The First Syzygy of Hibi Rings Associated with Planar distributive lattices

TL;DR

This paper investigates the structure of the first syzygy of Hibi rings associated with planar distributive lattices, providing explicit formulas and characterizations of linear syzygies.

Contribution

It describes the first syzygy of Hibi rings for planar distributive lattices and characterizes when the syzygy is linear, including a formula for the first Betti number.

Findings

Explicit description of the first syzygy for planar distributive lattices.

An exact formula for the first Betti number of such lattices.

Characterization of when the first syzygy is linear.

Abstract

Let $\mathcal{L}$ be a finite distributive lattice and $S=K[x_\alpha: \alpha \in \mathcal{L}]$ be a polynomial ring over a field $K$ and $I=\langle x_\alpha x_\beta - x_{\alpha\vee \beta} x_{\alpha\wedge\beta} : \alpha \nsim \beta,\alpha,\beta \in {\mathcal{L}} \rangle$ an ideal of $S$. In this article we describe the first syzygy of the Hibi ring $R[\mathcal{L}]=S/I$, for a planar distributive lattice $\mathcal{L}$. We also derive an exact formula for the first Betti number of a planar distributive lattice. We give a characterization of planar distributive lattices for which the first syzygy is linear.

Equations

$$S(f_{({\alpha }_1,{\beta }_1)},f_{({\alpha }_1,{\beta }_2)})=$$

$$=$$

$$S(f_{({\alpha }_1,{\beta }_1)},f_{({\alpha }_1,{\beta }_2)})$$

$$=x_{{\beta }_1}{x}_{{\alpha }_1\vee {\beta }_2}{x}_{{\alpha }_1\wedge {\beta }_2}-x_{{\beta }_2}{x}_{{\alpha }_1\vee {\beta }_1}{x}_{{\alpha }_1\wedge {\beta }_1}$$

$$=-x_{{\alpha }_1\wedge {\beta }_1}(x_{{\beta }_2}{x}_{{\alpha }_1\vee {\beta }_1}-x_{{\beta }_1}{x}_{{\alpha }_1\vee {\beta }_2})$$

$$=-x_{{\alpha }_1\wedge {\beta }_1}{f}_{({\beta }_2,{\alpha }_1\vee {\beta }_1)}$$

$$S(f_{({\alpha }_1,{\beta }_2)},f_{({\beta }_2,{\alpha }_1\vee {\beta }_1)})=$$

$$=$$

$$S(f_{({\alpha }_1,{\beta }_1)},f_{({\alpha }_1,{\beta }_2)})=$$

$$=$$

$$-x_{{\alpha }_1\wedge {\beta }_1}{x}_{{\beta }_1\vee ({\alpha }_1\wedge {\beta }_2)})$$

$$=$$

$$S(f_{({\alpha }_1,{\beta }_1)},f_{({\alpha }_1,{\beta }_2)})=$$

$$=$$

$$-x_{{\alpha }_1\wedge {\beta }_2}(x_{{\alpha }_1\vee {\beta }_1}{x}_{{\beta }_1\vee {\beta }_2}-x_{{\alpha }_1\vee {\beta }_2}{x}_{{\beta }_1})$$

$$=$$

$$S(f_{({\alpha }_1,{\beta }_2)},f_{({\beta }_1,{\beta }_2)})=$$

$$=$$

$$+x_{{\alpha }_1\wedge {\beta }_2}(x_{{\alpha }_1\vee {\beta }_1}{x}_{{\beta }_1\vee {\beta }_2}-x_{{\beta }_1}{x}_{{\alpha }_1\vee {\beta }_2})$$

$$=$$

$$S(f_{({\alpha }_1,{\beta }_1)},f_{({\alpha }_2,{\beta }_2)})=$$

$$=$$

$$-x_{{\alpha }_1\vee {\beta }_1}{x}_{{\alpha }_1\wedge {\beta }_1}(x_{{\alpha }_2}{x}_{{\beta }_2}-x_{{\alpha }_2\vee {\beta }_2}{x}_{{\alpha }_2\vee {\beta }_2})$$

$$=$$

$$n({\theta }_1,{\theta }_2)=$$

$$n(S)=$$

$$m({\theta }_1,{\theta }_2)=$$

$$n(L)=$$

$$+\sum _{{\theta }_i,{\theta }_j,{\theta }_k}(ht({\theta }_j\vee {\theta }_k)-ht({\theta }_i\vee {\theta }_j)(ht({\theta }_j\wedge {\theta }_k)-ht({\theta }_i\wedge {\theta }_j))(\genfrac{}{}{0ex}{}{r+1}{2}\left)\right(\genfrac{}{}{0ex}{}{s+1}{2}).$$

$$n(B)=$$

$${\beta }_1(\mathcal{L})=$$

$$L(2,n)$$

$$=\frac{n(n^2-1)}{3}.$$

$$L(m,n)$$

$$=\frac{m(m^2-1)}{6}L(2,n)$$

$$=\frac{L(2,m)L(2,n)}{2}.$$

Full text

Keywords: Free resolution, Syzygy, Hibi rings, Gröbner basis

1 Introduction

Let $\mathcal{L}$ be a finite distributive lattice and let $S=K[x_{\alpha}:\alpha\in\mathcal{L}]$ be a polynomial ring over a field $K$. For $\alpha,\beta\in\mathcal{L}$, $f_{(\alpha,\beta)}=x_{\alpha}x_{\beta}-x_{\alpha\vee\beta}x_{\alpha\wedge\beta}\in S$ is called a diamond relation if $\alpha\nsim\beta$. Let $I=\langle f_{(\alpha,\beta)}:\alpha\nsim\beta,\alpha,\beta\in\mathcal{L}\rangle\subset S$ be the associated ideal, called a Hibi ideal. The associated ring $R[\mathcal{L}]=S/I$ is called a Hibi ring. In [8], Hibi showed that $R[\mathcal{L}]$ is an algebra with straightening laws on $\mathcal{L}$ over a field $K$ and further in [5], Herzog-Hibi showed that the diamond relations $f_{(\alpha,\beta)}$ are Gröbner basis for the lexicographic term order on $S$ extending the order of $\mathcal{L}$. The ring appears in geometric context in Lakshmibai-Gonciulea [6] where it was proved that the Schubert varieties in $G_{d,n}$ degenerate flatly to the toric varieties $X(I_{d,n})(=X_{d,n}$ as in [6]) which are varieties associated to the lattices $I_{d,n}=\{(i_{1},\ldots,i_{d}):1\leq i_{1}<\ldots<i_{d}\leq n\}$. Using the degeneration, several geometric properties for the Schubert varieties were derived in [6], where the study of the singularities of $X(I_{d,n})$ were discussed. The question of singularity of the algebraic varieties associated to a lattice $\mathcal{L}$, namely $\operatorname*{Spec}K[\mathcal{L}]$ is dealt with in Wagner [12] and Lakshmibai-Mukherjee[11]. In a series of papers, as in Brown-Lakshmibai [3], [4], interesting multiplicity formulas and other applications were found for several classes of distributive lattice.

