Koszul complexes over Cohen-Macaulay rings

We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jorgensen about Gorenstein rings, showing that if a noetherian ring A is Cohen-Macaulay, and a(1), ..., a(n) is any sequence of elements in A, then the Koszul complex K(A; a(1), ..., 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 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 -> 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. (C) 2021 Elsevier Inc. All rights reserved.