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Let $K$ be a Gorenstein noetherian ring of finite Krull dimension, andconsider the category of cohomologically noetherian commutative differentialgraded rings $A$ over $K$, such that $H^0(A)$ is essentially of finite typeover $K$, and $A$ has finite flat dimension over $K$. We extend Grothendieck'stwisted inverse image pseudofunctor to this category by generalizing the theoryof rigid dualizing complexes to this setup. We prove functoriality results withrespect to cohomologically finite and cohomologically essentially smooth maps,and prove a perfect base change result for $f^{!}$ in this setting. Asapplication, we deduce a perfect derived base change result for the twistedinverse image of a map between ordinary commutative noetherian rings. Ourresults generalize and solve some recent conjectures of Yekutieli.

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