Analytic Spread of Filtrations and Symbolic Algebras
Steven Dale Cutkosky, Parangama Sarkar

TL;DR
This paper introduces and investigates the concept of analytic spread for filtrations in local rings, extending classical ideal results to filtrations, especially symbolic and divisorial ones, and examines their properties and non-Noetherian conditions.
Contribution
It defines the analytic spread of filtrations, explores its properties for divisorial and symbolic filtrations, and highlights differences from ideal cases, including non-Noetherian conditions.
Findings
Analytic spread bounds extend to filtrations, with some differences.
For symbolic powers of a height two prime ideal, the spread can be 0, 1, or 2.
The symbolic algebra is non-Noetherian if the spread of symbolic powers is maximal.
Abstract
In this paper we define and explore the analytic spread of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal , we have the classical bounds . The upper bound is true for filtrations , but the lower bound is not true for all filtrations. We show that for the filtration of symbolic powers of a height two prime ideal in a regular local ring of dimension three (a space curve singularity), so that and $\dim…
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Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic Geometry and Number Theory · Polynomial and algebraic computation
