Finiteness of complete intersection Hom-injective dimension after completion

Determine whether, for a local ring R and a bounded complex M with finitely generated homology, finiteness of the complete intersection Hom-injective dimension over the completion of R implies finiteness of the complete intersection Hom-injective dimension over R.

Background

The paper studies how the three injective analogues of complete intersection dimension behave under completion. It proves an inequality for complete intersection Hom-injective dimension after completion and equality when the original complete intersection Hom-injective dimension is finite.

The unresolved converse asks whether finiteness over the completed ring forces finiteness over the original local ring. A positive answer would complete the comparison of this dimension under completion in the setting of bounded complexes with finitely generated homology.

References

Let $R$ be a local ring and $M\in\Dfb$. If $CI_{ R} M$ is finite, is $CI_RM$ finite as well?

— Complete intersection and quasi-homological dimensions  (2609.26686 - Dey et al., 22 Sep 2026) in Question following Theorem 5.2, Section 5