On some properties of the asymptotic Samuel function
A. Bravo, S. Encinas, J. Guillán-Rial
TL;DR
The paper introduces and analyzes the asymptotic Samuel function $ar{ u}_I$ for arbitrary Noetherian rings, extending the resolution-related order concepts from regular local rings to equidimensional excellent rings containing a field. It establishes fundamental properties via finite-transversal projections and Hickel's method for computing $ar{ u}$, and defines the Samuel slope as an intrinsic invariant of a local ring, showing finiteness (rational values) for non-regular reduced rings and equality with the reduced case for non-reduced rings. It further proves that the Samuel slope is preserved under certain faithfully flat extensions, including local étale extensions and completions, and analyzes how slopes compare across prime ideals, providing a dense-open semicontinuity-type behavior on the maximal spectrum. These results connect intrinsically defined invariants to resolution data and demonstrate robustness under base changes, contributing to intrinsic, base-change-stable measures of singularities.
Abstract
The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. Here we explore some natural properties in the context of excellent, equidimensional rings containing a field. In addition, we establish some results regarding the Samuel slope of a local ring. This is an invariant related with algorithmic resolution of singularities of algebraic varieties. Among other results, we study its behavior after certain faithfully flat extensions.