$n$-term silting complexes in $K^b(proj(\Lambda))$

. Let Λ be an Artin algebra and K b (proj(Λ)) be the triangulated category of bounded co-chain complexes in proj(Λ) . It is well known [AIR14] that two-terms silting complexes in K b (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 K b (proj(Λ)) which are induced by Λ-modules. In order to do that, we introduce the notions of τ n -rigid, τ n -tilting and τ n,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 τ n -tilting modules, of the well known Bazzoni’s characterization for tilting modules [Ba04, Theorem 3.11]. We give some connections between n -terms presilting complexes in K b (proj(Λ)) and τ n -rigid Λ-modules. Moreover, a characterization is given to know when a τ n -tilting Λ-module is n -tilting. We also study more deeply the properties of the τ n,m -tilting Λ-modules and their connections of being m -tilting in some quotient algebras. We apply the developed τ n,m -tilting theory to the finitistic dimension and thus for n = m = 1 , we get as a particular case [Su21, 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 τ n,m -tilting theory.