Also Hibi in [8] proved that $I$ is prime if and only if $\mathcal{L}$ is distributive and that $R[\mathcal{L}]$ is Gorenstein when the set of join irreducible elements $J=\{z\in\mathcal{L}:x\vee y=z\Rightarrow x=z$ or $y=z\}$ is pure i.e. all maximal chains have the same length. Furthermore, Hibi also showed in [8] that $R[\mathcal{L}]$ is a Cohen-Macaulay normal domain. Hibi rings over a finite distributive lattice are further studied by several authors, cf. Aramova-Herzog-Hibi [1], Thomas [7], Ene [9] and Ene-Herzog-Saeedi-Madani [10], to mention but a few. The projective dimension and regularity of $I$ was found in Ene [9], Ene-Herzog-Saeedi-Madani [10] respectively. It was also proved in Ene-Herzog-Saeedi-Madani [10] that, for a finite distributive lattice, reg $R[\mathcal{L}]=\lvert P\rvert-\operatorname*{rank}P-1$, where $P$ is the poset of join irreducible elements of $\mathcal{L}$. This, in particular, implies that $R[\mathcal{L}]$ has a linear resolution if and only if $P$ is a direct sum of a chain and an isolated element, see Ene-Herzog-Saeedi-Madani [10]. Also in [2], the author gave an explicit minimal free resolution for monomial curves in $A_{K}^{4}$ using Gröbner basis technique. In [13], the author found the first syzygies of determinantal ideals using Gröbner basis theory. In this article we give a set of minimal generators for planar distributive lattice $\mathcal{L}$. Using the generator we also give a formula for the first Betti number for these lattices.

2 First syzygy of Hibi ring

Let $f_{1},f_{2},\ldots,f_{n}$ be all the diamond relations in $I$. Let us denote $F=\oplus_{i=1}^{n}S(-2)$ and $g_{1},g_{2},\ldots,g_{n}$ be the generators of $F$, $\phi:F\longrightarrow S$, $g_{i}\longmapsto f_{i}$, ker$(\phi)$ is the first syzygy of $R[\mathcal{L}]$. Let us denote ker$(\phi)$=Syz${}^{1}R[\mathcal{L}]$ and Syz${}^{1}_{1}R[\mathcal{L}]$ denote linear generators of the $S$-module Syz${}^{1}R[\mathcal{L}]$.

Let $P=\{\alpha,\beta,\gamma,\delta\}$ be a poset $\leq$ defined by $\alpha\leq\beta,\gamma,\delta$, let the lattice of the poset ideals of $P$ be called a cube lattice.

Lemma 2.1**.**

$\mathcal{L}$* is a planar distributive lattice if and only if every join irreducible (meet irreducible) $\beta\in\mathcal{L}$ is covered (covers) by at most two join irreducibles (meet irreducibles) in $\mathcal{L}$.*

Proof.

$\Rightarrow:$ Let $\mathcal{L}$ be a planar distributive lattice and $\beta\in\mathcal{L}$ be a join-irreducible element. Let us assume that it is covered by three join-irreducible elements $x,y,z$ in $\mathcal{L}$. Then $x\vee y,x\vee z,y\vee z$ are non-comparable. Hence there will be a cube sublattice in $\mathcal{L}$, which is non-planar. Hence it follows.

$\Leftarrow$: Let every join-irreducible $\beta$ in $\mathcal{L}$ covered by at most two join-irreducible in $\mathcal{L}$, let this propery be called (). Then every sublattice of $\mathcal{L}$ also has the property (). Now if $\mathcal{L}$ is not planar then there exist a cube sublattice in $\mathcal{L}$, then this sublattice vialates the propery (*), as the cube has three join-irreducible elements covering the minimal element which is a join-irreducible. Hence we arrive at a contradiction. By duality we get the result with meet irreducibles. ∎

Lemma 2.2**.**

$\mathcal{L}$* is a planar distributive lattice if and only if for all $x,y,z\in\mathcal{L},x\vee y\sim x\vee z$ and $x\wedge y\sim x\wedge z$.*

Proof.

$\Rightarrow$: Let $\mathcal{L}$ be a planar distributive lattice. Let $x,y,z\in\mathcal{L}$, then the sublattice generated by $x\wedge y,x\wedge z,y\wedge z$ are also planar. If $x\wedge y,x\wedge z,y\wedge z$ are not comparable to each other then we will get three join-irreducibles covering minimal element. Therefore by Lemma 2.1 we will arrive at a contradiction. Hence $x\wedge y\sim x\wedge z$ and by duality other also holds.

$\Leftarrow$: Let the given conditions hold. Then we have the cases $(i)\,x\vee y\leq x\vee z,\,x\wedge y\leq x\wedge z\,(ii)\,x\vee y\geq x\vee y,\,x\wedge y\leq x\wedge z\,(iii)\,x\vee y\geq x\vee y,\,x\wedge y\leq x\wedge z$. Then the only possibilities to get the sublattices as in Figures 2 ,1, 4, 3, which are all planar sublattices. Hence $\mathcal{L}$ is planar. ∎

If we have two relations $f_{(\alpha_{1},\beta_{1})}=x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}}$ and $f_{(\alpha_{2},\beta_{2})}=x_{\alpha_{2}}x_{\beta_{2}}-x_{\alpha_{2}\vee\beta_{2}}x_{\alpha_{2}\wedge\beta_{2}}$ coming from the diamonds $(\alpha_{1},\beta_{1})$ and $(\alpha_{2},\beta_{2})$ respectively. Then with respect to monomial ordering $>$, we have in$(f_{(\alpha_{1},\beta_{1})})=x_{\alpha_{1}}x_{\beta_{1}}$, in$(f_{(\alpha_{2},\beta_{2})})=x_{\alpha_{2}}x_{\beta_{2}}$. Then we have the following possibilities

$\alpha_{1}=\alpha_{2}$ or $\alpha_{1}=\beta_{2}$ or $\beta_{1}=\alpha_{2}$ or $\beta_{1}=\beta_{2}$. 2. 2.

