Complete intersection and quasi-homological dimensions

Abstract

We prove that, over a commutative Noetherian ring, every finitely generated module of finite complete intersection dimension has finite quasi-projective dimension. Our proof adapts Bergh's technique of reducing complexity, originally used to establish virtual smallness for complexes of finite complete intersection dimension over local rings, by successively enlarging the perfect locus of the cones and controlling their homology. This gives an affirmative answer to part of a question of Jorgensen, Takahashi, and the third author. We also establish a dual injective version of our result for rings admitting a dualizing complex. Finally, we show that, over such rings, the three injective analogues of complete intersection dimension appearing in the literature coincide for bounded complexes with finitely generated homology, while two of them coincide even without a dualizing complex. We also give an example showing that the finite-generation hypothesis is necessary, thereby providing a full answer to a question of Sather-Wagstaff. We conclude by studying the localization behavior of these dimensions, obtaining a partial answer to a question of Sather-Wagstaff and Totushek.