Almost Gorenstein Hibi rings

TL;DR

This paper provides criteria for identifying when Hibi rings and ladder determinantal rings are almost Gorenstein, based on their structural properties, extending understanding of their algebraic and combinatorial characteristics.

Contribution

It introduces new criteria linking the almost Gorenstein property of these rings to their defining poset or ladder shape, advancing classification methods.

Findings

Criteria for Hibi rings to be level, non-Gorenstein, and almost Gorenstein.

Criteria for Hibi rings to be non-level and almost Gorenstein.

Criteria for ladder determinantal rings to be non-Gorenstein and almost Gorenstein.

Abstract

In this paper, we state criteria of a Hibi ring to be level, non-Gorenstein and almost Gorenstein and to be non-level and almost Gorenstein in terms of the structure of the partially ordered set defining the Hibi ring. We also state a criterion of a ladder determinantal ring defined by 2-minors to be non-Gorenstein and almost Gorenstein in terms of the shape of the ladder.

Equations

$$\cdot$$

$$[[R]]=\frac{h_0+h_1\lambda +\cdots +h_s{\lambda }^s}{(1-\lambda {)}^d}.$$

$$[[M]]=\frac{F(\lambda )}{(1-\lambda {)}^u}.$$

$${\dim }_k[M/\mathfrak{m}M{]}_i\le c_i$$

$$0\to R\to K_R(-a)\to C\to 0$$

$$0\to R\stackrel{\phi }{\to }{K}_R(-a)\stackrel{\pi }{\to }C\to 0$$

$${\dim }_k[K_R(-a)/\mathfrak{m}{K}_R(-a){]}_i={\dim }_k[C/\mathfrak{m}C{]}_i+{\delta }_{0,i},$$

$$[[C]]=\frac{{\sum }_{j=0}^{s-1}((h_s+\cdots +h_{s-j})-(h_0+\cdots +h_j)){\lambda }^j}{(1-\lambda {)}^{d-1}}.$$

$${\dim }_k[C/\mathfrak{m}C{]}_i={\dim }_k[C/\mathfrak{m}C{]}_{s-1-i}$$

$${\dim }_k[K_R(-a)/\mathfrak{m}{K}_R(-a){]}_i={\dim }_k[K_R(-a)/\mathfrak{m}{K}_R(-a){]}_{s-1-i}$$

$${\dim }_k[K_R(-a)/\mathfrak{m}{K}_R(-a){]}_{s-1}={\dim }_k[K_R(-a)/\mathfrak{m}{K}_R(-a){]}_0-1$$

$$R=\underset{\nu \in \overline{\mathcal{T}}(P)}{⨁}k{T}^{\nu }$$

$$K_R=\underset{\nu \in \mathcal{T}(P)}{⨁}k{T}^{\nu }$$

$$\min \{i\mid [K_R{]}_i\ne 0\}=\mathrm{rank}{P}^{+}.$$

$$r(y_1,x_1,\dots ,y_t,x_t):=\sum _{i=1}^t(\mathrm{rank}[x_{i-1},y_i]-\mathrm{rank}[x_i,y_i])+\mathrm{rank}[x_t,\infty ],$$

$$0\to R\to K_R(-a)\to C\to 0$$

$$x_0<x_1<\cdots <x_{r-1}<x_r=\infty$$

$$\#(P^{+}\setminus \{x_0,x_1,\dots ,x_r\})=1.$$

$$s=a+d=d-r=\#P-\mathrm{rank}{P}^{+}=1.$$

$$z_{i_{j}}\mathrel{<\!\!\!\cdot}z^{\prime}_{i_{j}+1}\quad\mbox{for $j=1$, \ldots, $p$.}$$

$$\mathrm{rank}[x_i,z]+\mathrm{rank}[z,y_{i+1}]=\mathrm{rank}[x_i,y_{i+1}],$$

$$\mathrm{rank}[x_{i-1},z]+\mathrm{rank}[z,y_i]=\mathrm{rank}[x_{i-1},y_i],$$

$$\nu (x_0)\le r(y_1,x_1,\dots ,y_t,x_t).$$

$$0\to R\to K_R(-a)\to C\to 0$$

$${\dim }_k[C/\mathfrak{m}C{]}_1={\dim }_k[K_R(-a)/\mathfrak{m}{K}_R(-a){]}_1>0.$$

