TY - JOUR

T1 - Koszul complexes over Cohen-Macaulay rings

AU - Shaul, Liran

N1 - Funding Information:
The author would like to thank Sean Sather-Wagstaff for asking me if Corollary 4.6 is true, and Amnon Yekutieli for helpful remarks on a previous version of this manuscript, and for asking me if a version of Corollary 5.7 holds. The author is thankful to an anonymous referee for several suggestions that helped significantly improving this manuscript. This work has been supported by Charles University Research Centre program No. UNCE/SCI/022 , and by the grant GA ČR 20-02760Y from the Czech Science Foundation .
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/8/6

Y1 - 2021/8/6

N2 - 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 a1,…,an is any sequence of elements in A, then the Koszul complex K(A;a1,…,an) 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 H0(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.

AB - 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 a1,…,an is any sequence of elements in A, then the Koszul complex K(A;a1,…,an) 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 H0(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.

KW - Cohen-Macaulay ring

KW - DG-algebra

KW - Koszul complex

UR - http://www.scopus.com/inward/record.url?scp=85107156755&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.107806

DO - 10.1016/j.aim.2021.107806

M3 - Article

AN - SCOPUS:85107156755

SN - 0001-8708

VL - 386

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107806

ER -