The divisor class group of a discrete polymatroid

TL;DR

This paper introduces toric rings of multicomplexes, focusing on computing their divisor class groups and canonical modules, especially for discrete polymatroids, providing new insights into their algebraic structure.

Contribution

It develops methods to compute divisor class groups and canonical modules of toric rings associated with multicomplexes, with detailed analysis for discrete polymatroids.

Findings

Computed divisor class groups for normal toric rings of multicomplexes.

Analyzed the structure of toric rings for various classes of discrete polymatroids.

Provided explicit descriptions of canonical modules in these cases.

Abstract

In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.

Equations

$$R_{\mathcal{M}}=K[{\mathbf{x}}^{\mathbf{v}}t:\mathbf{v}\in \mathcal{M}],$$

$$P=(x^{\mathbf{v}}t:\mathbf{v}\in \mathcal{M}\text{ and }f(p_{\mathbf{v}})>0),$$

$${\omega }_{R_{\mathcal{M}}}=(\bigcap _{i=1}^r{P}_i)\cap (\bigcap _{j=1}^n{Q}_j).$$

$$[{\omega }_{R_{\mathcal{M}}}]=\sum _{j=1}^r(1-c_{1,j}-\cdots -c_{n,j})[P_j].$$

$$b_{j}=\langle{\mathbf{e}}_{j},{\mathbf{e}}_{i}\rangle=\left\{\begin{array}[]{ll}1,&\text{if $j=i$,}\\ 0,&\text{otherwise.}\\ \end{array}\right.$$

$$[x_i]=(\sum _{j=1}^r{c}_{i,j}[P_j])+[Q_i].$$

$$\rho (A)=\max \{\mathbf{v}(A):\mathbf{v}\in \mathcal{P}\}.$$

$$f_A(x)=-\sum _{i\in A}{x}_i+\rho (A)x_{n+1},$$

$$|A|+1=a\rho (A)$$

$$\mathcal{P}=\{\mathbf{w}\in {\mathbb{Z}}^n:\mathbf{w}\le \mathbf{v}\}.$$

$$\rho (A)=\max \{|A\cap F|:F\in M\}.$$

$${\rho }_{\mathcal{A}}(X)=|\{k\in [n]:X\cap A_k\ne \varnothing \}|,X\subseteq [n].$$

$$B_{\mathcal{A}}=\{e_{i_1}+\cdots +e_{i_d}:i_k\in A_k,1\le k\le d\}\subseteq {\mathbb{Z}}_{+}^n.$$

$$\rho_{n,i}(X)=\left\{\begin{array}[]{ll}{n\choose i}-{n-|X|\choose i}&(|X|\leq n-i);\\ {n\choose i}&(|X|>n-i).\end{array}\right.$$

$$\left(\genfrac{}{}{0ex}{}{7}{4}\right)-\left(\genfrac{}{}{0ex}{}{6}{4}\right)=20,\left(\genfrac{}{}{0ex}{}{7}{4}\right)-\left(\genfrac{}{}{0ex}{}{5}{4}\right)=30,\left(\genfrac{}{}{0ex}{}{7}{4}\right)-\left(\genfrac{}{}{0ex}{}{4}{4}\right)=34,\left(\genfrac{}{}{0ex}{}{7}{4}\right)=35.$$

$$\mathrm{Cl}(R_{{\mathcal{P}}_{7,4}})={\mathbb{Z}}^{r-1},$$

$$r=\left(\genfrac{}{}{0ex}{}{n}{1}\right)+\left(\genfrac{}{}{0ex}{}{n}{2}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{n}{n-i}\right)+1.$$

$$\left(\genfrac{}{}{0ex}{}{n}{i}\right)-\left(\genfrac{}{}{0ex}{}{n-1}{i}\right),\left(\genfrac{}{}{0ex}{}{n}{i}\right)-\left(\genfrac{}{}{0ex}{}{n-2}{i}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{i}\right)-\left(\genfrac{}{}{0ex}{}{n-(n-i)}{i}\right),\left(\genfrac{}{}{0ex}{}{n}{i}\right)$$

$$\mathrm{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})={\mathbb{Z}}^{r-1}\oplus \mathbb{Z}/d\mathbb{Z}.$$

$${\mathcal{P}}_{\mathbf{s},d}=\{\mathbf{v}\in {\mathbb{Z}}_{+}^n:v_i\le s_i,|\mathbf{v}|\le d\}.$$

$${\rho }_{\mathbf{s},d}(\{i\})=s_i=\frac{2}{a},{\rho }_{\mathbf{s},d}([n])=d=\frac{n+1}{a}.$$

Full text

Takayuki Hibi is partially supported by JSPS KAKENHI 19H00637. Ayesha Asloob Qureshi is supported by The Scientific and Technological Research Council of Turkey - TÜBITAK (Grant No: 122F128).

Introduction

In the previous paper [7] the authors introduced the toric face ring of a simplicial complex. In this paper we extend this concept to multicomplexes with a special focus on polymatroids.

In Section 1 we present the general frame of toric face rings of multicomplexes and show how to compute the divisor class group and the class of the canonical module when the toric face ring is normal. The methods for the proofs are similar to those used in [7]. It turns out that their divisor class group is an abelian group with just one relation.

In Section 2 the results of Section 1 are made more explicit when the multicomplex is a discrete polymatroid. We determine the canonical class, and as an application we recover the Gorenstein criterion discussed in [5, Example 7.4(b)].

Section 3 deals with transversal polymatroids. As one of the main results it is shown in Corollary 3.7 that for any integer $r\geq 1$ and any integer $d\geq 0$, there exists a transversal polymatroid for which the divisor class group of its toric ring is isomorphic to ${\mathbb{Z}}^{r-1}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$. In Theorem 3.10 those transversal polymatroids are characterized for which the divisor class group of its toric ring is a finite cyclic group, and in Theorem 3.12 those for which the divisor class group is isomorphic to ${\mathbb{Z}}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$. Other families of transversal polymatroids are also considered.

In the last section we classify all discrete polymatroids of Veronese type whose toric face ring is Gorenstein. For the base ring of discrete polymatroids of Veronese type this classification was achieved in [3].

1. The toric face ring of a multicomplex

In this section we study the class group of normal toric face rings of multicomplexes. We first recall the basic concepts which are relevant for this paper.

