New characterizations of restricted injective dimensions for complexes (Q828940)

Let $R$ be a commutative noetherian ring, and $\mathcal{D}(R)$ denote the derived category of $R$. For any $X\in \mathcal{D}_{\sqsubset}(R)$, the small restricted injective dimension of $X$ is defined as \[ \operatorname{rid}_{R}(X) = \sup \Big\{- \inf \left(\mathbf{R}\mathrm{Hom}_{R}(M,X)\right): M \text{ is a finitely generated } R\text{-module of finite projective dimension} \Big\}, \] and for any $X\in \mathcal{D}_{\sqsupset}(R)$, the small restricted flat dimension of $X$ is defined as \[ \operatorname{rfd}_{R}(X) = \sup \Big\{\sup \left(M\otimes_{R}^{\mathbf L} X\right): M \text{ is a finitely generated } R\text{-module of finite projective dimension} \Big\}. \] The authors of this paper give a new characterization of the small restricted injective dimension in their main theorem, and then derive two base change inequalities as a corollary to it. As an application, they further provide a characterization of almost Cohen-Macaulay rings as follows: Theorem. Let $R$ be a local ring. Then the following assertions are equivalent: \begin{itemize} \item[(i)] $R$ is almost Cohen-Macaulay, i.e. $\operatorname{cmd}(R):= \dim(R)- \operatorname{depth}(R) \leq 1$. \item[(ii)] $\operatorname{rfd}_{R}(X)= \operatorname{fd}_{R}(X)$ for every $X\in D_{\square}(R)$ of finite flat dimension. \item[(iii)] $\operatorname{rfd}_{R}(M)= \operatorname{fd}_{R}(M)$ for every $R$-module of finite flat dimension. \end{itemize}