Relative homology and maximal l-orthogonal modules

Magdalini Lada

paper

Relative homology and maximal l-orthogonal modules

arXiv:0804.2335

Abstract

Let $Ł$ be an artin algebra. Iyama conjectures that the endomorphism ring of any two maximal $l$ -orthogonal modules, $M_{1}$ and $M_{2}$ , are derived equivalent. He proves the conjecture for $l = 1$ , and for $l > 1$ he gives some orthogonality condition on $M_{1}$ and $M_{2}$ , such that the $\End_Ł(M_2)^\op$-$\End_Ł(M_1)$-bimodule $\Hom_Ł(M_2,M_1)$ is tilting, which implies that the rings $\End_Ł(M_2)$ and $\End_Ł(M_1)$ are derived equivalent (see \cite{H}). The purpose of this paper is to characterize tilting modules of the form $\Hom_Ł(M_2,M_1)$ in terms of the relative theories induced by the $Ł$ -modules $M_{1}$ and $M_{2}$ , thus getting a generilization of Iyama's result.