We denote the set of non-negative integers by ${\mathbb{Z}}_{+}$ and the set of non-negative real numbers by $R_{+}$. For two vectors ${\mathbf{u}}=(u_{1},\ldots,u_{n})$ and ${\mathbf{v}}=(v_{1},\ldots,v_{n})$ in ${\mathbb{R}}_{+}^{n}$ we write ${\mathbf{u}}\leq{\mathbf{v}}$ if $u_{i}\leq v_{i}$ for all $i$. Moreover, write ${\mathbf{u}}<{\mathbf{v}}$ if ${\mathbf{u}}\leq{\mathbf{v}}$ and ${\mathbf{u}}\neq{\mathbf{v}}$. Let ${\mathbf{e}}_{1},\ldots,{\mathbf{e}}_{n}$ be the standard basis of ${\mathbb{Z}}^{n}$. A multicomplex on the ground set $[n]$ is a nonempty finite set $\mathcal{M}\subset{\mathbb{Z}}_{+}^{n}$ such that

1. (1)

for any ${\mathbf{v}}\in\mathcal{M}$ and ${\mathbf{u}}\in{\mathbb{Z}}_{+}^{n}$ with ${\mathbf{u}}\leq{\mathbf{v}}$, one has ${\mathbf{u}}\in\mathcal{M}$. 2. (2)

${\mathbf{e}}_{i}\in\mathcal{M}$ for any $1\leq i\leq n$.

Note that a simplicial complex on $[n]$ is in fact a multicomplex consisting of $0,1$-vectors.

Let $\mathcal{M}$ be a multicomplex on $[n]$. A vector ${\mathbf{v}}\in\mathcal{M}$ is called a facet of $\mathcal{M}$ if there exists no ${\mathbf{w}}\in\mathcal{M}$ with ${\mathbf{v}}<{\mathbf{w}}$. The set of all facets of $\mathcal{M}$ is denoted by $\mathcal{F}(\mathcal{M})$. If ${\mathcal{F}}(\mathcal{M})=\{{\mathbf{v}}_{1},\ldots,{\mathbf{v}}_{m}\}$, then we write $\mathcal{M}=\langle{\mathbf{v}}_{1},\ldots,{\mathbf{v}}_{m}\rangle$. Let $K$ be a field. For a vector ${\mathbf{v}}=(v_{1},\ldots,v_{n})\in\mathcal{M}$, we define the monomial ${\mathbf{x}}^{\mathbf{a}}=\prod_{i=1}^{n}x_{i}^{v_{i}}$ in the polynomial ring $K[x_{1},\ldots,x_{n}]$. The toric face ring of $\mathcal{M}$ is defined to be the subalgebra

[TABLE]

of the polynomial ring $S=K[x_{1},\ldots,x_{n},t]$. The algebra $R_{\mathcal{M}}$ has a $K$-basis consisting of monomials of $S$. If $f=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}t^{k}$ belongs to $R_{\mathcal{M}}$, we set $\deg f=k$. By this grading $R_{\mathcal{M}}$ is a standard graded $K$-algebra.

Any monomial ${\mathbf{x}}^{{\mathbf{v}}}t^{b}\in K[x_{1},\ldots,x_{n},t]$ can be identified with its exponent vector $({\mathbf{v}},b)\in{\mathbb{Z}}^{n+1}$. Then the monomial $K$-basis of $R_{\mathcal{M}}$ corresponds to an affine semigroup $S\subset{\mathbb{Z}}^{n+1}$ which is generated by the lattice points $p_{\mathbf{v}}=\sum_{i=1}^{n}v_{i}{\mathbf{e}}_{i}+{\mathbf{e}}_{n+1}$ in ${\mathbb{Z}}^{n+1}$, where ${\mathbf{v}}\in\mathcal{M}$.

Let ${\mathbb{Z}}S$ be the smallest subgroup of ${\mathbb{Z}}^{n+1}$ containing $S$ and let ${\mathbb{R}}_{+}S\subset{\mathbb{R}}^{n+1}$ be the smallest cone containing $S$. In our case, ${\mathbb{Z}}S={\mathbb{Z}}^{n+1}$. Since we assume $R$ is normal, Gordon’s lemma [2, Proposition 6.1.2] guarantees that $S={\mathbb{Z}}^{n+1}\cap{\mathbb{R}}_{+}S$.

By [1, Corollary 4.35] all minimal prime ideals of a monomial ideal in $R_{\mathcal{M}}$ are monomial prime ideals. In particular, they are generated by subsets of the generators ${\mathbf{x}}^{\mathbf{v}}t$ of $R_{\mathcal{M}}$. Moreover, it follows from [1, Proposition 2.36 and Proposition 4.33] that $P\subset R_{\mathcal{M}}$ is a monomial prime ideal of $R_{\mathcal{M}}$ if and only if there exists a face ${\mathcal{F}}$ of ${\mathbb{R}}_{+}S$ such that $P=(x^{\mathbf{v}}t\colon p_{\mathbf{v}}\not\in{\mathcal{F}})$. In other words, $P$ is a monomial prime ideal, if and only if there exists a supporting hyperplane $H$ of ${\mathbb{R}}_{+}S$ such that

[TABLE]

where $f$ is a linear form defining $H$.

The supporting hyperplane $H$ of a facet is uniquely determined. Since $H$ is spanned by lattice points, a linear form $f=\sum_{i=1}^{n+1}c_{i}x_{i}$ defining $H$ has rational coefficients. By clearing denominators we may assume that all $c_{i}$ are integers, and then dividing $f$ by the greatest common divisor of the $c_{i}$, we may furthermore assume that $\gcd(c_{1},\ldots,c_{n+1})=1$. Then this normalized linear form $f$ is uniquely determined by $H$. It has the property that $f(H)=0$ and $f({\mathbb{Z}}^{n+1})\cap{\mathbb{Z}}_{+}={\mathbb{Z}}_{+}$. Indeed, since $\gcd(c_{1},\ldots,c_{n+1})=1$, there exist $p=(b_{1},\ldots,b_{n+1})\in{\mathbb{Z}}^{n+1}$ with $\sum_{i=1}^{n+1}b_{i}c_{i}=1$, which implies that $f(p)=1$.

If $P$ is a height $1$ monomial prime ideal, then $P=(x^{\mathbf{v}}t\colon p_{\mathbf{v}}\not\in{\mathcal{F}})$, where ${\mathcal{F}}$ is a facet of ${\mathbb{R}}_{+}S$. Let $H$ be the supporting hyperplane of ${\mathcal{F}}$. Then we call the normalized linear form which defines $H$, the support form associated to $P$.

The following theorem generalizes [7, Theorem 1.1] with exactly the same argument.

