[Paper Review] Regularity of joint-meet ideals of distributive lattices

Viviana Ene, Ayesha Asloob Qureshı+1 more|arXiv (Cornell University)|Jul 29, 2013

Commutative Algebra and Its Applications4 references4 citations

TL;DR

This paper computes the regularity of Hibi rings associated with finite distributive lattices, providing an exact formula for planar lattices and bounds for non-planar ones using combinatorial invariants like the maximal size of a set of pairwise incomparable join-irreducible elements. The key result characterizes when the Hibi ring has a linear resolution: precisely when the lattice is the divisor lattice of $2 \cdot 3^a$ for some $a \geq 0$. This resolves a long-standing question about linear resolutions in this class of rings.

ABSTRACT

Let $L$ be a distributive lattice and $R(L)$ the associated Hibi ring. We compute $ eg R(L)$ when $L$ is a planar lattice and give a lower bound for $ eg R(L)$ when $L$ is non-planar, in terms of the combinatorial data of $L.$ As a consequence, we characterize the distributive lattices $L$ for which the associated Hibi ring has a linear resolution.

Motivation & Objective

• To determine the regularity of the Hibi ring $R(L)$ for a finite distributive lattice $L$.
• To establish a precise formula for $\operatorname{reg}R(L)$ when $L$ is planar, based on lattice combinatorics.
• To provide bounds for $\operatorname{reg}R(L)$ in the non-planar case, using the maximal size of pairwise incomparable join-irreducible elements.
• To characterize all distributive lattices $L$ for which $R(L)$ has a linear resolution.
• To resolve the open problem of identifying which Hibi rings admit linear resolutions by linking it to the structure of divisor lattices of $2 \cdot 3^a$.

Proposed method

• The paper uses the Hilbert series of $R(L)$, which coincides with that of the Stanley-Reisner ring of the order complex $\Delta(L)$, to relate regularity to the $h$-vector of $\Delta(L)$.
• It applies the identity $\operatorname{reg}R(L) = \deg h_{R(L)}$, valid because $R(L)$ is Cohen-Macaulay and admits a regular sequence of linear forms.
• The $h$-vector coefficients are interpreted via the number of linear extensions of the poset $P$ of join-irreducible elements with a given descent set $S$, using equation (1): $h_{K[\Delta(L)]}(t) = \sum_{S \subset [d]} \beta(S) t^{|S|}$.
• For planar lattices, the regularity is computed via the maximal descent set size over all maximal chains, using EL-labelings and cyclic sublattices.
• For non-planar lattices, the method bounds $\operatorname{reg}R(L)$ from below by $|Q| - 1$, where $Q$ is a maximal set of pairwise incomparable join-irreducible elements, via a comparison with Boolean lattices.
• The proof relies on a poset ordering $\prec$ on join-irreducible elements that preserves order and allows comparison of $\beta(S)$ values, leading to the inequality $\operatorname{reg}R(L) \geq \operatorname{reg}R(B_r)$.

Experimental results

Research questions

• RQ1 What is the exact value of $\operatorname{reg}R(L)$ for a planar distributive lattice $L$?
• RQ2 How can one bound $\operatorname{reg}R(L)$ for non-planar distributive lattices in terms of their combinatorial structure?
• RQ3 Which distributive lattices $L$ yield Hibi rings $R(L)$ with a linear resolution?
• RQ4 What is the relationship between the maximal size of a set of pairwise incomparable join-irreducible elements and the regularity of $R(L)$?
• RQ5 Can the characterization of linear resolutions in Hibi rings be reduced to a specific class of lattices?

Key findings

• For planar distributive lattices, $\operatorname{reg}R(L)$ is equal to the maximum size of the descent set of any maximal chain in $L$, computed via EL-labelings.
• For non-planar lattices, $\operatorname{reg}R(L) \geq \max\{|Q| : Q \text{ is a set of pairwise incomparable join-irreducible elements}\} - 1$, with equality possible.
• The regularity of the Boolean lattice $B_n$ is $n-1$, which serves as a lower bound benchmark for non-planar lattices.
• The Hibi ring $R(L)$ has a linear resolution if and only if $L$ is the divisor lattice of $2 \cdot 3^a$ for some $a \geq 0$, a complete characterization.
• The regularity of $R(L)$ is bounded above by $|P| - 1$, where $P$ is the set of join-irreducible elements, and this bound is sharp.
• Example 9 shows that both inequalities in the non-planar bound can be strict: $\operatorname{reg}R(L) = 3$ while the maximal incomparable set size is 3, and $|P| - 1 = 4$, so $3 < 4$ and $3 = 3$.