Simultaneous finiteness of cid(φ) and cid_R(R_φ)

Determine whether, for a local ring homomorphism φ: R → R, the two invariants cid(φ) (the complete intersection dimension attached to a Cohen factorization of φ) and cid_R(R_φ) (the complete intersection dimension of the R-module R with scalar action restricted along φ) are simultaneously finite.

Background

The paper studies ghost maps via André–Quillen homology and applies the resulting ghost lemma to derive criteria for regularity and complete intersection properties. In comparing their results with those of Majadas and others, the author highlights invariants that measure singularity in terms of complete intersection dimension (CI-dimension).

Blanco–Majadas and Majadas use two closely related finiteness conditions: cid(φ), defined via CI-dimension of a Cohen factorization of the local homomorphism φ, and cid_R(R_φ), the CI-dimension of the module R under the scalar action induced by φ. Establishing whether these notions are simultaneously finite is identified as an open problem that would bridge existing results and the paper’s ghost-map-based criteria.