Theorem 1.1**.**

Let $\mathcal{M}$ be a multicomplex on $[n]$ such that $R=R_{\mathcal{M}}$ is normal, and let $P_{1},\ldots,P_{r}$ be the minimal prime ideals of $(t)$, where $t\in R$ is the element corresponding to the zero vector in $\mathcal{M}$. Then $\operatorname{Cl}(R)$ is generated by the classes $[P_{i}]$, $i=1,\ldots,r$. Since $R_{P_{i}}$ is a discrete valuation ring, we have $tR_{P_{i}}=P_{i}^{a_{i}}R_{P_{i}}$ with $a_{i}\in{\mathbb{Z}}$ for $i=1,\ldots,r$. Then $\sum_{i=1}^{r}a_{i}[P_{i}]=0$ is the only generating relation among these generators of $\operatorname{Cl}(R)$.

The following lemma and proposition are needed for studying the canonical class of $R_{\mathcal{M}}$.

Lemma 1.2**.**

*(see [7, Lemma 1.6])

Let $\mathcal{M}$ be a multicomplex on $[n]$ such that $R=R_{\mathcal{M}}$ is normal, and let $P$ be a monomial prime ideal of $R$ of height one. Furthermore, let $f$ be the support form associated with $P$, and let ${\mathbf{v}}_{f}=(c_{1},\ldots,c_{n+1})$ be the coefficient vector of $f$. Then the following holds:*

If $u\in Q(R)$ is a monomial with exponent vector ${\mathbf{v}}_{u}$, then $uR_{P}=P^{a}R_{P}$, where $a=\langle{\mathbf{v}}_{f},{\mathbf{v}}_{u}\rangle.$ Here $\langle-,-\rangle$ denotes the standard inner product in ${\mathbb{R}}^{n+1}$.

Proposition 1.3**.**

Let $\mathcal{M}$ be a multicomplex on the ground set $[n]$. For $i=1,\ldots,n$, let $Q_{i}=({\mathbf{x}}^{\mathbf{v}}t\colon{\mathbf{v}}\in\mathcal{M},\ v_{i}>0)$. Then $\{Q_{1},\ldots,Q_{n}\}$ is the set of height one monomial prime ideals of $R_{\mathcal{M}}$ which do not contain $t$.

Proof.

For $i=1,\ldots,n$, let $f_{i}({\mathbf{x}})=x_{i}$ and $H_{i}=\{{\mathbf{x}}\colon f_{i}({\mathbf{x}})=0\}$. We claim that $H_{i}$ is a supporting hyperplane of a facet of ${\mathbb{R}}_{+}S$. For all ${\mathbf{v}}\in\mathcal{M}$, we have $f_{i}(p_{\mathbf{v}})=v_{i}\geq 0$. Hence $H_{i}$ is a supporting hyperplane, and the points ${\mathbf{e}}_{1},\ldots,\widehat{{\mathbf{e}}_{i}},\ldots,{\mathbf{e}}_{n+1}$ lie on the hyperplane. This shows that $H_{i}$ is a supporting hyperplane of a facet of ${\mathbb{R}}_{+}S$. Hence $P=({\mathbf{x}}^{\mathbf{v}}t\colon{\mathbf{v}}\in\mathcal{M},\ f_{i}(p_{\mathbf{v}})>0)$ is a height one monomial prime ideal of $R_{\mathcal{M}}$ and $P=Q_{i}$.

Now, let $P$ be a height one monomial prime ideal of $R_{\mathcal{M}}$ with $t\notin P$. First we claim that $x_{i}t\in P$, for some $i$. Suppose that this is not the case. Then there exists a nonzero ${\mathbf{v}}\in\mathcal{M}$ such that ${\mathbf{x}}^{\mathbf{v}}t\in P$ and for any ${\mathbf{u}}<{\mathbf{v}}$, ${\mathbf{x}}^{\mathbf{u}}t\notin P$. Let $j$ be an integer with $v_{j}>0$ and let ${\mathbf{w}}={\mathbf{v}}-{\mathbf{e}}_{j}$. Then $(x^{\mathbf{w}}t)(x_{j}t)=({\mathbf{x}}^{\mathbf{v}}t)(t)\in P$, while $x^{\mathbf{w}}t\notin P$ and $x_{j}t\notin P$, a contradiction. So the claim is proved and there exists $i$ such that $x_{i}t\in P$. Let ${\mathbf{v}}\in\mathcal{M}$ with $v_{i}>0$. Then $({\mathbf{x}}^{\mathbf{v}}t)(t)=(x_{i}t)({\mathbf{x}}^{{\mathbf{v}}-e_{i}}t)\in P$. Since $t\notin P$, we have ${\mathbf{x}}^{\mathbf{v}}t\in P$. This shows that $Q_{i}\subseteq P$. Since $P$ and $Q_{i}$ both have height one, we obtain $P=Q_{i}$.

Let $\omega_{R_{\mathcal{M}}}$ be the canonical module of $R_{\mathcal{M}}$. By [2, Corollary 3.3.19], $\omega_{R_{{\mathcal{M}}}}$ is a divisorial ideal and corresponds to the relative interior of the cone ${\mathbb{R}}_{+}S$, see [2, Theorem 6.3.5(b)]. Let $P_{1}.\ldots,P_{r}$ be the height one monomial prime ideals of $R_{\mathcal{M}}$ which contain $t$, and for each $j$ let $f_{j}$ be the support form associated with $P_{j}$. Let ${\mathbf{v}}_{f_{j}}=(c_{1,j},\ldots,c_{n+1,j})$ be the coefficient vector of $f_{j}$. Then by Proposition 1.3, $P_{1},\ldots,P_{r},Q_{1},\ldots,Q_{n}$ are the set of all height one monomial prime ideals of $R_{\mathcal{M}}$. By [2, Theorem 6.3.5(b)] we have

[TABLE]

Theorem 1.4**.**

Let $\mathcal{M}$ be a multicomplex on the ground set $[n]$ such that $R_{\mathcal{M}}$ is normal. Then with the notation introduced above, we have

[TABLE]

Proof.

To simplify the notation, we set $R=R_{\mathcal{M}}$. It follows from (1) that $[\omega_{R}]=\sum_{j=1}^{r}[P_{j}]+\sum_{k=1}^{n}[Q_{k}]$. For a fixed integer $1\leq i\leq n$, let $[x_{i}]=\sum_{j=1}^{r}a_{j}[P_{j}]+\sum_{k=1}^{n}b_{k}[Q_{k}]$. By [7, Lemma 1.6] we have $a_{j}=\langle{\mathbf{v}}_{f_{j}},{\mathbf{e}}_{i}\rangle=c_{i,j}$, where ${\mathbf{e}}_{i}\in{\mathbb{Z}}^{n+1}$ is a standard basis element. For each $1\leq j\leq n$, the associated support form of $Q_{j}$ is $f^{\prime}_{j}(x)=x_{j}$ (see Proposition 1.3). Hence ${\mathbf{v}}_{f^{\prime}_{j}}=e_{j}$ and

[TABLE]

Hence

[TABLE]

from which we conclude that $[Q_{i}]=-\sum_{j=1}^{r}c_{i,j}[P_{j}]$. This implies the desired equality.

