Asymptotic behaviour of Vasconcelos invariants for products and powers of graded ideals

Abstract

Let $R$ be a commutative Noetherian $\mathbb{N}$-graded ring. Let $N\subseteq M$ be finitely generated $\mathbb{Z}$-graded $R$-modules. Let $I_1,\dots ,I_r$ be nonzero proper homogeneous ideals of $R$. Denote ${\mathbf{I}}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$ for $\underline{n}=(n_1,\dots ,n_r)\in {\mathbb{N}}^r$. In this paper, we prove that the (local) Vasconcelos invariant of ${\mathbf{I}}^{\underline{n}}M/{\mathbf{I}}^{\underline{n}}N$ is eventually the minimum of finitely many linear functions in $\underline{n}$. The same holds for $M/{\mathbf{I}}^{\underline{n}}N$ under certain conditions. Some specific examples are provided, where these functions are not eventually linear in $\underline{n}$. However, when $R$ is a polynomial ring over a field, we show that the global Vasconcelos invariants of $R/{\mathbf{I}}^{\underline{n}}$ and ${\mathbf{I}}^{\underline{n}}/{\mathbf{I}}^{\underline{n}+\underline{1}}$ are, in fact, asymptotically linear in $\underline{n}$ with the leading coefficients given by the initial degrees of $I_1,\dots ,I_r$. The last result is surprising: It differs from the Castelnuovo-Mumford regularity, which is not always linear even over polynomial rings, as shown by Bruns-Conca.

Revised version, Journal ref. Journal of Algebraic Combinatorics. Added Section 2, Remark 3.2, Paragraph 4.1 and Example 4.8. Using Lemma 2.10, we have elaborated the proof of the 2nd part of Theorem 3.1