Stability of CI-dimension under iteration of a local endomorphism

Establish whether, for a local ring endomorphism φ: R → R, finiteness of the complete intersection dimension cid_R(R_φ) implies finiteness of cid_R(R_{φ^n}) for every positive integer n, where R_{φ^n} denotes R regarded as an R-module via the n-th iterate of φ.

Background

The paper proves complete intersection criteria assuming finiteness of CI-dimension for some iterate φ^n, but it notes that connecting these conditions to Majadas’ approach requires understanding how CI-dimension behaves under iteration of the endomorphism.

Specifically, determining whether the CI-dimension finiteness of R_φ propagates to R_{φ^n} for n ≥ 1 would align module-theoretic finiteness conditions with endomorphism-based invariants and clarify the relationship between different regularity and complete intersection tests.