[Paper Review] On linear resolution of powers of an ideal

Keivan Borna|arXiv (Cornell University)|Oct 6, 2008

Commutative Algebra and Its Applications9 references4 citations

TL;DR

This paper provides a general criterion using the Rees algebra to determine when high powers of a graded ideal have linear resolutions. It proves that for two specific ideals $J$ and $J_1$, all powers except $k=2$ have linear resolution in characteristic zero, with regularity $\mathrm{reg}(I^k) = dk$ for $k \neq 2$, resolving an open question on asymptotic regularity behavior.

ABSTRACT

In this paper we give a generalization of a result of Herzog, Hibi, and Zheng providing an upper bound for regularity of powers of an ideal. As the main result of the paper, we give a simple criterion in terms of Rees algebra of a given ideal to show that high enough powers of this ideal have linear resolution. We apply the criterion to two important ideals $J,J_{1}$ for which we show that $J^{k},$ and $J_{1}^{k}$ have linear resolution if and only if $k eq 2.$ The procedures we include in this work is encoded in computer algebra package CoCoA.

Motivation & Objective

• To generalize a result by Herzog, Hibi, and Zheng on regularity bounds for powers of ideals.
• To develop a computational criterion based on the Rees algebra to determine when high powers of an ideal have linear resolution.
• To resolve the open question on the regularity behavior of powers of two specific ideals $J$ and $J_1$, particularly whether $\mathrm{reg}(I^k) = 3k$ for $k \neq 2$.
• To provide explicit computational verification using the CoCoA package for the regularity and Betti numbers of $J^k$ and $J_1^k$.

Proposed method

• Use Römer's formula relating the regularity of $I^k$ to the $x$-regularity of the Rees algebra $R(I)$: $\mathrm{reg}(I^k) \leq kd + \mathrm{reg}_x(R(I))$.
• Apply a bi-homogeneous change of coordinates $g = g_1 \times g_2$ to transform the defining ideal of the Rees ring into a form where initial ideals can be analyzed.
• Compute the initial ideal $\mathrm{in}(g(P))$ of the Rees ring's defining ideal $P$, decomposing it as $G + B$ where $G$ generates the $x$-part and $B$ the $t$-part.
• Verify that $I_{(k,\star)} = G_{(k,\star)}$ for $k > 2$ by checking that certain monomials in $B$ multiply into $G$, ensuring $\mathrm{reg}(I^k) = dk$.
• Use the CoCoA computer algebra system to perform explicit Gröbner basis computations and verify the conditions for linear resolution.
• Leverage multigraded Hilbert series equality between $J$ and $J_1$ to show identical Betti numbers for all powers.

Experimental results

Research questions

• RQ1 Under what conditions on an ideal $I$ do all sufficiently high powers $I^k$ have linear resolution?
• RQ2 For the ideal $J$ corresponding to the triangulation of $\mathbb{P}^2$, what is the precise regularity of $J^k$ for all $k$?
• RQ3 Does there exist an ideal $Q$ of degree $d$ such that $\mathrm{reg}(Q^k) = dk$ for all $k \neq 3$ or $k \neq 2,3$?
• RQ4 How can the Rees algebra of an ideal be used to algorithmically determine the regularity of its powers?

Key findings

• For the ideal $J$ defined by 10 cubic monomials on 6 variables, $\mathrm{reg}(J^k) = 3k$ holds for all $k \neq 2$ in characteristic zero.
• For the related ideal $J_1$, $\mathrm{reg}(J_1^k) = 3k$ for all $k \neq 2$, with the same regularity behavior as $J^k$.
• The second power $J^2$ has $\mathrm{reg}(J^2) = 7 > 6 = 2 \times 3$, showing a regularity jump at $k=2$, which is the only exception.
• The Rees rings of $J$ and $J_1$ have identical multigraded Hilbert series, implying $\beta_{i,j}(J^k) = \beta_{i,j}(J_1^k)$ for all $i,j,k$.
• The criterion based on $\mathrm{reg}_x(R(I)) = 0$ is both necessary and sufficient for all high powers of $I$ to have linear resolution.
• Computational verification via CoCoA confirms that $I_{(k,\star)} = G_{(k,\star)}$ for $k > 2$ in both $J$ and $J_1$, validating the linear resolution claim.