From the paper: For a large class of local homomorphisms \(\varphi:R\to S\), including those of finite \(G\)-dimension studied by L. L. Avramov and H.-B. Foxby [Proc. Lond. Math. Soc., III. Ser. 15, No. 2, 241–270 (1997; Zbl 0901.13011)], we assign a new numerical invariant called the quasi Cohen-Macaulay defect of \(\varphi\).
Definition. Let \(\varphi:R\to S\) be a local homomorphism. If the dualizing complexes \(D^R\) and \(D^S\) exist, \(D^\varphi=:{\mathbf R} \text{Hom}_R(D^R,D^S).\)
Definition. Let \(\varphi\) be a local homomorphism of finite Gorenstein dimension (\(G\)-dimension). The quasi dimension of \(\varphi\), denoted \({\mathbf q}\, \text{dim}\,\varphi\), is defined as \({\mathbf q}\,\text{dim}\,\varphi=\sup D^{\widehat\varphi}\), and the quasi Cohen-Macaulay defect of \(\varphi\), denoted \({\mathbf q}\text{cmd}\,\varphi\), is defined as \({\mathbf q}\text{cmd}\,\varphi= \text{amp}\,D^{\widehat \varphi}\). Here \(^\wedge\) means completion and for an \(R\)-complex \(X\): \(\sup X=\sup\{i\in\mathbb{Z}\mid H_i(X)\neq 0\}\), \(\inf X=\inf\{i\in\mathbb{Z}\mid H_i(X)\neq 0\}\) and \(\text{amp}\,X=\sup X-\inf X\).
A local homomorphism is called quasi Cohen-Macaulay if it is of finite \(G\)-dimension and has trivial quasi Cohen-Macaulay defect. We show among other things the following:
Ascent-descent theorem. Let \(\varphi:R\to S\) be a local homomorphism.
(A) If \(R\) is Cohen-Macaulay and \(\varphi\) is quasi Cohen-Macaulay, then \(S\) is Cohen-Macaulay.
(D) If \(S\) is Cohen-Macaulay and \(G\)-dim\(\varphi\) is finite, then \(\varphi\) is quasi Cohen-Macaulay.
If, furthermore, the map of spectra Spec\(\widehat S\to\text{Spec}\,\widehat R\) is surjective, one also has:
\((\text{D}')\) If \(S\) is Cohen-Macaulay and \(G\)-dim\(\varphi\) is finite, then \(\varphi\) is quasi Cohen-Macaulay, and \(R\) is Cohen-Macaulay.