$\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ all are different.

Without a loss of generality, let us suppose that $\alpha_{1}=\alpha_{2}$, for the case $(1)$. Then we have in$(f_{(\alpha_{1},\beta_{1})})=x_{\alpha_{1}}x_{\beta_{1}}$, in$(f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{2}}$ and therefore, Lcm$(f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{1}}x_{\beta_{2}}$. So,

[TABLE]

Then either $\beta_{1}\sim\beta_{2}$, let us say $\beta_{1}\leq\beta_{2}$ or $\beta_{1}\nsim\beta_{2}$. Also by Lemma 2.2, for $\alpha_{1},\beta_{1},\beta_{2}$ we have $\alpha_{1}\wedge\beta_{1}\sim\alpha_{1}\wedge\beta_{2}$ and $\alpha_{1}\vee\beta_{1}\sim\alpha_{1}\vee\beta_{2}$. Then we have the only possibilities

$\alpha_{1}\wedge\beta_{1}\leq\alpha_{1}\wedge\beta_{2}$, $\alpha_{1}\vee\beta_{1}\leq\alpha_{1}\vee\beta_{2}$. 2. 2.

$\alpha_{1}\wedge\beta_{1}\geq\alpha_{1}\wedge\beta_{2}$, $\alpha_{1}\vee\beta_{1}\leq\alpha_{1}\vee\beta_{2}$. 3. 3.

$\alpha_{1}\wedge\beta_{1}\leq\alpha_{1}\wedge\beta_{2}$, $\alpha_{1}\vee\beta_{1}\geq\alpha_{1}\vee\beta_{2}$.

Now we verify all the above cases

• •

Case I: When $\beta_{1}\sim\beta_{2}$.

• –

Check for $(1)$: We will get the following sublattice

• –

Check for $(2)$: We will get the following sublattice

• –

Check for $(3)$: We will get the following sublattice

• •

Case I: When $\beta_{1}\nsim\beta_{2}$.

• –

Check for $(1)$: This subcase cannot consider since we will get $\beta_{1}\leq\beta_{2}$, which will give a contradiction.

• –

Check for $(2)$: We will get the following sublattice

• –

Check for $(3)$: We will get the same sublattice like Figure 4 by replacing $\beta_{1}$ and $\beta_{2}$ by $\beta_{2}$ and $\beta_{1}$ respectively.

Now using the conditions of the cases (for which lattice exists), we prove that the following are the types of the generators of the first syzygy of the Hibi ring. Let us denote $\mathcal{D}_{2}=\{(\alpha,\beta):\alpha\nsim\beta,\alpha,\beta\in\mathcal{L}\}$. Infact, $\mathcal{D}_{2}=Hom(F_{2},\mathcal{L})$ non trivial homomorphism where $F_{2}$=the free lattice generated by $\alpha$ and $\beta$ in $\mathcal{L}$.

Lemma 2.3**.**

$(\textbf{Strip-type})$* Let $(\alpha_{1},\beta_{1}),(\alpha_{1},\beta_{2})\in\mathcal{D}_{2}$, for every sublattice isomorphic to Figure 2 with conditions $\alpha_{1}\vee\beta_{1}\neq\alpha_{1}\vee\beta_{2},\alpha_{1}\wedge\beta_{1}=\alpha_{1}\wedge\beta_{2},\beta_{1}\leq\beta_{2},\beta_{1}\leq\alpha_{1}\vee\beta_{2},\beta_{1}\geq\alpha_{1}\wedge\beta_{2},\beta_{2}\nsim\alpha_{1}\vee\beta_{1},\beta_{2}\geq\alpha_{1}\wedge\beta_{1}$ the following are the generators of the first syzygy*

$S_{1}=-x_{\beta_{2}}g_{(\alpha_{1},\beta_{1})}+x_{\beta_{1}}g_{(\alpha_{1},\beta_{2})}-x_{\alpha_{1}\wedge\beta_{1}}g_{(\beta_{2},\alpha_{1}\vee\beta_{1})}$. 2. 2.

$S_{2}=x_{\alpha_{1}\vee\beta_{2}}g_{(\alpha_{1},\beta_{1})}-x_{\alpha_{1}\vee\beta_{1}}g_{(\alpha_{1},\beta_{2})}+x_{\alpha_{1}}g_{(\beta_{2},\alpha_{1}\vee\beta_{1})}$.

Proof.

Let $f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{1},\beta_{2})}$ be the relations corresponding to $(\alpha_{1},\beta_{1}),(\alpha_{1},\beta_{2})$ respectively such that $f_{(\alpha_{1},\beta_{1})}=x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}},f_{(\alpha_{1},\beta_{2})}=x_{\alpha_{1}}x_{\beta_{2}}-x_{\alpha_{1}\vee\beta_{2}}x_{\alpha_{1}\wedge\beta_{2}}$. Then with respect to monomial ordering $>$, in$(f_{(\alpha_{1},\beta_{1})})=x_{\alpha_{1}}x_{\beta_{1}}$, in$(f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{2}}$. Lcm$(f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{1}}x_{\beta_{2}}$.

[TABLE]

As $\beta_{2}\nsim\alpha_{1}\vee\beta_{1}$. Hence $-x_{\beta_{2}}g_{(\alpha_{1},\beta_{1})}+x_{\beta_{1}}g_{(\alpha_{1},\beta_{2})}-x_{\alpha_{1}\wedge\beta_{1}}g_{(\beta_{2},\alpha_{1}\vee\beta_{1})}$ is a generator of the first syzygy. 2. 2.

