Partial cotilting modules and the lattices induced by them

Abstract

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P1P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rejp-torsion parts, and the elements of L′ by their Trp-torsion-free parts.