Intersection theorems over DG-rings revisited

In this work we generalize two recently proved intersection theorems for DG-rings.

The Derived Improved New Intersection Theorem concerns the length of semi-free DG-modules over DG-rings and it was recently proved by the second author.

We show that it holds under weaker hypotheses.

Foxby's Intersection Theorem was generalized to DG-rings by Yang and we improve the inequality that they provided.

As an application we prove a DG version of the classic result that finite length modules of finite projective dimension only exist over Cohen-Macaulay rings, generalizing another result of Yang.