The Mittag-Leffler condition descents via pure monomorphisms

TL;DR

This paper clarifies the proof that the Mittag-Leffler property and projectivity descend via pure monomorphisms of commutative rings, and extends the result to pure-projectivity, enhancing understanding of module properties under ring homomorphisms.

Contribution

It provides a clearer proof of the descent of the Mittag-Leffler property and projectivity via pure monomorphisms, and extends these results to pure-projectivity.

Findings

Mittag-Leffler property descends via pure monomorphisms

Projectivity descends via pure monomorphisms

Pure-projectivity also descends via pure monomorphisms

Abstract

This notes aims to clarify the proof given by Raynaud and Gruson that the Mittag-Leffler property descents via pure rings monomorphism of commutative rings. A consequence of that is that projectivity dencents via such ring homomorphisms, a revision of the proof also allows to prove that the property of being pure-projective also descents via pure monomorphisms between commutative rings}.