Riccardo Colpi

Partial cotilting modules and the lattices induced by them

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.

Tilting objects in abelian categories and quasitilted rings

D. Happel, I. Reiten and S. Smarlo initiated an investigation of quasitilted artin K-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring K. Here, employing a notion of *-objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

Classes of generalized ∗-modules

We study the class, STAR , of all ∗-modules by means of the classes, Sλ, of all ∗λ-modules, λ> 0 being a cardinal. Since STAR, equals the intersection of the decreasing chain Sλ, λ > 0, our approach ‘from above’ complements the usual approach ‘from below’ consisting in the study of quasi-progenerators and tilting modules. We present relations between categorical properties of ∗-;modules and those of ∗-modules. Answering a question of Menini, we use the solution of the Artins problem to show that the chain Sλ,λ > No, is strictly decreasing in general.

Equivalences between projective and injective modules and morita duality for artinian rings

Given two rings A and R, we study the equivalences between all projective right A-modules and all injective right R-modules. We prove that such equivalences exist if and only if AA and RR are Artinian with a Morita duality. This naturally generalizes a well known result on quasi-Frobenius rings.