Injective Modules under Faithfully Flat Ring Extensions

TL;DR

This paper investigates conditions under which the injectivity of modules is preserved under faithfully flat ring extensions, providing a near-characterization involving Ext vanishing and Hom functors.

Contribution

It establishes a new criterion linking injectivity over a ring and its extension via Ext vanishing, extending known results in commutative algebra.

Findings

Hom(S,N) injective over S implies N injective over R under certain Ext conditions

Ext^i(S,N)=0 for all i>0 characterizes injectivity of N when Hom(S,N) is injective

Provides a partial converse to the known fact about injective modules and their extensions

Abstract

Let R be a commutative ring and S be an R-algebra. It is well-known that if N is an injective R-module, then Hom(S,N) is an injective S-module. The converse is not true, not even if R is a commutative noetherian local ring and S is its completion, but it is close: It is a special case of our main theorem that in this setting, an R-module N with Ext^i(S,N)=0 for all i>0 is injective if Hom(S,N) is an injective S-module.