Huneke’s Uniform Artin–Rees Conjecture

Establish that every excellent Noetherian ring R of finite Krull dimension has the Uniform Artin–Rees Property: for every ideal J ⊂ R there exists a constant A_J ∈ ℕ such that for all ideals I ⊂ R and all n ∈ ℕ, the containment J ∩ I^{n+A_J} ⊆ J I^n holds.

Background

The paper recalls the definitions of the Uniform Artin–Rees and Uniform Briançon–Skoda properties and cites Huneke’s conjectures asserting these properties for very general classes of excellent rings. While significant progress has been made under additional hypotheses (e.g., essentially of finite type over a local ring, F-finite, etc.), the general conjectural assertions remain open.

These uniform properties are central to the paper’s techniques and applications. The authors use proven uniform results under certain conditions, but they explicitly note Huneke’s original conjectural scope beyond those settings.