Pure morphisms are effective for modules

TL;DR

This paper provides a new proof that a morphism of commutative rings is an effective descent morphism for modules if and only if it is pure, highlighting limitations of Moerdijk's descent criterion.

Contribution

It offers a novel proof of the equivalence between effective descent morphisms and pure morphisms for modules, and demonstrates that this result cannot be obtained from Moerdijk's criterion.

Findings

New proof of the equivalence between pure morphisms and effective descent for modules

Shows the result cannot be derived from Moerdijk's descent criterion

Clarifies the relationship between purity and descent in module theory

Abstract

Yet another proof of the result asserting that a morphism of commutative rings is an effective descent morphism for modules if and only if it is pure is given. Moreover, it is shown that this result cannot be derived from Moerdijk's descent criterion.