Koszul complexes over Cohen-Macaulay rings

Abstract

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.

26 pages, final version, to appear in Advances in Mathematics

References in corpus (2)

• The Cohen-Macaulay property in derived commutative algebra
• On commutative differential graded algebras

Cited by in corpus (5)

• Sequence-regular commutative DG-rings
• Prismatic Kunz's theorem
• Smooth flat maps over commutative DG-rings
• Maximal Cohen-Macaulay DG-complexes
• Categorical properties of reduction functors over non-positive DG-rings