On the structure of the fiber cone of ideals with analytic spread one

TL;DR

This paper investigates the algebraic structure of the fiber cone of ideals with analytic spread one in local rings, providing a detailed module-theoretic description and linking it to key invariants like multiplicity and regularity.

Contribution

It offers a complete module-theoretic description of the fiber cone for ideals with analytic spread one, connecting its structure to important algebraic invariants.

Findings

Fiber cone has a module structure over a polynomial ring in one variable.

Complete description of the fiber cone as a module over its Noether normalization.

Characterization of invariants like multiplicity and Castelnuovo-Mumford regularity.

Abstract

Foa a given local ring, we study the fiber cone of ideals with analytic spread one. In this case, the fiber cone has a strucure as a module over its Noether normalization which is a polynomial ring in one variable over the residue field. One may then apply the structure theorem for graded modules over a graded principal domain to get a complete descriptionof the fiber cone as a module. We analyze this structure in order to study and characterize in terms of the ideal itself the aritmetical properties and other numerical invariants of the fiber cone as multiplicity, reduction number or Castelnuovo-Mumford regularity.