On an Instance of the Small Cohen-Macaulay Conjecture II

Abstract

We show that any $d$-dimensional local ring $A$ with a dualizing complex, $\mathrm{depth}A=d-1$, 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 $A\to K^{d-1}(A)$. This gives new cases of the small Cohen--Macaulay conjecture. When $A$ is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module ${\omega }_{A/xA}$, for any $x\in {\mathrm{ann}}_A{K}^{d-1}(A)$ that is regular on $A$. This recovers a theorem of Tavanfar and Shimomoto for quasi-Gorenstein deformations of a 3-dimensional quasi-Gorenstein local ring $A$ with $K^2(A)\cong k$. We also construct examples of section rings satisfying the hypotheses of our theorem.

arXiv comment11 pages, final version