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

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.

38 pages. Final version, to appear in Advances in Mathematics

References in corpus (5)

• Reduction of Hochschild cohomology over algebras finite over their center
• Relations between derived Hochschild functors via twisting
• Duality and Tilting for Commutative DG Rings
• An improvement on the base-change theorem and the functor $f^{!}$
• Compactification for essentially finite-type maps

Cited by in corpus (7)

• Injective DG-modules over non-positive DG-rings
• Completion and torsion over commutative DG rings
• The Cohen-Macaulay property in derived commutative algebra
• Sequence-regular commutative DG-rings
• Open loci results for commutative DG-rings
• Smooth flat maps over commutative DG-rings
• Maximal Cohen-Macaulay DG-complexes