Let $f_{(\alpha_{1},\beta_{2})},f_{(\beta_{2},\alpha_{1}\vee\beta_{1})}$ be the relations corresponding to $(\alpha_{1},\beta_{2}),(\beta_{2},\alpha_{1}\vee\beta_{1})$ respectively, therefore $f_{(\alpha_{1},\beta_{2})}=x_{\alpha_{1}}x_{\beta_{2}}-x_{\alpha_{1}\vee\beta_{2}}x_{\alpha_{1}\wedge\beta_{2}},f_{(\beta_{2},\alpha_{1}\vee\beta_{1})}=x_{\beta_{2}}x_{\alpha_{1}\vee\beta_{1}}-x_{\alpha_{1}\vee\beta_{2}}x_{\beta_{1}}$. Then with respect to monomial ordering $>$, in$(f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{2}}$, in$(f_{(\beta_{2},\alpha_{1}\vee\beta_{1})})=x_{\beta_{2}}x_{\alpha_{1}\vee\beta_{1}}$. Lcm$(f_{(\alpha_{1},\beta_{1})},\\ f_{(\beta_{2},\alpha_{1}\vee\beta_{1})})=x_{\alpha_{1}}x_{\beta_{2}}x_{\alpha_{1}\vee\beta_{1}}$. Now,

[TABLE]

Hence, $x_{\alpha_{1}\vee\beta_{2}}g_{(\alpha_{1},\beta_{1})}-x_{\alpha_{1}\vee\beta_{1}}g_{(\alpha_{1},\beta_{2})}+x_{\alpha_{1}}g_{(\beta_{2},\alpha_{1}\vee\beta_{1})}$ is a generator of the first syzygy.

∎

Lemma 2.4**.**

$(\textbf{L-type})$* Let $(\alpha_{1},\beta_{1}),(\alpha_{1},\beta_{2})\in\mathcal{D}_{2}$, for every sublattice isomorphic to Figure 1 with conditions $\alpha_{1}\wedge\beta_{1}\neq\alpha_{1}\wedge\beta_{2},\alpha_{1}\vee\beta_{1}\neq\alpha_{1}\vee\beta_{2},\beta_{1}\leq\beta_{2},\beta_{1}\leq\alpha_{1}\vee\beta_{2},\beta_{1}\nsim\alpha_{1}\wedge\beta_{2},\beta_{2}\nsim\alpha_{1}\vee\beta_{1},\beta_{2}\geq\alpha_{1}\wedge\beta_{1}$ the following is a generator of the first syzygy*

$L=-x_{\beta_{2}}g_{(\alpha_{1},\beta_{1})}+x_{\beta_{1}}g_{(\alpha_{1},\beta_{1})}+x_{\alpha_{1}\vee\beta_{2}}g_{(\beta_{1},\alpha_{1}\wedge\beta_{2})}-x_{\alpha_{1}\wedge\beta_{1}}g_{(\beta_{2},\alpha_{1}\vee\beta_{1})}$.

Proof.

Let $f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{1},\beta_{2})}$ be the relations corresponding to $(\alpha_{1},\beta_{1}),(\alpha_{1},\beta_{2})$ respectively such that $f_{(\alpha_{1},\beta_{1})}=x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}},f_{(\alpha_{1},\beta_{2})}=x_{\alpha_{1}}x_{\beta_{2}}-x_{\alpha_{1}\vee\beta_{2}}x_{\alpha_{1}\wedge\beta_{2}}$. Then with respect to monomial ordering $>$, in$(f_{(\alpha_{1},\beta_{1})})=x_{\alpha_{1}}x_{\beta_{1}}$, in$(f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{2}}$. Lcm$(f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{1}}x_{\beta_{2}}$.

[TABLE]

where the third equality follows as $\beta_{1}\vee(\alpha_{1}\wedge\beta_{2})=\beta_{2}\wedge(\alpha_{1}\vee\beta_{1})$. Hence $-x_{\beta_{2}}g_{(\alpha_{1},\beta_{1})}+x_{\beta_{1}}g_{(\alpha_{1},\beta_{2})}+x_{\alpha_{1}\vee\beta_{2}}g_{(\beta_{1},\alpha_{1}\wedge\beta_{2})}-x_{\alpha_{1}\wedge\beta_{1}}g_{(\beta_{2},\alpha_{1}\vee\beta_{1})}$ is a generator of the first syzygy. ∎

Lemma 2.5**.**

$(\textbf{Box-type})$* Let $(\alpha_{1},\beta_{1}),(\alpha_{1},\beta_{2})\in\mathcal{D}_{2}$, for every sublattice isomorphic to Figure 4 with conditions $\alpha_{1}\wedge\beta_{1}\neq\alpha_{1}\wedge\beta_{2},\alpha_{1}\vee\beta_{1}\neq\alpha_{1}\vee\beta_{2},\beta_{1}\nsim\beta_{2},\beta_{1}\leq\alpha_{1}\vee\beta_{2},\beta_{1}\geq\alpha_{1}\wedge\beta_{2},\beta_{2}\nsim\alpha_{1}\vee\beta_{1},\beta_{2}\nsim\alpha_{1}\wedge\beta_{1}$ the following are the generators of the first syzygy*

$B_{1}=-x_{\beta_{2}}g_{(\alpha_{1},\beta_{1})}+x_{\beta_{1}}g_{(\alpha_{1},\beta_{2})}-x_{\alpha_{1}\vee\beta_{1}}g_{(\beta_{2},\alpha_{1}\wedge\beta_{1})}-x_{\alpha_{1}\wedge\beta_{2}}g_{(\alpha_{1}\vee\beta_{1},\beta_{1}\vee\beta_{2})}$. 2. 2.

$B_{2}=-x_{\beta_{1}}g_{(\alpha_{1},\beta_{2})}+x_{\alpha_{1}}g_{(\beta_{1},\beta_{2})}+x_{\beta_{1}\vee\beta_{2}}g_{(\alpha_{1},\beta_{1}\wedge\beta_{2})}+x_{\alpha_{1}\wedge\beta_{2}}g_{(\alpha_{1}\vee\beta_{1}),(\beta_{1}\vee\beta_{2})}$.

Proof.

Let $f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{1},\beta_{2})}$ be the relations corresponding to $(\alpha_{1},\beta_{1}),(\alpha_{1},\beta_{2})$ respectively such that $f_{(\alpha_{1},\beta_{1})}=x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}},f_{(\alpha_{1},\beta_{2})}=x_{\alpha_{1}}x_{\beta_{2}}-x_{\alpha_{1}\vee\beta_{2}}x_{\alpha_{1}\wedge\beta_{2}}$. Then with respect to monomial ordering $>$, in$(f_{(\alpha_{1},\beta_{1})})=x_{\alpha_{1}}x_{\beta_{1}}$, in$(f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{2}}$. Lcm$(f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{1}}x_{\beta_{2}}$.

[TABLE]

Hence the lemma follows. 2. 2.

