Injective modules under faithfully flat ring extensions (Q2790177)

Let \(\mathbf{R}\text{Hom}_R(-, N)\) and \(\text{id}_R(N)\) denote the right derived homomorphism functor and the injective dimension of \(N\), respectively. Also, assume that \(S\) is a flat \(R\)-algebra in which \(R\) is a Noetherian ring. In the paper under review, the authors prove that if every flat module has finite projective dimension, then NEWLINE\[NEWLINE\text{id}_R(N)\geq \text{id}_S \mathbf{R}\text{Hom}_R(S, N)\geq \text{id}_R \mathbf{R}\text{Hom}_R(S, N);NEWLINE\]NEWLINE moreover, equalities hold if \(S\) is a faithfully flat \(R\)-algebra. So, in particular, the injectivity of \(\text{Hom}_R(S, N)\) and the vanishing of the functor \(\text{Ext}_R^n(-,N)\) on \(S\), for every \(n\geq 1\) implies the injectivity of \(N\).