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

Abstract

Let $\Lambda$ be an Artin algebra and $K^b(proj(\Lambda))$ be the triangulated category of bounded co-chain complexes in $proj(\Lambda).$ It is well known that two-terms silting complexes in $K^b(proj(\Lambda))$ are described by the $\tau$-tilting theory. The aim of this paper is to give a characterization of certain $n$-term silting complexes in $K^b(proj(\Lambda))$ which are induced by $\Lambda$-modules. In order to do that, we introduce the notions of $\tau_n$-rigid, $\tau_n$-tilting and $\tau_{n,m}$-tilting $\Lambda$-modules. The latter is both a generalization of $\tau$-tilting and tilting in $mod(\Lambda).$ It is also stated and proved some variant, for $\tau_n$-tilting modules, of the well known Bazzoni's characterization for tilting modules. We give some connections between $n$-terms presilting complexes in $K^b(proj(\Lambda))$ and $\tau_n$-rigid $\Lambda$-modules. Moreover, a characterization is given to know when a $\tau_n$-tilting $\Lambda$-module is $n$-tilting. We also study more deeply the properties of the $\tau_{n,m}$-tilting $\Lambda$-modules and their connections of being $m$-tilting in some quotient algebras. We apply the developed $\tau_{n,m}$-tilting theory to the finitistic dimension of $\Lambda.$ 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 $\tau_{n,m}$-tilting theory.