Remark. Theorem 1.1 holds true, when we replace $\mathcal{M}$ by any finite set of vectors in ${\mathbb{Z}}_{+}^{n}$ which includes $e_{1},\ldots,e_{n}$ and the zero vector.

2. The class group and the canonical class of the toric face ring of a discrete polymatroid

Discrete polymatroids are special, but important classes of multicomplexes. In this section we apply the results of Section 1 to obtain the class group and the canonical class of discrete polymatroids.

For a vector ${\mathbf{v}}=(v_{1},\ldots,v_{n})\in{\mathbb{Z}}^{n}$ we set $|{\mathbf{v}}|=\sum_{i=1}^{n}v_{i}$, and for a subset $A\subset[n]$ we set ${\mathbf{v}}(A)=\sum_{i\in A}v_{i}$. In particular, ${\mathbf{v}}(\emptyset)=0$ and ${\mathbf{v}}([n])=|{\mathbf{v}}|$.

A discrete polymatroid on the ground set $[n]$ is a multicomplex ${\mathcal{P}}$ satisfying the following property: if ${\mathbf{v}}=(v_{1},\ldots,v_{n})$ and $w=(w_{1},\ldots,w_{n})$ belong to ${\mathcal{P}}$ and $|{\mathbf{v}}|<|{\mathbf{w}}|$, then there exists $i\in[n]$ with $v_{i}<w_{i}$ such that ${\mathbf{v}}+{\mathbf{e}}_{i}\in{\mathcal{P}}$.

Discrete polymatroids are particularly nice multicomplexes. Indeed, by [4] one has

Theorem 2.1** (Edmonds).**

Let ${\mathcal{P}}$ be a discrete polymatroid. Then $R_{\mathcal{P}}$ is normal.

The ground set rank function $\rho\colon 2^{[n]}\rightarrow{\mathbb{Z}}_{+}$ of a discrete polymatroid ${\mathcal{P}}$ is defined to be

[TABLE]

The ground set rank function $\rho$ of ${\mathcal{P}}$ has the following properties

1. (i)

$\rho(A)\leq\rho(B)$, if $A\subseteq B\subseteq[n]$. 2. (ii)

$\rho(A)+\rho(B)\geq\rho(A\cup B)+\rho(A\cap B)$ for all $A,B\subset[n]$.

A nonempty set $A\subseteq[n]$ is called $\rho$-closed, if $\rho(A)<\rho(B)$ for all $B\subseteq[n]$ properly containing $A$, and $A$ is called separable, if there exist nonempty subset $A_{1}$ and $A_{2}$ of $A$ with $A=A_{1}\cup A_{2}$ and $A_{1}\cap A_{2}=\emptyset$ such that $\rho(A)=\rho(A_{1})+\rho(A_{2})$. The set $A$ is called inseparable if it is not separable.

Let $S\subset{\mathbb{Z}}^{n+1}$ be the affine semigroup which is generated by the lattice points corresponding to the generators $x^{\mathbf{v}}t$ of $R_{\mathcal{P}}$. We need to determine the hyperplane defining the facets of the cone ${\mathbb{R}}_{+}S$.

For each $A\subseteq[n]$ which is $\rho$-closed and $\rho$-inseparable, we consider the hyperplane $H_{A}$ defined by the linear form

[TABLE]

and for $i=1,\ldots,n$, let $H_{i}$ be the hyperplane defined by the linear form $f_{i}(x)=x_{i}$.

The following result is crucial for our considerations

Theorem 2.2** (Edmonds).**

The hyperplanes $H_{A}$ and the hyperplanes $H_{i}$ introduced above are the supporting hyperplanes of the facets of the cone ${\mathbb{R}}_{+}S$ attached to the polymatroid ${\mathcal{P}}$.

We denote by $P_{A}$ the monomial prime ideals of $R_{\mathcal{P}}$ determined by the hyperplanes $H_{A}$ and by $Q_{i}$ the monomial prime ideals determined by the hyperplanes $H_{i}$.

Now we may apply the results of Section 1 and obtain

Theorem 2.3**.**

Let ${\mathcal{P}}$ be a polymatroid on the ground set $[n]$, and let $\rho$ be its ground set rank function. Let ${\mathcal{A}}$ be the set of $\rho$-closed and $\rho$-inseparable subsets of $[n]$. Then $\operatorname{Cl}(R_{\mathcal{P}})$ is generated by the classes $[P_{A}]$ with $A\in{\mathcal{A}}$. Moreover, $\sum_{A\in{\mathcal{A}}}\rho(A)[P_{A}]=0$ is the only generating relation among these generators of $\operatorname{Cl}(R_{\mathcal{P}})$.

In particular, $\operatorname{Cl}(R_{\Delta})\cong{\mathbb{Z}}^{r-1}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$, where $r=|{\mathcal{A}}|$ and $d=\gcd\{\rho(A)\colon\;A\in{\mathcal{A}}\}$.

Proof.

The prime ideals $P_{A}$ are precisely the minimal prime ideals of $(t)$. Therefore, the divisor classes $[P_{A}]$ with $A\in{\mathcal{A}}$ generate $\operatorname{Cl}(R_{\mathcal{P}})$. By Edmond’s theorem, the coefficient vector of the support form of $P_{A}$ is ${\mathbf{v}}_{A}=-\sum_{i\in A}e_{i}+\rho(A)e_{n+1}$. Therefore, by Theorem 1.1 and Lemma 1.2, the generating relation of $\operatorname{Cl}(R_{\mathcal{P}})$ is $\sum_{A\in{\mathcal{A}}}\rho(A)[P_{A}]=0$, as asserted.

The resulting group structure of $\operatorname{Cl}(R_{\mathcal{P}})$ is an immediate consequence of the statements before.

For the canonical class of $R=R_{\mathcal{P}}$ we have the following presentation.

Theorem 2.4**.**

$[\omega_{R}]=\sum_{A\in{\mathcal{A}}}(|A|+1)[P_{A}].$**

Proof.

Let ${\mathbf{v}}_{A}=(c_{1,A},\ldots,c_{n+1,A})$ be the coefficient vector of the linear form $f_{A}$. It follows from (2) that $c_{i,A}=-1$ if $i\in A$, and $c_{i,A}=0$ if $i\not\in A$. Therefore, Theorem 1.4 yields the desired result.

