Tilting in Module Categories

Let M be a module over an associative ring R and σ[M ] the category of M -subgenerated modules. Generalizing the notion of a projective generator in σ[M ], a module P ∈ σ[M ] is called tilting in σ[M ] if (i) P is projective in the category of P -generated modules, (ii) every P -generated module is P presented, and (iii) σ[P ] = σ[M ]. We call P self-tilting if it is tilting in σ[P ]. Examples of (not self-small) tilting modules are I Q/ZZ in the category of torsion ZZ-modules, I Q⊕ I Q/ZZ in the category ZZ-Mod, certain divisible modules over integral domains, and also cohereditary coalgebras C over a QF-ring in the category of comodules over C. Self-small tilting modules P in σ[M ] are finitely presented in σ[M ]. For M = P , they are just the ∗-modules introduced by C. Menini and A. Orsatti, and for M = R, they are the usual tilting modules considered in representation theory. Notice that our techniques and most of our results also apply to locally finitely generated Grothendieck categories.