2022/06/23 by Luis Martı́nez, Martinez, Luis, Octavio Mendoza +1
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2206.11755
Let Λ be an Artin algebra and Kb(proj(Λ)) be the triangulated category of bounded co-chain complexes in proj(Λ). It is well known that two-terms silting complexes in Kb(proj(Λ)) are described by the τ-tilting theory. The aim of this paper is to give a characterization of certain n-term silting complexes in Kb(proj(Λ)) which are induced by Λ-modules. In order to do that, we introduce the notions of τn-rigid, τn-tilting and τn,m-tilting Λ-modules. The latter is both a generalization of τ-tilting and tilting in mod(Λ). It is also stated and proved some variant, for τn-tilting modules, of the well known Bazzoni's characterization for tilting modules. We give some connections between n-terms presilting complexes in Kb(proj(Λ)) and τn-rigid Λ-modules. Moreover, a characterization is given to know when a τn-tilting Λ-module is n-tilting. We also study more deeply the properties of the τn,m-tilting Λ-modules and their connections of being m-tilting in some quotient algebras. We apply the developed τn,m-tilting theory to the finitistic dimension of Λ. Finally, at the end of the paper we discuss and state some open questions (conjectures) that we consider crucial for the future develop of the τn,m-tilting theory.