$${\dim }_k[K_R(-a)/\mathfrak{m}{K}_R(-a){]}_{s-2}\ge {\dim }_k[C/\mathfrak{m}C{]}_{s-2}={\dim }_k[C/\mathfrak{m}C{]}_1>0$$

$$r_{\max }\ge s-2-a=a+d-2-a=d-2.$$

$$u_0+v_1+\cdots +v_t+u_t-2$$

$$n\mathrel{:=}\max\{j\mid\alpha_{0j}<\alpha_{10}\},\ m^{\prime}\mathrel{:=}\min\{j\mid\alpha_{1j}>\alpha_{0u_{0}}\},\$$

$$n^{\prime }:=u_1-m^{\prime }\text{mbox}andm:=u_0-n.$$

Full text

Keywords: almost Gorenstein, Hibi ring, level ring, ladder determinantal ring

MSC:13H10, 13F50, 13A02

1 Introduction

Cohen-Macaulay and Gorenstein properties are traditional and very important notions in commutative ring theory. First, Cohen-Macaulay property of a Noetherian ring is defined as a ring which satisfies unmixedness theorem and/or other equivalent conditions. After that, Bass [Bas] defined Gorenstein property for Noetherian rings. A Noetherian local ring is by definition Gorenstein if its self injective dimension is finite and a general Noetherian ring is Gorenstein if the localized ring by any prime ideal of it is a Gorenstein local ring.

Since then, researchers have accumulated the theories concerning Cohen-Macaulay and Gorenstein rings and had feeling that in some theories, assuming only Cohen-Macaulay property is to weak to deduce interesting results while Gorenstein property is to strong so that only trivial results can be deduced. Therefore, there are efforts to find and formulate a nice class of rings to fill the gap between Cohen-Macaulay and Gorenstein properties.

The notion of almost Gorenstein property is an outcome of these efforts. First Barucci and Fröberg [BF] defined almost Gorenstein property for 1-dimensional analytically unramified local rings. After that, Goto, Matsuoka and Phuong [GMP] generalized the notion of almost Gorenstein property to arbitrary 1-dimensional local rings. Further, Goto, Takahashi and Taniguchi [GTT] generalized the notion of almost Gorenstein property to arbitrary local rings and graded rings.

In this paper, we study almost Gorenstein property of Hibi rings and completely characterize non-Gorenstein almost Gorenstein graded Hibi rings in terms of the structure of partially ordered sets. Let $k$ be a field, $H$ a finite distributive lattice, $P$ the set of join-irreducible elements of $H$ and ${\mathcal{R}}_{k}(H)$ the Hibi ring over $k$ on $H$. Then ${\mathcal{R}}_{k}(H)$ is non-Gorenstein and almost Gorenstein if and only if, after deleting elements $x\in P$ with ${\mathrm{star}}_{P}(x)=P$, $P$ is one of the following forms.

1. (1)

Disjoint union of an element and a chain with length at least 1. 2. (2)

A partially ordered set obtained by adding one of the following covering relations to disjoint union of 2 chains with the same length of at least 2.

1. (i)

The bottom element of each chain is covered by the top element of the other chain. 2. (ii)

The bottom element of the second chain is covered by the top element of the first chain and there are integers $i_{1}$, …, $i_{p}$ such that the $i_{j}$-th element of the first chain from the bottom is covered by the $(i_{j}+1)$-th element of the second chain from the bottom for each $j=1$, …, $p$ ($p$ may be 0 in this case).

[TABLE]

See Theorems 3.1 and 4.1.

We also state a criterion of non-Gorenstein almost Gorenstein property of ladder determinantal rings defined by 2-minors, since (ladder) determinantal rings defined by 2-minors have structures of Hibi rings. See Theorem 5.1.

This paper is organized as follows. After establishing basic materials used in the main argument of this paper in Section 2, we state a criterion of a Hibi ring to be level, non-Gorenstein and almost Gorenstein graded ring in Section 3. Further, we state a criterion of a Hibi ring to be non-level and almost Gorenstein graded ring in Section 4. We show that (1) above corresponds to the level case and (2) to the non-level case. Finally in Section 5, we state a criterion of a ladder determinantal ring defined by 2-minors to be non-Gorenstein and almost Gorenstein.

2 Preliminaries

In this section, we collect some known facts and state basic facts in order to prepare the main argument of this paper.

