Koszul complexes over Cohen-Macaulay rings

TL;DR

This paper proves that Koszul complexes over Cohen-Macaulay rings are Cohen-Macaulay DG-rings, extends the result to commutative DG-rings, and applies these findings to scheme morphisms and flatness theorems.

Contribution

It introduces a new technique using Cohen-Macaulay DG-rings to analyze the dimension theory of noetherian rings and extends classical results to derived algebraic geometry.

Findings

Koszul complexes over Cohen-Macaulay rings are Cohen-Macaulay DG-rings

Homotopy fibers of certain scheme morphisms are Cohen-Macaulay

Generalization of the miracle flatness theorem

Abstract

We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-J{\o}rgensen 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.