Quasi-injective dimension without a dualizing complex

Determine whether, for a local ring R not necessarily admitting a dualizing complex and a finitely generated R-module M with finite upper complete intersection injective dimension, the equality $CI^*\text{-}id_R M=qid_R M$ holds; more generally, establish whether quasi-injective dimension satisfies flat descent.

Background

The paper establishes the equality between upper complete intersection injective dimension and quasi-injective dimension for finitely generated modules over local rings admitting a dualizing complex. The proof uses Grothendieck duality and a global construction based on the dualizing complex.

The authors explicitly leave open whether the equality remains valid when no dualizing complex exists. They identify a possible route through completion and formulate the broader unresolved issue of flat descent for quasi-injective dimension.

References

Does \Cref{cor:ci*id&qid} hold for rings that do not necessarily have a dualizing complex? A possible approach would be to reduce to the completion by provig that $qid_{ R} M<\infty$ implies $qid_RM<\infty$. More generally we ask whether flat descent holds for the quasi-injective dimension.

— Complete intersection and quasi-homological dimensions  (2609.26686 - Dey et al., 22 Sep 2026) in Question following Corollary 4.3, Section 4