The twisted inverse image pseudofunctor over commutative DG rings and perfect base change

TL;DR

This paper extends Grothendieck's twisted inverse image pseudofunctor to cohomologically noetherian commutative DG rings over a Gorenstein noetherian base, establishing functoriality and base change results, and addressing recent conjectures.

Contribution

It generalizes the theory of rigid dualizing complexes to DG rings and proves functoriality and base change theorems for the twisted inverse image functor in this setting.

Findings

Extended the twisted inverse image pseudofunctor to DG rings over Gorenstein bases.

Proved functoriality for cohomologically finite and smooth maps.

Established a perfect base change theorem for the functor.

Abstract

Let $K$ be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings $A$ over $K$, such that $H^0(A)$ is essentially of finite type over $K$, and $A$ has finite flat dimension over $K$. We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for $f^{!}$ in this setting. As application, we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings. Our results generalize and solve some recent conjectures of Yekutieli.