Corollary 2.5**.**

$R_{\mathcal{P}}$* is Gorenstein if and only if there exists an integer $a$ such that*

[TABLE]

for all $\rho$-closed and $\rho$-inseparable $A\subseteq[n]$.

Proof.

The ring $R_{\mathcal{P}}$ is Gorenstein if and only if $\omega_{R_{\mathcal{P}}}$ is a principal ideal, which by Theorem 2.4 is the case if and only if $\sum_{A}(|A|+1)[P_{A}]=0$. Thus the desired result follows from Theorem 2.3.

As a first example consider for any integer $d>0$ the discrete polymatroid ${\mathcal{P}}_{d}=\{v\colon\;|v|\leq d\}$ on the ground set $[n]$. For ${\mathcal{P}}_{d}$ the only $\rho$-closed and $\rho$-inseparable set is $[n]$ with $\rho([n])=d$. Thus, Theorem 2.3 implies that $\operatorname{Cl}(R_{P_{d}})\cong{\mathbb{Z}}/d{\mathbb{Z}}$, and from Corollary 2.5 it follows that $R_{{\mathcal{P}}_{d}}$ is Gorenstein if and only if $d$ divides $n+1$.

Another simple example is given in

Example 2.6**.**

Let ${\mathbf{v}}=(v_{1},\ldots,v_{n})\in{\mathbb{Z}}^{n}$ be a vector with $v_{i}\neq 0$ for all $i$, and let

[TABLE]

For a nonempty subset $A\subseteq[n]$, we have $\rho(A)=\sum_{i\in A}v_{i}$. Hence $\rho(A)<\rho(B)$ for any set $B\supsetneq A$. Thus $A$ is $\rho$-closed. If $|A|>1$, then $\rho(A)=\rho(A\setminus\{i\})+\rho(\{i\})$, which implies that $A$ is $\rho$-separable. Therefore the $\rho$-closed and $\rho$-inseparable subsets of $[n]$ are $\{1\},\{2\},\ldots,\{n\}$. Moreover $\rho(\{i\})=v_{i}$ for $1\leq i\leq n$.

Then by Theorem 2.3, $\operatorname{Cl}(R_{\mathcal{P}})={\mathbb{Z}}^{n-1}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$, where $d=\gcd(v_{1},\ldots,v_{n})$. Moreover, by Corollary 2.5, $R_{\mathcal{P}}$ is Gorenstein if and only if $1\leq v_{1}=v_{2}=\cdots=v_{n}\leq 2$. **

The following corollary gives a necessary condition on a matroid $M$ for $R_{M}$ to be Gorenstein. Recall that a graph $G$ is called unmixed if all its maximal independent sets have the same cardinality.

Corollary 2.7**.**

Let $M$ be a matroid on $[n]$ and let $G_{M}$ be the graph on $[n]$ whose edge set is the $1$-skeleton of $M$. If $R_{M}$ is Gorenstein, then $G$ is unmixed.

Proof.

Let $\rho\colon 2^{[n]}\rightarrow{\mathbb{Z}}_{+}$ be the ground set rank function of $M$, where

[TABLE]

We show that all maximal independent sets of $G_{M}$ are $\rho$-closed and $\rho$-inseparable. Let $A$ be a maximal independent set of $G_{M}$. Then $|A\cap F|\leq 1$ for any $F\in M$. Hence $\rho(A)=1$. For any set $B\subseteq[n]$ with $A\subsetneq B$, there exists $F\in M$ with $|F|=2$ such that $F\subseteq B$. Thus $\rho(B)\geq 2>\rho(A)$. This shows that $A$ is $\rho$-closed.

Now, let $A=X\cup Y$ with $X\cap Y=\emptyset$, where $X\neq\emptyset$ and $Y\neq\emptyset$. Since $\rho(X)\geq 1$ and $\rho(Y)\geq 1$, we have $\rho(A)<\rho(X)+\rho(Y)$. Therefore $A$ is $\rho$-inseparable. By Corollary 2.5 if $R_{M}$ is Gorenstein, then there exists an integer $a$ such that for any maximal independent set $A$ of $G_{M}$, $|A|+1=a\rho(A)=a$. This means that $G_{M}$ is unmixed.

3. Classes of transversal polymatroids

Let ${\mathcal{A}}=(A_{1},\ldots,A_{d})$ be a family of nonempty subsets of $[n]$ and suppose that $[n]=A_{1}\cup\ldots\cup A_{d}$. It is not required that $A_{i}\neq A_{j}$ if $i\neq j$. One defines the integer valued nondecreasing function $\rho_{\mathcal{A}}:2^{[n]}\rightarrow{\mathbb{Z}}_{+}$ by setting

[TABLE]

It follows from [6, Exercise 12.2] that $\rho_{\mathcal{A}}$ is submodular and the set of bases of the discrete polymatroid ${\mathcal{P}}_{\mathcal{A}}\subset{\mathbb{Z}}_{+}^{n}$ arising from $\rho_{\mathcal{A}}$ is

[TABLE]

One says that the discrete polymatroid ${\mathcal{P}}_{\mathcal{A}}$ is the transversal polymatroid presented by ${\mathcal{A}}$.

Theorem 3.1**.**

Let ${\mathcal{P}}_{\mathcal{A}}$ be the transversal polymatroid presented by ${\mathcal{A}}=(A_{1},\ldots,A_{d})$ and suppose that there is $i$ with $A_{i}\not\subseteq\cup_{i\neq j}A_{j}$. Then ${\operatorname{Cl}}(R_{{\mathcal{P}}_{\mathcal{A}}})$ is free.

Proof.

Let $X=A_{i}\setminus\cup_{i\neq j}A_{j}$. Then $X\neq\emptyset$ is $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable. Since $\rho_{\mathcal{A}}(X)=1$, it follows that ${\operatorname{Cl}}(R_{{\mathcal{P}}_{\mathcal{A}}})$ is torsion free.

Now, fix $1<i<n$ and write ${[n]\choose i}$ for the set of all $i$-element subsets of $[n]$. Let ${\mathcal{P}}_{n,i}$ denote the transversal polymatroid presented by ${[n]\choose i}$, and let $\rho_{n,i}$ denote the rank function of ${\mathcal{P}}_{n,i}$.

Lemma 3.2**.**

Let $X\subseteq[n]$. Then

[TABLE]

Proof.

Let $A\in{[n]\choose i}$. It is clear that $X\cap A\neq\emptyset$ if $|X|>n-i$. Let $|X|=a\leq n-i$. One can assume that $X=\{1,\ldots,a\}$. It follows that $X\cap A=\emptyset$ if and only if $A\subset\{a+1,\ldots,n\}$. Hence $\rho_{n,i}(X)={n\choose i}-{n-a\choose i}$, as desired.

