Uniform arithmetic in local rings via ultraproducts
摘要
We reinterpret various properties of Noetherian local rings via the existence of some n-ary numerical function satisfying certain uniform bounds. We provide such characterizations for seminormality, weak normality, generalized Cohen–Macaulayness, and F-purity, among others. Our proofs that such numerical functions exist are nonconstructive and rely on the transference of the property in question from a local ring to its ultrapower or catapower.