Let $f_{(\alpha_{1},\beta_{2})},f_{(\beta_{1},\beta_{2})}$ be the relations corresponding to $(\alpha_{1},\beta_{2}),(\beta_{1},\beta_{2})$ respectively. Therefore, $f_{(\alpha_{1},\beta_{2})}=x_{\alpha_{1}}x_{\beta_{2}}-x_{\alpha_{1}\vee\beta_{2}}x_{\alpha_{1}\wedge\beta_{2}}$ and $f_{(\beta_{1},\beta_{2})}=x_{\beta_{1}}x_{\beta_{2}}-x_{\beta_{1}\vee\beta_{2}}x_{\beta_{1}\wedge\beta_{2}}$. Then with respect to monomial ordering $>$, in$(f_{(\alpha_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{2}}$, in$(f_{(\beta_{1},\beta_{2})})=x_{\beta_{1}}x_{\beta_{2}}$. Lcm$(f_{(\alpha_{1},\beta_{2})},f_{(\beta_{1},\beta_{2})})=x_{\alpha_{1}}x_{\beta_{1}}x_{\beta_{2}}$. Now,

[TABLE]

Hence the lemma follows.

∎

Theorem 2.6** **(Main Theorem 1).

Syz${}^{1}_{1}R[\mathcal{L}]$ is generated by the types $S_{1},S_{2},L,B_{1},B_{2}$.

Proof.

Let $f_{(\alpha_{1},\beta_{1})}=x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}},f_{(\alpha_{2},\beta_{2})}=x_{\alpha_{2}}x_{\beta_{2}}-x_{\alpha_{2}\vee\beta_{2}}x_{\alpha_{2}\wedge\beta_{2}}$ be the relation coming from the diamonds $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})$ . Therefore with respect to monomial ordering $>$, in$(f_{(\alpha_{1},\beta_{1})})=x_{\alpha_{1}}x_{\beta_{1}}$, in$(f_{(\alpha_{2},\beta_{2})})=x_{\alpha_{2}}x_{\beta_{2}}$. Now we have that the $S$-polynomial is found from the sublattices described in the Figures 3, 1, 4 and we proved in all Lemmas 2.3, 2.4, 2.5, that these are the only linear generators of the first syzygy which is coming from the above said figures. Hence the theorem follows. ∎

Lemma 2.7**.**

$(\textbf{Diamond-type})$* Let $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})\in\mathcal{D}_{2}$, where $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ all are distincts and $f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{2},\beta_{2})}$ be the relations of the pairs $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})$ respectively. If there does not exist $a,b\in\mathcal{L}$ such that $a\wedge b=\beta_{1},a\vee b=\beta_{2}$ then the following is a generator of first syzygy and we call it as a diamond type*

$D=(x_{\alpha_{2}}x_{\beta_{2}}-x_{\alpha_{2}\vee\beta_{2}}x_{\alpha_{2}\wedge\beta_{2}})g_{(\alpha_{1},\beta_{1})}-(x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}})g_{(\alpha_{2},\beta_{2})}$.

Proof.

Let $f_{(\alpha_{1},\beta_{1})}=x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}}$ and $f_{(\alpha_{2},\beta_{2})}=x_{\alpha_{2}}x_{\beta_{2}}-x_{\alpha_{2}\vee\beta_{2}}x_{\alpha_{2}\wedge\beta_{2}}$. Then in$(f_{(\alpha_{1},\beta_{1})})=x_{\alpha_{1}}x_{\beta_{1}}$, in$(f_{(\alpha_{2},\beta_{2})})=x_{\alpha_{2}}x_{\beta_{2}}$ and Lcm$(f_{(\alpha_{1},\beta_{1})},f_{(\alpha_{2},\beta_{2})})=x_{\alpha_{1}}x_{\alpha_{2}}x_{\beta_{1}}x_{\beta_{2}}$. So,

[TABLE]

Hence $(x_{\alpha_{2}}x_{\beta_{2}}-x_{\alpha_{2}\vee\beta_{2}}x_{\alpha_{2}\wedge\beta_{2}})g_{(\alpha_{1},\beta_{1})}-(x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}})g_{(\alpha_{2},\beta_{2})}$ is a generator of the syzygy. ∎

Let us denote the $S$-module Syz${}^{2}_{1}R[\mathcal{L}]$ be the set of all diamond type syzygies of $R[\mathcal{L}]$.

Theorem 2.8**.**

Syz${}^{1}R[\mathcal{L}]$=Syz${}^{1}_{1}R[\mathcal{L}]\oplus$Syz${}^{2}_{1}R[\mathcal{L}]$.

Proof.

It follows from the Theorem 2.6 and Lemma 2.7. ∎

Theorem 2.9** **(Main Theorem 2).

Syz${}^{2}_{1}R[\mathcal{L}]=0$, if for $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})\in\mathcal{D}_{2}$, where $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ all are distincts and $\alpha_{1},\beta_{1}<\alpha_{2},\beta_{2}$. If there exists $a,b\in\mathcal{L}$ such that $a\wedge b=\beta_{1},a\vee b=\beta_{2},\alpha_{1},\alpha_{2},a\nsim b,\alpha_{2}\wedge b\nsim\alpha_{1}\vee\beta_{1}$, then the generator of the first syzygy from these two diamonds is a combination of other types of the generators.

Proof.

The proof follows from the following Lemmas 2.10 and 2.11. ∎

Lemma 2.10**.**

With the same notation as Lemma 2.9, if $a$ and $b$ satisfy the given conditions then there exists non-trivial $x,y,z\in\mathcal{L}$ such that the following (5) is a sublattice of $\mathcal{L}$.

Proof.

In Figure 5, we have $a\wedge b=\beta_{1}$ and $a\vee b=\beta_{2}$. Now we prove that there exist $x,y,z\in\mathcal{L}$ such that the following claim holds.

Claim: there exist $x$ such that $x=\alpha_{2}\wedge b=\alpha_{2}\wedge\beta_{2}\wedge b$ and $\beta_{1}<x<b$.

Let $\alpha_{2}\wedge b=x_{1},(\alpha_{2}\wedge\beta_{2})\wedge b=x$. Now since $x_{1}\vee\beta_{2}=(\alpha_{2}\wedge b)\vee\beta_{2}=\beta_{2},x\vee\beta_{2}=((\alpha_{2}\wedge\beta_{2})\wedge b)\vee\beta_{2}=\beta_{2},x_{1}\wedge\beta_{2}=(\alpha_{2}\wedge b)\wedge\beta_{2}=\alpha_{2}\wedge b,x\wedge\beta_{2}=((\alpha_{2}\wedge\beta_{2})\wedge b)=\alpha_{2}\wedge b$ then it implies $x=x_{1}$ and since we have $x=(\alpha_{2}\wedge\beta_{2})\wedge b$. Let $(\alpha_{2}\wedge\beta_{2})\wedge b=b$. Then $\alpha_{2}\wedge\beta_{2}\geq b$. Since $\alpha_{1}>a,\beta_{1}>b$, then $\alpha_{1}\wedge\beta_{1}>a\vee b=\beta_{2}$. Which is a contradiction. Hence it follows. 2. 2.

