On generalized Narita ideals

TL;DR

This work extends Narita-type vanishings of Hilbert coefficients to higher dimensions by introducing Ratliff–Rush filtrations and the L-construction for filtrations, then proving that an $\mathfrak{m}$-primary generalized Narita ideal $I$ in a CM local ring $A$ forces $e_i^I(M)=0$ for all $i\ge2$ and all MCM $A$-modules $M$. It shows that $\widetilde G_I(M)$ is CM with minimal multiplicity, that $G_{I^n}(M)$ is CM for large $n$, and that $G_I(M)$ is generalized CM, along with a uniform bound on ${\rm reg}\,G_I(M)$ depending only on $A$ and $I$. The paper develops a robust framework (Ratliff–Rush filtration, L-construction, superficial elements) to transfer Cohen–Macaulayness properties from $A$ to $G_I(M)$ and its variants, and provides explicit examples in dimension three illustrating existence of generalized Narita ideals. Collectively, these results clarify the asymptotic behavior of Hilbert functions and blow-up algebras in higher dimensions and yield practical regularity bounds for associated graded structures.

Abstract

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$. An $\mathfrak{m}$-primary ideal $I$ is said to be a generalized Narita ideal if $e_i^I(A) = 0$ for $2 \leq i \leq d$. If $I$ is a generalized Narita ideal and $M$ is a maximal Cohen-Macaulay $A$-module then we show $e_i^I(M) = 0$ for $2 \leq i \leq d$. We also have $G_I(M)$ is generalized Cohen-Macaulay. Furthermore we show that there exists $c_I$ (depending only on $A$ and $I$) such that $\text{reg} \ G_I(M) \leq c_I$.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Proposition 3.2
• proof
• Lemma 5.1
• Proposition 5.3
• proof
• Proposition 5.5