Koszul complexes over Cohen-Macaulay rings

We prove a Cohen-Macaulay version of result by Avramov-Golod and Frankild-J{\o}rgensen about Gorenstein rings, showing that if noetherian ring $A$ is Cohen-Macaulay, $a_1,\dots,a_n$ any sequence elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ DG-ring. further generalize this result, it also holds for commutative DG-rings. In process proving this, we develop new technique to study dimension theory finding DG-ring $B$ such $\mathrm{H}^0(B) = A$, using structure deduce results $A$. As application, $f:X \to Y$ morphism schemes, where $X$ $Y$ nonsingular, homotopy fiber $f$ at every point Cohen-Macaulay. another miracle flatness theorem. Generalizations these applications derived algebraic geometry are given.