Relative homology and maximal l-orthogonal modules

Abstract

Let $\L$ 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 $\End_\L(M_2)^\op$-$\End_\L(M_1)$-bimodule $\Hom_\L(M_2,M_1)$ is tilting, which implies that the rings $\End_\L(M_2)$ and $\End_\L(M_1)$ are derived equivalent (see \cite{H}). The purpose of this paper is to characterize tilting modules of the form $\Hom_\L(M_2,M_1)$ in terms of the relative theories induced by the $\L$-modules $M_1$ and $M_2$, thus getting a generilization of Iyama's result.