Claim: $x\nsim\alpha_{1}\vee\beta_{1},(\alpha_{1}\vee\beta_{1})\wedge x=\beta_{1}$.

By the given conditions, we have $x\nsim\alpha_{1}\vee\beta_{1},(\alpha_{1}\vee\beta_{1})\wedge x=(\alpha_{1}\wedge x)\vee(\beta_{1}\wedge x)=(\alpha_{1}\wedge\beta_{1})\vee\beta_{1}=\beta_{1}$. 3. 3.

Claim: $a\vee x=\alpha_{2}\wedge\beta_{2}$ and $a\wedge x=\beta_{1}$.

$a\vee x=a\vee[(\alpha_{2}\wedge\beta_{2})\wedge b]=(a\vee b)\wedge[a\vee(\alpha_{2}\wedge\beta_{2})]=\beta_{2}\wedge(\alpha_{2}\wedge\beta_{2})=\alpha_{2}\wedge\beta_{2}$ and $a\wedge x=a\wedge[(\alpha_{2}\wedge\beta_{2})\wedge b]=\beta_{1}\wedge(\alpha_{2}\wedge\beta_{2})=\beta_{1}$. 4. 4.

Claim: $y\wedge b=x$, where $y=(\alpha_{1}\vee\beta_{1})\vee x$.

$[(\alpha_{1}\vee\beta_{1})\vee x]\wedge b=[(\alpha_{1}\vee\beta_{1})\wedge b]\vee(x\wedge b)=\beta_{1}\vee x=x$. 5. 5.

Claim: $a\vee y=\alpha_{2}\wedge\beta_{2}$ and $a\wedge y=\alpha_{1}\wedge\beta_{1}$.

$a\vee[(\alpha_{1}\vee\beta_{1})\vee x]=[a\vee(\alpha_{1}\vee\beta_{1})\vee x=a\vee x=\alpha_{2}\wedge\beta_{2}$. and $a\wedge[(\alpha_{1}\vee\beta_{1})\vee x]=(a\wedge x)\vee[a\wedge(\alpha_{1}\vee\beta_{1})]=\beta_{1}\vee(\alpha_{1}\vee\beta_{1})=\alpha_{1}\vee\beta_{1}$. 6. 6.

Claim: $(\alpha_{2}\wedge\beta_{2})\wedge z=y$ and $(\alpha_{2}\wedge\beta_{2})\vee z=\beta_{2}$, where $z=(\alpha_{1}\vee\beta_{1})\vee b$.

$(\alpha_{2}\wedge\beta_{2})\wedge[(\alpha_{1}\vee\beta_{1})\vee b]=[(\alpha_{2}\wedge\beta_{2})\wedge(\alpha_{1}\vee\beta_{1})]\vee[(\alpha_{2}\wedge\beta_{2})\wedge b]=(\alpha_{1}\vee\beta_{1})\vee x=y$ and $(\alpha_{2}\wedge\beta_{2})\vee[(\alpha_{1}\vee\beta_{1})\vee b]=(\alpha_{2}\wedge\beta_{2})\vee(\alpha_{1}\vee\beta_{1})=(\alpha_{2}\wedge\beta_{2})\vee b=\beta_{2}$.

∎

Lemma 2.11**.**

In the following Figure 6, if $f_{(2,3)},f_{(11,12)}$ be the relations coming from the diamonds $(2,3),(11,12)$ respectively. Then the diamond type arrising out of these two diamonds can be expressed as a combination of $L$-shape syzygies.

Proof.

Let $f_{(2,3)}=x_{2}x_{3}-x_{1}x_{4}$ and $f_{(11,12)}=x_{11}x_{12}-x_{9}x_{13}$ be the relations coming from the diamonds $(2,3)$ and $(11,12)$ respectively. Then with respect to monomial ordering $>$ we have in$(f_{(2,3)})=x_{2}x_{3}$ and in$(f_{(11,12)})=x_{11}x_{12}$. So Lcm$(f_{(2,3)},f_{(11,12)})=x_{2}x_{3}x_{11}x_{12}$. $S(f_{(2,3)},f_{(11,12)})=x_{11}x_{12}f_{(2,3)}-x_{2}x_{3}f_{(11,12)}=x_{2}x_{3}x_{9}x_{13}-x_{1}x_{4}x_{11}x_{12}=x_{9}x_{13}f_{1}-x_{1}x_{4}f_{6}$. Therefore $(x_{11}x_{12}-x_{9}x_{13})g_{(2,3)}-(x_{2}x_{3}-x_{1}x_{4})g_{(11,12)}$ gives a element of the first syzygy. Now this expression can be written as $(x_{11}x_{12}-x_{9}x_{13})g_{(2,3)}-(x_{2}x_{3}-x_{1}x_{4})g_{(11,12)}=x_{11}(x_{12}g_{(2,3)}-x_{6}g_{(2,8)}+x_{2}g_{(6,8)}-x_{1}g_{(6,10)})-x_{13}(x_{9}g_{(2,3)}-x_{6}g_{(2,5)}+x_{2}g_{(5,6)}-x_{1}g_{(6,7)})+x_{6}(x_{13}g_{(2,5)}-x_{11}g_{(2,8)}+x_{2}g_{(8,11)}-x_{1}g_{(10,11)})+x_{2}(x_{13}g_{(5,6)}-x_{11}g_{(6,8)}+x_{6}g_{(8,11)}-x_{3}g_{(11,12)})-x_{1}(x_{13}g_{(6,7)}-x_{11}g_{(6,10)}+x_{6}g_{(10,11)}-x_{4}g_{(11,12)})$ which shows that it is a combination of four $L$-type elements. Hence the lemma follows for this case. ∎

Remark: The Theorem 2.9, tells us when the first syzygy is linear.

