Huneke’s Uniform Artin–Rees Conjecture

Establish that every excellent Noetherian ring of finite Krull dimension possesses the Uniform Artin–Rees Property: for every ideal J ⊆ R there exists an integer A ≥ 0 (depending on J) such that for all ideals I ⊆ R and all n ∈ N one has J ∩ I^{n+A} ⊆ J I^n.

Background

The paper recalls the definition: an ideal J in a Noetherian ring R has the Uniform Artin–Rees Property if there exists a constant A such that for all ideals I and all n, J ∩ I^{n+A} ⊆ J I^n; the ring R has the property if every ideal of R does. The authors cite Huneke’s conjecture that every excellent Noetherian ring of finite Krull dimension should have this property.

They also note supporting results: under additional hypotheses (e.g., R essentially of finite type over a local ring, or of prime characteristic and F-finite, or essentially of finite type over Z), the property is known to hold (Theorem cited to Huneke). The general conjecture remains an outstanding problem beyond these cases.