AI 中文总结
本文针对具有对偶复形、depth A=d-1 及循环缺损模 K^{d-1}(A) 的 d 维局部环，构造出其极大 Cohen-Macaulay 模，在拟-Gorenstein 环情形下得到与已有定理一致结果，并给出满足假设的截面环实例。
AI 中文摘要
我们证明：任何具有对偶复形的 d 维局部环 A，若满足 depth A = d-1 且循环缺损模 K^{d-1}(A)，则它存在极大 Cohen-Macaulay 模。该模由 A→K^{d-1}(A) 的满射诱导的导出态射的锥的唯一非零上同调模构造而成。当 A 是拟-Gorenstein 环时，对任意在 A 上正则且属于 ann_A K^{d-1}(A) 的 x，该模可等同于典范模 ω_{A/xA} 的第一合冲。这在 3 维拟-Gorenstein 环且 K^2(A)≅k 的情形下，重新得到 Tavanfar 与 Shimomoto 的定理。我们还给出满足本文定理假设的截面环例子。
英文摘要
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 $A\to K^{d-1}(A)$. When $A$ is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module $ω_{A/xA}$, for any $x\in\operatorname{ann}_A K^{d-1}(A)$ 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.
Comments9 pages, comments welcome