Lemma 3.3**.**

A subset $X\subseteq[n]$ is $\rho$-closed and $\rho$-inseparable if and only if $1\leq|X|\leq n-i$ or $X=[n]$.

Proof.

Lemma 3.2 says that $X\subseteq[n]$ is $\rho$-closed if and only if $1\leq|X|\leq n-i$ or $X=[n]$. It is clear that if $|X|=1$, then $X$ is $\rho$-inseparable. Let $1<|X|\leq n-i$ or $X=[n]$ and $X=Y\cup Z$, where $Y\neq\emptyset,Z\neq\emptyset$ and $Y\cap Z=\emptyset$. Let say, $1\in Y,2\in Z$. Then $\{1,2,\ldots,i\}$ intersects both $Y$ and $Z$. Hence $\rho_{n,i}(Y)+\rho_{n,i}(Z)>\rho_{n,i}(X).$ Thus $X$ is $\rho$-inseparable.

Example 3.4**.**

Let $n=7,i=4$. Then

[TABLE]

Hence

[TABLE]

where $r={7\choose 1}+{7\choose 2}+{7\choose 3}+1=64$.

As an immediate consequence of Theorem 2.3 together with Lemma 3.2 and Lemma 3.3 we obtain

Theorem 3.5**.**

${\operatorname{Cl}}(R_{{\mathcal{P}}_{n,i}})={\mathbb{Z}}^{r-1},$* where*

[TABLE]

Proof.

The result follows once we note that the greatest common divisor of the numbers

[TABLE]

is $1$.

Theorem 3.6**.**

Let ${\mathcal{A}}=(\,\underbrace{A_{1},\ldots,A_{1}}_{k_{1}}\,,\ldots,\,\underbrace{A_{r},\ldots,A_{r}}_{k_{r}}\,)$, where $A_{1}\subsetneq\cdots\subsetneq A_{r}=[n]$, and let ${\mathcal{P}}_{\mathcal{A}}$ be the transversal polymatroid presented by ${\mathcal{A}}$. Then $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})={\mathbb{Z}}^{r-1}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$, where $d=\gcd(k_{1},\ldots,k_{r})$.

Proof.

Let $X\subseteq[n]$ be a $\rho_{\mathcal{A}}$-closed set and let $i$ be the smallest integer such that $X\cap A_{i}\neq\emptyset$. Since $X\subseteq[n]\setminus A_{i-1}$ and $\rho_{\mathcal{A}}([n]\setminus A_{i-1})=\rho_{\mathcal{A}}(X)$, we have $X=[n]\setminus A_{i-1}$. Hence $[n],[n]\setminus A_{1},\ldots,[n]\setminus A_{r-1}$ are the $\rho_{\mathcal{A}}$-closed sets. Furthermore, each set $[n]\setminus A_{i}$ is $\rho_{\mathcal{A}}$-inseparable. Indeed one has $\rho_{\mathcal{A}}([n]\setminus A_{i})=k_{i+1}+\cdots+k_{r}$. Let $[n]\setminus A_{i}=X\cup Y$ with $X\cap Y=\emptyset$, $X\neq\emptyset$ and $Y\neq\emptyset$. We may assume that $X\cap A_{i+1}\neq\emptyset$. Then $\rho_{\mathcal{A}}(X)=\rho_{\mathcal{A}}([n]\setminus A_{i})$ and $\rho_{\mathcal{A}}(Y)\geq k_{r}$. Hence $\rho_{\mathcal{A}}(X)+\rho_{\mathcal{A}}(Y)>\rho_{\mathcal{A}}([n]\setminus A_{i})$, as desired. Hence $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})={\mathbb{Z}}^{r-1}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$, where $d=\gcd(k_{1},\ldots,k_{r})$.

By choosing $k_{1},\ldots,k_{r}=d$ in Theorem 3.6 we obtain the following fundamental result in the present section.

Corollary 3.7**.**

Given $r>0$ and $d>0$, there exists a transversal polymatroid ${\mathcal{P}}_{\mathcal{A}}$ for which

[TABLE]

Example 3.8**.**

Let $n\geq 3$, and let $G$ be a finite simple connected graph on $[n]$ with the edge set $\{e_{1},\ldots,e_{s}\}$. Suppose that $G$ is not the star graph $K_{1,n-1}$ (which is equivalent to $\cap_{i=1}^{s}e_{i}=\emptyset$). Let $A_{i}=[n]\setminus e_{i}$ and ${\mathcal{A}}=(A_{1},\ldots,A_{s})$. Let ${\mathcal{P}}_{\mathcal{A}}$ be the transversal polymatroid presented by ${\mathcal{A}}$. For any set $X\subseteq[n]$ we have $\rho_{\mathcal{A}}(X)>0$. Moreover, if $|X|>2$, then $\rho_{\mathcal{A}}(X)=s$. If $i$ is a leaf of $G$, then $\rho_{\mathcal{A}}(\{i\})=s-1$. Otherwise $\rho_{\mathcal{A}}(\{i\})\leq s-2$. Let $i<j$. If $\{i,j\}$ is an edge of $G$, then $\rho_{\mathcal{A}}(\{i,j\})=s-1$. If $\{i,j\}$ is not an edge of $G$, then $\rho_{\mathcal{A}}(\{i,j\})=s$. Hence a set $\{i\}$ is $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable if and only if $\deg_{G}(i)\geq 2$. Furthermore, $\{i,j\}$ is $\rho_{\mathcal{A}}$-closed if and only if $\{i,j\}$ is an edge of $G$. Let $\{i,j\}$ be an edge of $G$ and suppose that $\{i,j\}$ is not $\rho_{\mathcal{A}}$-inseparable. Then $\rho_{\mathcal{A}}(\{i,j\})=\rho_{\mathcal{A}}(\{i\})+\rho_{\mathcal{A}}(\{j\})$, which says that each edge contains either $i$ or $j$. In other words the set $\{i,j\}$ is a vertex cover of $G$. Hence a set $\{i,j\}$ is $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable if and only if $\{i,j\}$ is and edge of $G$ and it is not a vertex cover of $G$. Clearly, $[n]$ is $\rho_{\mathcal{A}}$-closed. Let $[n]=X\cup Y$ with $X\neq\emptyset$, $Y\neq\emptyset$ and $X\cap Y=\emptyset$. If $n\geq 5$, then either $|X|>2$ or $|Y|>2$. Hence $\rho_{\mathcal{A}}(X)=s$ or $\rho_{\mathcal{A}}(Y)=s$. Since $\rho_{\mathcal{A}}(X)>0$ and $\rho_{\mathcal{A}}(Y)>0$, we get $\rho_{\mathcal{A}}(X)+\rho_{\mathcal{A}}(Y)>s=\rho_{\mathcal{A}}([n])$. Hence $[n]$ is $\rho_{\mathcal{A}}$-inseparable. If $n=3$, since $G\neq K_{1,2}$ and $G$ is connected we have $G=K_{3}$. Then for $X=\{1,2\}$ and $Y=\{3\}$ we have $\rho_{\mathcal{A}}([3])=\rho_{\mathcal{A}}(X)+\rho_{\mathcal{A}}(Y)$. Hence $[n]$ is $\rho_{\mathcal{A}}$-separable. Now, let $n=4$ and consider a nontrivial partition $[n]=X\cup Y$. If $|X|>2$ or $|Y|>2$, then $\rho_{\mathcal{A}}(X)+\rho_{\mathcal{A}}(Y)>\rho_{\mathcal{A}}([n])$. Let $|X|=|Y|=2$. Then $\rho_{\mathcal{A}}(X)\geq s-1$ and $\rho_{\mathcal{A}}(Y)\geq s-1$ and then $\rho_{\mathcal{A}}(X)+\rho_{\mathcal{A}}(Y)\geq 2s-2$. Since $G$ is connected, $s\geq n-1=3$. Hence $2s-2>s=\rho_{\mathcal{A}}([n])$. Then we conclude that $[n]$ is $\rho_{\mathcal{A}}$-inseparable if and only if $n>3$. Hence for $n>3$, $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})$ is free, since $s-1$ and $s$ are relatively prime and there exists at least one edge which is not a vertex cover of $G$. Moreover, the rank of $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})$ is $n-l+m$, where $l$ is the number of leaves of $G$ and $m$ is the number of edges of $G$ which are not vertex covers of $G$. Finally, if $n=3$, then $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})$ is free of rank $2$. **

