Derived quotients of Cohen-Macaulay rings

It is well known that if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is an $A$-regular sequence, then the quotient ring $A/(a_1,\dots,a_n)$ is also a Cohen-Macaulay ring. In this talk we explain that by deriving the quotient operation, if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is any sequence of elements in $A$, the derived quotient of $A$ with respect to $(a_1,\dots,a_n)$ is Cohen-Macaulay. As an application, we generalize the miracle flatness theorem to derived algebraic geometry. As another application, given a morphism $f:X\to Y$ from a Cohen-Macaulay scheme to a nonsingular scheme, we show that the homotopy fiber of $f$ at every point is Cohen-Macaulay.