[Paper Review] On the structure of the fiber cone of ideals with analytic spread one

Teresa Cortadellas, Santiago Zarzuela|ArXiv.org|Mar 2, 2006

Commutative Algebra and Its Applications21 references3 citations

TL;DR

This paper studies the fiber cone of ideals with analytic spread one in a Noetherian local ring, leveraging module structure over a polynomial ring (its Noether normalization) to fully characterize the fiber cone’s properties. The key contribution is a complete algebraic description of the fiber cone as a module, enabling explicit computation of invariants like multiplicity, reduction number, Castelnuovo-Mumford regularity, and Cohen-Macaulay/Gorenstein properties in terms of the ideal itself.

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.

Motivation & Objective

• To understand the structure of the fiber cone of ideals with analytic spread one in a Noetherian local ring.
• To characterize arithmetical and numerical invariants of the fiber cone—such as multiplicity, reduction number, and Castelnuovo-Mumford regularity—using the ideal's intrinsic properties.
• To determine conditions under which the fiber cone is Cohen-Macaulay, Gorenstein, or Buchsbaum, based on ideal-theoretic data.
• To provide a complete module-theoretic description of the fiber cone over its Noether normalization, a polynomial ring in one variable.

Proposed method

• Utilize the structure theorem for finitely generated graded modules over a principal ideal domain to describe the fiber cone as a module over its Noether normalization, which is a polynomial ring in one variable over the residue field.
• Apply the graded module decomposition to analyze the fiber cone’s invariants, including Hilbert function, multiplicity, and reduction number.
• Use the isomorphism $ F(I) o F(I)/(a_1^0, \\/dots, a_{l-1}^0) $ to reduce the problem to the case of analytic spread one.
• Leverage the equality $ \lambda(I^2 / JI) = 1 $ as a key condition for Cohen-Macaulayness in $ \mathfrak{m} $-primary ideals.
• Apply the formula $ \sum_{i=0}^{d} (-1)^i \binom{d}{i} \mu(I^{n+1-i}) = 1 $ to characterize the Hilbert function of the fiber cone under specific conditions.
• Use the isomorphism $ I_{l-1}^n / (\mathfrak{m}_{l-1} I_{l-1}^n + a_l I_{l-1}^{n-1}) \cong I^n / (\mathfrak{m} I^n + J I^{n-1}) $ to relate the fiber cone’s structure to the ideal’s minimal generators.

Experimental results

Research questions

• RQ1 What conditions on an ideal $ I $ with analytic spread one ensure that its fiber cone $ F(I) $ is Cohen-Macaulay?
• RQ2 How can the Castelnuovo-Mumford regularity of $ F(I) $ be computed from the ideal $ I $?
• RQ3 What is the precise relationship between the multiplicity of $ F(I) $ and the number of generators of powers of $ I $?
• RQ4 Can the reduction number and Hilbert function of $ F(I) $ be fully determined by invariants of $ I $?
• RQ5 Under what conditions is $ F(I) $ Gorenstein or Buchsbaum, and how can these be detected from $ I $?

Key findings

• The fiber cone $ F(I) $ is Cohen-Macaulay if and only if $ \mathfrak{m}I^2 = J\mathfrak{m}I $, where $ J $ is a minimal reduction of $ I $, under the condition $ \lambda(I^2 / JI) = 1 $.
• When $ \lambda(I^2 / JI) = 1 $, the Hilbert function satisfies $ \sum_{i=0}^{d} (-1)^i \binom{d}{i} \mu(I^{n+1-i}) = 1 $ for $ 1 \leq n \leq r(I) - 1 $ if $ F(I) $ is Cohen-Macaulay.
• The Castelnuovo-Mumford regularity of $ F(I) $ equals the reduction number $ r(I) $, i.e., $ \mathrm{reg}(F(I)) = r(I) $.
• The multiplicity of $ F(I) $ is given by $ e(F(I)) = \sum_{i=0}^{l-1} \binom{l-1}{i} \mu(I^{r-i}) $, where $ l $ is the analytic spread and $ r $ the reduction number.
• The fiber cone $ F(I) $ is Cohen-Macaulay if and only if $ \lambda(I^n / (\mathfrak{m}I^n + JI^{n-1})) = \sum_{i=0}^{l} (-1)^i \binom{l}{i} \mu(I^{n-i}) $ for all $ 1 \leq n \leq r(I) $.
• The structure of $ F(I) $ as a module over its Noether normalization allows full reconstruction of its invariants from the ideal $ I $, particularly in the analytic spread one case.