Lemma 3.9**.**

Let ${\mathcal{P}}_{\mathcal{A}}$ be the transversal polymatroid presented by ${\mathcal{A}}=(A_{1},\ldots,A_{s})$. Then there is a unique $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subset of $[n]$ if and only if $A_{1}=\cdots=A_{s}=[n]$.

Proof.

If $A_{1}=\cdots=A_{s}=[n]$, then $[n]$ is the unique $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subset of $[n]$.

Suppose that there is a unique $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subset of $[n]$. Then $[n]$ must be $\rho_{\mathcal{A}}$-inseparable.

In fact, if $[n]$ is not $\rho_{\mathcal{A}}$-inseparable, then $[n]=X\cup Y$ with $X\cap Y=\emptyset$, where $X\neq\emptyset,Y\neq\emptyset$, and with $\rho_{\mathcal{A}}([n])=\rho_{\mathcal{A}}(X)+\rho_{\mathcal{A}}(Y)$. Hence each $A_{i}$ is contained in either $X$ or $Y$. Thus in particular each of $X$ and $Y$ is $\rho_{\mathcal{A}}$-closed. By continuing these procedure, one can find a $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subset of $X$ and $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subset of $Y$. This contradicts the uniqueness of $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subset. Hence $[n]$ is $\rho_{\mathcal{A}}$-inseparable.

Now, suppose there is $i$ with $A_{i}\neq[n]$. Let $j\not\in A_{i}$ and $Z$ a maximal subset of $[n]$ with $j\in Z$ and $\rho_{\mathcal{A}}(\{j\})=\rho_{\mathcal{A}}(Z)$. Then $\rho_{\mathcal{A}}(Z)<s$ and $Z$ is $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable. Hence both $[n]$ and $Z$ are $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable, a contradiction. Hence $A_{1}=\cdots=A_{s}=[n]$, as desired.

Lemma 3.9 guarantees that

Theorem 3.10**.**

Let ${\mathcal{P}}_{\mathcal{A}}$ be the transversal polymatroid presented by ${\mathcal{A}}=(A_{1},\ldots,A_{s})$. Then $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})={\mathbb{Z}}/d{\mathbb{Z}}$ with $d>0$ if and only if $A_{1}=\cdots=A_{s}=[n]$ and $s=d$.

Lemma 3.11**.**

Let ${\mathcal{P}}_{\mathcal{A}}$ be the transversal polymatroid presented by ${\mathcal{A}}=(A_{1},\ldots,A_{s})$, where $(A_{1},\ldots,A_{s})\neq([n],\ldots,[n])$. Then there are exactly two $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subsets of ${\mathcal{P}}_{\mathcal{A}}$ if and only if one of the following conditions is satisfied:

• (i)

There is a decomposition $[n]=B\cup C$, where $B\neq\emptyset,C\neq\emptyset,B\cap C=\emptyset$, for which each $A_{i}$ is equal to either $B$ or $C$.

• (ii)

There is a nonempty subset $\Omega\subsetneq[n]$, for which each $A_{i}$ is equal to either $[n]\setminus\Omega$ or $[n]$.

Proof.

Suppose there are exactly two $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subsets of ${\mathcal{P}}_{\mathcal{A}}$.

First, suppose that $[n]$ is not $\rho_{\mathcal{A}}$-inseparable. Then there is a decomposition $[n]=B\cup C$, where $B\neq\emptyset,C\neq\emptyset,B\cap C=\emptyset$, for which each $A_{i}$ is contained in either $B$ or $C$. Each of $B$ and $C$ is $\rho_{\mathcal{A}}$-closed. We claim that each of $B$ and $C$ is $\rho_{\mathcal{A}}$-inseparable. If, say, $B$ is not $\rho_{\mathcal{A}}$-inseparable, then there is a decomposition $B=D\cup E$, where $D\neq\emptyset,E\neq\emptyset,D\cap E=\emptyset$, for which each $A_{i}\subset B$ is contained in either $D$ or $E$. Let $k\in D$ and $Y$ the unique maximal subset of $[n]$ with $k\in Y$ and with $\rho_{{\mathcal{P}}_{\mathcal{A}}}(\{k\})=\rho_{{\mathcal{P}}_{\mathcal{A}}}(Y)$. Then $Y$ is $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable with $Y\subset D$. This observation guarantees that each of $D,E$ and $C$ contains a $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subset of ${\mathcal{P}}_{\mathcal{A}}$, a contradiction. Hence both $B$ and $C$ are $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable.

Now, let, say, $A_{i}\subsetneq B$ and $j\in B\setminus A_{i}$. Let $X$ denote the unique maximal subset of $B$ with $j\in X$ and with $\rho_{{\mathcal{P}}_{\mathcal{A}}}(\{j\})=\rho_{{\mathcal{P}}_{\mathcal{A}}}(X)$. Then $X$ is $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable with $X\cap A_{i}=\emptyset$. Hence $B,C$ and $X$ are $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable, a contradiction.

