Partial tilting modules over $m$-replicated algebras

TL;DR

This paper characterizes when partial tilting modules over m-replicated hereditary algebras are tilting modules, shows they have complements, and proves the tilting quiver is connected, with applications to duplicated algebras.

Contribution

It provides a criterion for partial tilting modules to be tilting modules over m-replicated algebras and establishes the connectedness of the tilting quiver.

Findings

Partial tilting modules are tilting iff they have maximal indecomposable summands.

Every partial tilting module has at least one complement.

The tilting quiver of the algebra is connected.

Abstract

Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $\delta (T)$ the number of non isomorphic indecomposable summands of $T$. In this paper, we prove that a partial tilting $A^{(m)}$-module $T$ is a tilting $A^{(m)}$-module if and only if $\delta (T)=\delta (A^{(m)})$, and that every partial tilting $A^{(m)}$-module has complements. As an application, we deduce that the tilting quiver $\mathscr{K}_{A^{(m)}}$ of $A^{(m)}$ is connected. Moreover, we investigate the number of complements to almost tilting modules over duplicated algebras.