On an Instance of the Small Cohen-Macaulay Conjecture II

We show that any $d$-dimensional local ring $A$ with a dualizing complex, $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}A=d-1$, and cyclic deficiency module $K^{d-1}(A)$ admits a maximal Cohen–Macaulay module. It is constructed as the unique nonzero cohomology module of the cone of the derived morphism induced by a surjection $A\to K^{d-1}(A)$. When $A$ is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module ${\omega }_{A\mathrm{/}xA}$, for any $x\in {\mathrm{ann}}_A{K}^{d-1}(A)$ that is regular on $A$. This recovers a theorem of Tavanfar and Shimomoto in the $3$-dimensional quasi-Gorenstein case with $K^2(A)\cong k$. We also give examples of section rings satisfying the hypotheses of our theorem.