We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring A is Cohen-Macaulay, and $a_1,\dots ,a_n$ is any sequence of elements in A, then the Koszul complex $K(A;a_1,\dots ,a_n)$ is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring A, by finding a Cohen-Macaulay DG-ring B such that ${\mathrm{H}}^0(B)=A$, and using the Cohen-Macaulay structure of B to deduce results about A. As application, we prove that if $f:X\to Y$ is a morphism of schemes, where X is Cohen-Macaulay and Y is nonsingular, then the homotopy fiber of f at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given.

我们证明了 Avramov-Golod 和 Frankild-Jørgensen 关于 Gorenstein 环的结果的 Cohen-Macaulay 版本，表明如果诺特环A是 Cohen-Macaulay，并且${\mathrm{一种}}_1,\dots ,{\mathrm{一种}}_n$是A中元素的任意序列，那么 Koszul 复数$钾(\mathrm{一种};{\mathrm{一种}}_1,\dots ,{\mathrm{一种}}_n)$是 Cohen-Macaulay DG 环。我们进一步概括了这个结果，表明它也适用于可交换的 DG 环。在证明这一点的过程中，我们开发了一种新技术来研究诺特环A的维数理论，通过找到 Cohen-Macaulay DG-环B使得${\mathrm{H}}^0(乙)=\mathrm{一种}$，并使用B的 Cohen-Macaulay 结构来推导A 的结果。作为应用，我们证明如果$F：X\to 是$是方案的态射，其中X是 Cohen-Macaulay，Y是非奇异的，则f在每一点的同伦纤维是 Cohen-Macaulay。作为另一个应用，我们概括了奇迹平坦度定理。还给出了这些应用对派生代数几何的推广。