3 First Betti number of planar distributive lattice

Let $C_{m+1}$ be the chain $1<2<\ldots<m+1$, $G(m,n)=C_{m+1}\times C_{n+1}$ be the product lattice, we call $G(m,n)$ an $m\times n$ a grid lattice. Let $\mathcal{L}$ be a planar distributive lattice and $JM$ be the set of all join-meet irreducible elements of $\mathcal{L}$. Let $D_{(\theta_{i},\theta_{j})}=[\theta_{i}\wedge\theta_{j},\theta_{i}\vee\theta_{j}]$, the interval for $\theta_{i}\nsim\theta_{j},\theta_{i},\theta_{j}\in JM$. One can write the lattice $\mathcal{L}$ as a union of these intervals. Let $n(\theta_{1},\theta_{2})$ be the number of strip type generator in $D_{(\theta_{1},\theta_{2})}$. Then we have the following

Lemma 3.1**.**

For $\theta_{1},\theta_{2}\in JM$,

[TABLE]

where $S(.,.)$ is the number of strip type generator from the grid lattice.

Proof.

Since the grid lattice formed by $D_{(\theta_{1},\theta_{2})}$ is the lattice $G(ht(\theta_{1})-ht(\theta_{1}\wedge\theta_{2}),ht(\theta_{2})-ht(\theta_{1}\wedge\theta_{2}))$. Hence by the Lemma 4.3 it follows. ∎

Lemma 3.2**.**

For a planar distributive lattice $\mathcal{L}$, the number of strip type generator of $R[\mathcal{L}]$, we denote it by $n(S)$ is the following

[TABLE]

Proof.

From the Figure 7 it follows. ∎

Let $m(\theta_{1},\theta_{2})$ be the number of $L$-type generator in $D_{(\theta_{1},\theta_{2})}$. Then we have the following

Lemma 3.3**.**

For $\theta_{1},\theta_{2}\in JM$,

[TABLE]

where $L(.,.)$ denotes the number of $L$-shape generator from the grid lattice.

Proof.

This lemma is clearly follows from the Lemma 4.5. ∎

Lemma 3.4**.**

For a planar distributive lattice $\mathcal{L}$, the number of $L$-type generator of $R[\mathcal{L}]$, we denote it by $n(L)$ is the following

[TABLE]

where $r=ht(\theta_{i}\vee\theta_{j})-ht(\theta_{j}),s=ht(\theta_{j})-ht(\theta_{j}\wedge\theta_{k})$.

Proof.

The number of $L$-type for each $D_{(\theta_{i},\theta_{j})}$ is $m(\theta_{i},\theta_{j})$. For each $\theta_{i},\theta_{j},\theta_{k}$ there is double counting $m(r,s)$ where $r=ht(\theta_{i}\vee\theta_{j})-ht(\theta_{j}),s=ht(\theta_{j})-ht(\theta_{j}\wedge\theta_{k})$. Also for each $\theta_{i},\theta_{j},\theta_{k}$ there is a $L$-type (see the dotted line in the the Figure 8) and the number is $(ht(\theta_{j}\vee\theta_{k})-ht(\theta_{i}\vee\theta_{j})(ht(\theta_{j}\wedge\theta_{k})-ht(\theta_{i}\wedge\theta_{j}))\binom{r+1}{2}\binom{s+1}{2}$. Since for $(\theta_{i},\theta_{j}),(\theta_{j},\theta_{k}),(\theta_{k},\theta_{l})$ thete is no $L$-type. Hence the lemma follows. ∎

Let $B(\theta_{1},\theta_{2})$ be the number of box type in $D_{(\theta_{1},\theta_{2})}$ then we have the following:

Lemma 3.5**.**

For a planar distributive lattice $\mathcal{L}$, the number of box type generator of $R[\mathcal{L}]$, $n(B)$ is given by

[TABLE]

Proof.

For each $\theta_{i},\theta_{j}$ the number of box types is $B(\theta_{i},\theta_{j})$ and for each $\theta_{i},\theta_{j},\theta_{k}$ there is double countings $B(\theta_{i}\vee(\theta_{j}\wedge\theta_{k}),\theta_{j})$ in numbers in their intersections. Hence the lemma follows. ∎

Definition: Let $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})\in\mathcal{D}_{2}$. Then we say these two diamonds are comparable if $\alpha_{1}\leq\alpha_{2},\alpha_{1}\leq\beta_{2},\beta_{1}\leq\alpha_{2},\beta_{1}\leq\beta_{2}$. If at least one of these pairs is not comparable we say $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})$ are non- comparable.

Lemma 3.6**.**

Let $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})\in\mathcal{D}_{2}$, the diamond type arrising out of these two diamonds is expressed by strip, $L$, box types (infact only $L$-type) if only if

The diamonds are non-comparable or, 2. 2.

There exist a diamond $(\alpha_{3},\beta_{3})\in\mathcal{D}_{2}$ such that there is a sublattice 9

Proof.

$\Leftarrow$: This follows from the Lemma 2.9.

$\Rightarrow$: Let there does not exist $(\alpha_{3},\beta_{3})\in\mathcal{D}_{2}$ such that there is no sublattice 9 and let $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})$ are comparable, then we prove that $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})$ are not expressible. The element $(x_{\alpha_{2}}x_{\beta_{2}}-x_{\alpha_{2}\vee\beta_{2}}x_{\alpha_{2}\wedge\beta_{2}})g_{(\alpha_{1},\beta_{1})}-(x_{\alpha_{1}}x_{\beta_{1}}-x_{\alpha_{1}\vee\beta_{1}}x_{\alpha_{1}\wedge\beta_{1}})g_{(\alpha_{2},\beta_{2})}$ can be expressed by other type of generators if there exist a diagram like Figure 10, by Lemma 2.9. Now we will show for maximal sublattices this can not be express. Let either the diamond $(a)$ or $(b)$ is missing, let say $(a)$ is missing. Then $x_{\alpha_{2}}g_{(\alpha_{1},\beta_{1})}$ or $x_{\beta_{2}}g_{(\alpha_{1},\beta_{1})}$ is a term of a syzygy generator. But from the figure we see that the coefficient of $g_{(\alpha_{1},\beta_{1})}$ could only be the points which are bolded in the figure and we see that $x_{\alpha_{2}}$ or $x_{\beta_{2}}$ is not there. Thus this element cannot be expressed as a combination of other type of syzygy genrators. Similarly, we can show that if the diamond $(b)$ is not there then the element cannot be expressed in terms of other types of generators. Hence the lemma follows. ∎

Remark: This lemma will help us to find the number of diamond type generator for a planar distributive lattice. Let $n(D)$ be the number of diamond type generators. Then we have the following formula for the first Betti number.

