[Paper Review] On the value-semigroup of a simple complete ideal in a two-dimensional regular local ring

Silvio Greco, K. Kiyek|ArXiv.org|Dec 28, 2007

Commutative Algebra and Its Applications10 references3 citations

TL;DR

This paper establishes the equivalence of three invariants—value-semigroup, multiplicity sequence, and formal characteristic sequence—for a simple complete $π$-primary ideal in a two-dimensional regular local ring under residual rationality. Using the Hamburger-Noether tableau, it provides a new proof that the value-semigroup is symmetric and derives explicit formulas for its conductor in terms of the tableau data.

ABSTRACT

Let R be a two-dimensional regular local ring with maximal ideal \mathfrak m, and let \wp be a simple complete \mathfrak m-primary ideal which is residually rational. Let R_0:= R\subsetneqq ...\subsetneqq R_r be the quadratic sequence associated to \wp, let Γ_\wp be the value-semigroup associated to \wp, and let ((e_j(\wp))_{0\leq j\leq r} be the multiplicity sequence of \wp. We associate to \wp a sequence of natural integers, the formal characteristic sequence of \wp, and we show that the value-semigroup, the multiplicity sequence and the formal characteristic sequence are equivalent data. Furthermore, we give a new proof that Γ_\wp is symmetric, and give a formula for c_\wp, the conductor of Γ_\wp, in terms of entries of the Hamburger-Noether tableau of \wp.

Motivation & Objective

• To establish the equivalence between the value-semigroup, multiplicity sequence, and formal characteristic sequence of a simple complete $π$-primary ideal in a two-dimensional regular local ring.
• To provide a new proof that the value-semigroup is symmetric under the assumption of residual rationality.
• To derive explicit formulas for the conductor $c_{\wp}$ of the value-semigroup in terms of the Hamburger-Noether tableau of the ideal.
• To clarify and correct omissions in prior work, particularly in the construction of Hamburger-Noether tableaux and their use in semigroup computations.

Proposed method

• Utilizes the Hamburger-Noether tableau of the ideal $\wp$ as the central computational and theoretical tool.
• Constructs the formal characteristic sequence $(\gamma_i(\wp))_{0\leq i\leq g}$ recursively from the multiplicity sequence and divisibility conditions among multiplicities.
• Applies the HN-tableau to derive the semigroup sequence $(r_i)$ and divisor sequence $(\theta_i)$, which strictly generate the value-semigroup $\Gamma_\wp$.
• Uses recursive relations between successive $\gamma_i$ values to relate the formal characteristic sequence to the HN-tableau structure.
• Employs the length formula of Hoskin-Deligne to express the number of 1s in the multiplicity sequence as a function of the other multiplicities.
• Compares conductor formulas across different HN-tableaux (e.g., $\operatorname{HN}(\wp;x,y)$ and $\operatorname{HN}(\wp^{R_1};x_1,y_1)$) to verify consistency and derive the main conductor formula.

Experimental results

Research questions

• RQ1 Are the value-semigroup, multiplicity sequence, and formal characteristic sequence of a residually rational simple complete ideal in a 2D regular local ring equivalent data?
• RQ2 Can the symmetry of the value-semigroup $\Gamma_\wp$ be re-proven using the Hamburger-Noether tableau?
• RQ3 What is the precise formula for the conductor $c_\wp$ of $\Gamma_\wp$ in terms of the Hamburger-Noether tableau data?
• RQ4 How do the multiplicity sequence and the formal characteristic sequence relate structurally through the HN-tableau?
• RQ5 What corrections or clarifications are needed in earlier constructions of Hamburger-Noether tableaux for semigroup computations?

Key findings

• The value-semigroup $\Gamma_\wp$, the multiplicity sequence $(e_j(\wp))_{0\leq j\leq r}$, and the formal characteristic sequence $(\gamma_i(\wp))_{0\leq i\leq g}$ are equivalent data for a residually rational simple complete $\mathfrak{m}$-primary ideal.
• The value-semigroup $\Gamma_\wp$ is symmetric, and this is proven anew using the Hamburger-Noether tableau.
• The conductor $c_\wp$ of $\Gamma_\wp$ is given by the formula $c_\wp = -r_0 + 1 + \sum_{j=1}^h r_j(\theta_j/\theta_{j+1} - 1)$, where $r_j$ and $\theta_j$ are the semigroup and divisor sequences from the HN-tableau.
• An alternative expression for $c_\wp$ is $c_\wp = (q_1 - 1)(\theta_1 - 1) + \sum_{j=2}^h q_j(\theta_j - 1)$, where $q_j$ are the quotients from the HN-tableau.
• A third equivalent formula is $c_\wp = (p_1 - 1)(c_1 - 1) + \sum_{i=2}^\epsilon p_i(c_i - 1)$, using the $p_i$ and $c_i$ entries of the HN-tableau.
• The paper corrects an omission in [3] by adding the condition $\theta_1 > \cdots > \theta_{g+1} = 1$ to ensure the HN-tableau is valid.