Injective modules under faithfully flat ring extensions

Abstract:

Let $R$ be a commutative ring and let $S$ be an $R$-algebra. It is well-known that if $N$ is an injective $R$-module, then $\operatorname {Hom}_R(S,N)$ 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 $\operatorname {Ext}^{>0}_R(S,N) =0$ is injective if $\operatorname {Hom}_R(S,N)$ is an injective $S$-module.