## 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},…,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.

Original language | English |
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Article number | 107806 |

Journal | Advances in Mathematics |

Volume | 386 |

DOIs | |

State | Published - 6 Aug 2021 |

Externally published | Yes |

## Keywords

- Cohen-Macaulay ring
- DG-algebra
- Koszul complex

## ASJC Scopus subject areas

- General Mathematics