论文标题
科恩·麦克劳（Cohen-Macaulay）环上的koszul复合物
Koszul complexes over Cohen-Macaulay rings
论文作者
论文摘要
我们证明了Avramov-Golod和Frankild-Jørgensen关于Gorenstein响起的Cohen-Macaulay版本，关于Gorenstein响起，表明，如果Noetherian ring $ a $ a $是Cohen-Macaulay和$ a_1，$ a_1，\ dots，\ dots，a_n $，a_n $是$ a___ $ a $ a $ a $ $ $ $ k（a_n $ a $ k（a）; Cohen-Macaulay DG环。我们进一步概括了这一结果，表明它也适用于交换性DG环。在证明这一点的过程中，我们开发了一种新技术来研究Noetherian Ring $ a $的维度理论，通过找到Cohen-Macaulay dg-Ring $ b $，使得$ \ mathrm {h}^0（b）= a $，并使用$ b $ $ b $的cohen-macaulay结构来推出$ a $ a $ a $ a $。作为应用程序，我们证明，如果$ f：x \ to y $是方案的形态，其中$ x $是cohen-macaulay，而$ y $是非词的，那么每个点的同型$ f $都是cohen-macaulay。作为另一个应用程序，我们概括了奇迹平坦定理。还给出了这些应用程序对派生代数几何形状的概括。
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.
