced isomorphisms and G-matrices
• From objects finitely presented by a rigid object in a triangulated category to 2-term complexes
• Classification of silted algebras for two quivers of Dynkin type \mathbbD_n

Related

• Tilting and silting theory of noetherian algebras
• Silting modules
• Tilting theory for extended module categories
• Silting and cosilting classes in derived categories
• Extending (τ-)tilting subcategories and (co)silting modules