Dualities Between Finitely Closed Subcategories of Modules

Abstract

<p>The thesis studies dualities between categories of modules which are finitely closed, i.e. closed under submodules, factor modules and finite direct sums. Omitting the requirement that the categories contain all finitely generated modules from the classical Morita situation provides a generalization which substantially increases the number of rings that posses such a duality.</p> <p>In Chapter II we prove that a duality between two finitely closed categories A and B of modules is representable if and only if A and B consists of linearly compact modules. While a linearly compact finitely closed category of modules is always an AB5*- category with no infinite direct sums, we demonstrate the converse in Chapter III for certain rings including all commutative ones, thus simplifying our characterization of representable dualities in these cases; we were however unable to prove this in general or to give a counterexample. In Chapter IV we show that a duality between two arbitrary finitely closed categories of modules over commutative rings may be decomposed into representable dualities between finitely closed categories of modules over local rings.</p>