Second, suppose that $[n]$ is $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable. Let $\Omega$ denote the set of integers $j\in[n]$ with $j\not\in A_{i}$ for some $i$. Then $\Omega\neq\emptyset$. For each $j\in\Omega$, one writes $X_{j}$ for a unique maximal subset of $[n]$ with $j\in X_{j}$ and with $\rho_{{\mathcal{P}}_{\mathcal{A}}}(\{j\})=\rho_{{\mathcal{P}}_{\mathcal{A}}}(X_{j})$. Then $X_{j}\subsetneq[n]$ and $X_{j}$ is $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable. It follows from the assumption that $X_{j}=X_{j^{\prime}}$ for $j,j^{\prime}\in\Omega$. In particular, each $A_{i}$ satisfies either $\Omega\cap A_{i}=\emptyset$ or $A_{i}\subset\Omega$. It then follows that for each $i$ one has either $A_{i}=[n]$ or $A_{i}=[n]\setminus\Omega$. Since $[n]=A_{1}\cup\cdots\cup A_{s}$, there is $i$ with $A_{i}=[n]$.

On the other hand, if (i) or (ii) is satisfied, then clearly there are exactly two $\rho_{\mathcal{A}}$-closed and $\rho_{\mathcal{A}}$-inseparable subsets of ${\mathcal{P}}_{\mathcal{A}}$.

Theorem 3.12**.**

Let ${\mathcal{P}}_{\mathcal{A}}$ be the transversal polymatroid presented by ${\mathcal{A}}=(A_{1},\ldots,A_{s})$. Then $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})={\mathbb{Z}}$ or ${\mathbb{Z}}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$ if and only if one of the following conditions is satisfied:

• (i)

There is a decomposition $[n]=B\cup C$, where $B\neq\emptyset,C\neq\emptyset,B\cap C=\emptyset$ for which, for some $1\leq q<s$, one has $A_{1}=\cdots=A_{q}=B$ and $A_{q+1}=\cdots=A_{s}=C$.

• (ii)

There is a nonempty subset $A\subsetneq[n]$ for which, for some $1\leq q<s$, one has $A_{1}=\cdots=A_{q}=A$ and $A_{q+1}=\cdots=A_{s}=[n]$.

Moreover, $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})={\mathbb{Z}}$ if and only if $q$ and $s-q$ are relatively prime. If the greatest common divisor of $q$ and $s-q$ is $d>1$, then $\operatorname{Cl}(R_{{\mathcal{P}}_{\mathcal{A}}})={\mathbb{Z}}\oplus{\mathbb{Z}}/d{\mathbb{Z}}$.

4. Gorenstein polymatroids of Veronese type

Fix an integer $d$ and a sequence ${\mathbf{s}}=(s_{1},\ldots s_{n})$ of integers with $1\leq s_{1}\leq\cdots\leq s_{n}\leq d$ and $d<\sum_{i=1}^{n}s_{i}$. The discrete polymatroid of Veronese type $({\mathbf{s}},d)$ is the discrete polymatroid

[TABLE]

Let $\rho_{{\mathbf{s}},d}=\rho_{{\mathcal{P}}_{{\mathbf{s}},d}}$ denote the rank function of ${\mathcal{P}}_{{\mathbf{s}},d}$.

Lemma 4.1**.**

The $\rho_{{\mathbf{s}},d}$-closed and $\rho_{{\mathbf{s}},d}$-inseparable subsets are each $\{i\}$ and $[n]$. Furthermore, $\rho_{{\mathbf{s}},d}(\{i\})=s_{i}$ and $\rho_{{\mathbf{s}},d}([n])=d$.

Proof.

If $\emptyset\neq A\subset[n]$, then $\rho_{{\mathbf{s}},d}(A)=\min\{\sum_{i\in A}s_{i},d\}$. Let $A\subset[n]$ with $1<|A|<n$. If $\rho_{{\mathbf{s}},d}(A)=\sum_{i\in A}s_{i}$, then $A$ cannot be $\rho_{{\mathbf{s}},d}$-inseparable. If $\rho_{{\mathbf{s}},d}(A)=d$, then $A$ cannot be $\rho_{{\mathbf{s}},d}$-closed.

First, one shows that $[n]$ is $\rho_{{\mathbf{s}},d}$-closed and $\rho_{{\mathbf{s}},d}$-inseparable. Clearly $[n]$ is $\rho_{{\mathbf{s}},d}$-closed. Let $[n]=A\cup B$ with $A\neq\emptyset,B\neq\emptyset,A\cap B=\emptyset$. Then $\rho_{{\mathbf{s}},d}(A)=\min\{\sum_{i\in A}s_{i},d\}$ and $\rho_{{\mathbf{s}},d}(B)=\min\{\sum_{i\in B}s_{i},d\}$. Hence $\rho_{{\mathbf{s}},d}([n])<\rho_{{\mathbf{s}},d}(A)+\rho_{{\mathbf{s}},d}(B)$. Thus $[n]$ is $\rho_{{\mathbf{s}},d}$-inseparable.

Second, one shows that each $\{i\}$ is $\rho_{{\mathbf{s}},d}$-closed and $\rho_{{\mathbf{s}},d}$-inseparable. Clearly $\{i\}$ is $\rho_{{\mathbf{s}},d}$-inseparable. Let $j\neq i$. Then $\rho_{{\mathbf{s}},d}(\{i,j\})=\min\{s_{i}+s_{j},d\}$. Since $s_{i}<d$ and $s_{j}>0$, one has $\rho_{{\mathbf{s}},d}(\{i\})<\rho_{{\mathbf{s}},d}(\{i,j\})$. Hence $\{i\}$ is $\rho_{{\mathbf{s}},d}$-closed.

Now, Corollary 2.5 enables us to classify Gorenstein discrete polymatroids of Veronese type.

Theorem 4.2**.**

The toric ring $R_{{\mathcal{P}}_{{\mathbf{s}},d}}$ is Gorenstein if and only if one of the following conditions is satisfied:

• (i)

each $s_{i}=2$ and $n=d-1\geq 2$;

• (ii)

each $s_{i}=1$ and $n=2d-1\geq 3$.

Proof.

It follows that $R_{{\mathcal{P}}_{{\mathbf{s}},d}}$ is Gorenstein if and only if there is an integer $a\geq 1$ for which

[TABLE]

Let $a=1$. Then each $s_{i}=2$ and $n=d-1\geq 2$. Let $a=2$. Then each $s_{i}=1$ and $n=2d-1\geq 3$, as desired.