Covariant functors commuting with direct limits

TL;DR

This paper establishes conditions under which covariant right exact functors commute with direct limits, linking their behavior on projective modules to their general commutativity with all direct limits.

Contribution

It provides a criterion connecting the functor's commutativity with direct limits of projective modules to its overall commutativity with all direct limits.

Findings

Functor commutes with direct limits of projective modules

Conditions ensure functor commutes with all direct limits

Theoretical framework for covariant functors and direct limits

Abstract

In this work we state conditions on a covariant right exact functor so that it commutes with direct limits. These conditions are related to the commutativity of the functor under direct limits of projective modules. We prove that if the functor commutes with direct limits of projective modules, then the functor commutes with direct limits.