N-Term silting complexes in [formula omitted

Abstract

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 [1] 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 [12, Theorem 3.11]. 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 and thus for n=m=1, we get as a particular case [22, Theorem 2.5]. 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.