Relative homology and maximal l-orthogonal modules

TL;DR

This paper explores the conditions under which the endomorphism rings of maximal l-orthogonal modules over artin algebras are derived equivalent, generalizing Iyama's conjecture and characterizing associated tilting modules.

Contribution

It characterizes tilting modules of the form Hom(M2,M1) using relative theories, extending Iyama's results on derived equivalences of endomorphism rings.

Findings

Provides a characterization of tilting modules in terms of relative theories.

Generalizes Iyama's conjecture for l > 1.

Establishes conditions for derived equivalence of endomorphism rings.

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.