Theorem 3.7**.**

Let $\mathcal{L}$ be a planar distributive lattice such that the condition of the Lemma 3.6 holds. Then the first syzygy of the Hibi ring $R[\mathcal{L}]$ is linear.

Proof.

This theorem follows from the Lemma 3.6. Since any diamond type generator can be expressed by other type of generators (linear). ∎

Theorem 3.8**.**

The first Betti number $\beta_{1}{\mathcal{(}L)}$ of $R[\mathcal{L}]$ is

[TABLE]

Proof.

It follows from the Lemmas 3.2,3.4,3.5 and 3.6. ∎

Let $k(\mathcal{L})=\#\{(\theta_{i},\theta_{j}):\theta_{i}\nsim\theta_{j},\theta_{i},\theta_{j}\in JM\}$.

Theorem 3.9** **(Main Theorem 3).

With above notations we have the following

When $k(\mathcal{L})=1$, the first syzygy of $R[\mathcal{L}]$ is linear. 2. 2.

When $k(\mathcal{L})=2$,

1. (a)

If the lattice is 11, then first syzygy is non linear. 2. (b)

Else if the lattice is 12 then first syzygy is linear if and only if $ht(\theta_{2}\wedge\theta_{3})-ht(\theta_{1}\wedge\theta_{2})=1$ or $ht(\theta_{2}\vee\theta_{3})-ht(\theta_{1}\vee\theta_{2})=1$, where $\theta_{1},\theta_{2},\theta_{3}\in JM$. 3. 3.

If $k(\mathcal{L})\geq 3$ then the first syzygy is non linear.

Proof.

Since $k(\mathcal{L})=1$ then it is always a grid lattice. By Lemma 3.6 it is linear.

3. (a)

If the lattice is 11, then the first syzygy have diamond type generators such as $(x_{\theta_{3}}x_{\theta_{4}}-x_{\theta_{3}\vee\theta_{4}}x_{\theta_{3}\wedge\theta_{4}})g_{(\theta_{1},\theta_{2})}-(x_{\theta_{1}}x_{\theta_{2}}-x_{\theta_{1}\vee\theta_{2}}x_{\theta_{1}\wedge\theta_{2}})g_{(\theta_{3},\theta_{4})}$ and it cannot be expressed by other types of generators, by Lemma 3.6. Hence it is non linear.

2. (b)

$\Rightarrow$: If the lattice is 12 and since first syzygy is linear then by Lemma 3.6 there is a sublattice 9 and therefore $ht(\theta_{2}\wedge\theta_{3})-ht(\theta_{1}\wedge\theta_{2})=1$ or $ht(\theta_{2}\vee\theta_{3})-ht(\theta_{1}\vee\theta_{2})=1$, where $\theta_{1},\theta_{2},\theta_{3}\in JM$.

$\Leftarrow$: Let the condition holds then all diamond type elements of the first syzygy can be expressed by other types. Hence the first syzygy is linear.

If $k(\mathcal{L})\geq 3$ then we have a figure like 13, then the first syzygy element coming from the lower diamond and upper diamond cannot be expressed by other type of syzygies. Hence diamond type elements will occur. Thus for this case the first syzygy is not linear.

∎

4 Betti number of the Grid lattice

Theorem 4.1**.**

The first syzygy of Hibi ring for an $m\times n$ grid lattice is generated by strip type, $L$-type, box type.

Proof.

We have the generators except strip type, $L$-type, box type and diamond type are coming from non planer lattices. But by Theorem 2.9 we have that diamond type will not appear in the $m\times n$ grid lattice. Hence the theorem follows. ∎

Now we give an exact formula for the first Betti number of the Hibi ring $R[\mathcal{L}]$, where $\mathcal{L}$ is an $m\times n$ grid distributive lattice.

Lemma 4.2**.**

Let $T(n)$ be the number of strip type generators for $R[\mathcal{L}]$, where $\mathcal{L}$ is a $1\times n$ grid lattice. Then

$T(n)=2\binom{n+1}{3}$.

Proof.

So we have the lattice $1\times n$ grid. We know that strip type generators from a lattice coming from every two diamonds which have a common side. Therefore, we have the following

$T(n)=2T(n-1)-T(n-2)+2(n-1)$

So, clearly $T(n)=2\binom{n+1}{3}$. ∎

Now we find strip type generators for the $m\times n$ grid lattice.

Lemma 4.3**.**

For the $m\times n$ grid lattice the total number of strip type generators we denote it by $S(m,n)$ is the following

$S(m,n)=\binom{m+1}{2}T(n)+\binom{n+1}{2}T(m)$,

where $T(m)$ and $T(n)$ are the number of srtip type generators for the lattices $1\times n$ and $1\times m$ respectively.

Proof.

The strip type generators from $m\times n$ grid lattice is

$(m+(m-1)+\ldots+1)T(n)+(n+(n-1)+\ldots+1)T(m)=\binom{m+1}{2}T(m)+\binom{n+1}{2}T(m)$.

∎

Now we find the number of $L$-type generators for the $2\times n$ grid lattice and using this we will find the number of $L$-type generators for $m\times n$ grid lattice.

Lemma 4.4**.**

Let $L(2,n)$ be the number of $L$-type generators for $2\times n$ grid lattice. Then

$L(2,n)=\frac{n(n^{2}-1)}{3}$.

Proof.

We see that

[TABLE]

∎

Lemma 4.5**.**

The total number of $L$-type generators we denote it by $L(m,n)$ for the $m\times n$ grid lattice is the following

$L(m,n)=\frac{L(2,m)L(2,n)}{2}$.

Proof.

We see that the total number is

[TABLE]

∎

Lemma 4.6**.**

The total number of box type generators we denote it by $B(m,n)$ for the $m\times n$ grid lattice is the following

$B(m,n)=\frac{B(2,m)B(2,n)}{2}$.

Proof.

Since in a $2\times 2$ grid lattice the number of box type is same as number of $L$-type generators, hence the lemma follows. ∎

Lemma 4.7**.**

The Betti number $\beta_{1}$ of $R[\mathcal{L}]$ for the $m\times n$ grid lattice is the following

$\beta_{1}=S(m,n)+L(m,n)+B(m,n)$.

Proof.

As by above lemma, first syzygy is generated by $L$-type, box type and strip type, so the total betti number is the sum of all numbers. Hence the lemma follows. ∎

Acknowledgement

The corresponding author thanks University Grants Commission(UGC) for financial support and Dept. of Mathematics of Bits-Pilani Goa campus for hospitality.