Modular QFD lattices with applications to Grothendieck categories and torsion theories

Abstract

A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We extend some results about QFD modules to upper continuous modular lattices by using Lemonnier's Lemma. One result says that QFD for a compactly generated lattice L is equivalent to Condition (C): for every m∈L, there exists a compact element t of L such that t∈[0,m] and [t,m[ has no maximal element. If L is not compactly generated, then QFD and (C) separate into two distinct conditions, which are analyzed and characterized for upper continuous modular lattices. We also extend to upper continuous modular lattices some characterizations of QFD modules with Gabriel dimension. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.