Partial cotilting modules and the lattices induced by them

ABSTRACTWe 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 = [CogenP,I.P] of torsion-free classes. Similarly, each partial tilting m o d u l e P ' d e f i n e s a l a t t i c e L ' = en en P ' , P " ] ] o f t o r s i o n c l a s s e s . G e n e r a h z i n g 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.