In this paper, all rings and algebras are assumed to be commutative with an identity element. We denote by $\mathbb{N}$ the set of non-negative integers, by $\mathbb{Z}$ the set of integers. Let $k$ be a field, $R$ a Cohen-Macaulay standard graded $k$-algebra, i.e., $R=\bigoplus_{n\geq 0}R_{n}$, $R_{0}=k$ and $R=k[R_{1}]$. We set $d=\dim R$, $a=a(R)$, where $a(R)$ is the $a$-invariant of $R$. We denote the irrelevant maximal ideal $\bigoplus_{n>0}R_{n}$ of $R$ by ${\mathfrak{m}}$, the graded canonical module of $R$ by $K_{R}$.

For a graded finitely generated $R$-module $M$, we denote the Hilbert series $\sum_{i}(\dim_{k}M_{i})\lambda^{i}$ by $[[M]]$. We set $(h_{0},h_{1},\ldots,h_{s})$, $h_{s}\neq 0$, the $h$-vector of $R$, i.e.,

[TABLE]

It is known that $s=a+d$. We also denote the number of elements in a minimal system of generators of $M$ by $\mu(M)$ and the multiplicity of $M$ with respect to ${\mathfrak{m}}$ by $e(M)$. It is known that $\mu(M)\leq e(M)$. Further the following fact is known.

Lemma 2.1

Let $M$ be a Cohen-Macaulay non-negatively graded $R$-module with $\dim M=u$. Then there is a polynomial $F(\lambda)\in\mathbb{Z}[\lambda]$ such that

[TABLE]

Further if we set $F(\lambda)=\sum_{i}c_{i}\lambda^{i}$, then

[TABLE]

for any $i$. Moreover, if $\mu(M)=e(M)$, then the equality holds for any $i$.

Now we recall the definition of an almost Gorenstein ring in our case.

Definition 2.2** ([GTT])**

$R$ is an almost Gorenstein graded ring if there exist an exact sequence

[TABLE]

of graded $R$-modules such that $C=0$ or $\mu(C)=e(C)$. (We always assume a homomorphism between graded $R$-modules is of degree 0.)

In [GTT], notions of almost Gorenstein local rings and almost Gorenstein graded rings are defined. In this paper, we treat only almost Gorenstein graded rings and just call them almost Gorenstein rings.

Note that if

[TABLE]

is an exact sequence, then $\varphi(1)$ is a generator of $K_{R}(-a)$. Set $\xi_{1}=\varphi(1)$ and take a minimal homogeneous generating system $\xi_{1}$, $\xi_{2}$, …, $\xi_{p}$ of $K_{R}(-a)$. Then $\pi(\xi_{2})$, …, $\pi(\xi_{p})$ is a minimal homogeneous generating system of $C$. In particular, we see the following fact.

Lemma 2.3

In the above situation,

[TABLE]

where $\delta_{0,i}$ is the Kronecker’s delta.

Note that if $C\neq 0$, then $C$ is a Cohen-Macaulay $R$-module of dimension $d-1$ [GTT, Lemma 3.1]. We also recall the following fact.

Fact 2.4** ([Hig, Proposition 2.4])**

If $C\neq 0$, then

[TABLE]

By Lemmas 2.1 and 2.3 and Fact 2.4, we see the following

Corollary 2.5

Suppose that $C\neq 0$ and $\mu(C)=e(C)$. Then

[TABLE]

for $0\leq i\leq s-1$,

[TABLE]

for $1\leq i\leq s-2$ and

[TABLE]

if $s\geq 2$.

Next we recall some basic facts about Hibi rings. First we fix notation and terminology on partially ordered set (poset for short).

Let $Q$ be a finite poset. A chain $X$ in $Q$ is a totally ordered subset of $Q$. The length of a chain $X$ is by definition $\#X-1$, where $\#X$ is the cardinality of $X$. The rank of $Q$, denoted $\mathrm{rank}Q$, is the maximum length of chains in $Q$. If every maximal chain (with respect to the inclusion relation) has length $\mathrm{rank}Q$, we say that $Q$ is pure. A subset $I$ of $Q$ such that $x\in Q$, $y\in I$ and $x<y$ imply $x\in I$ is called a poset ideal of $Q$. For $x$, $y\in Q$ with $x\leq y$ we define $[x,y]_{Q}\mathrel{:=}\{z\in Q\mid x\leq z\leq y\}$. We denote $[x,y]_{Q}$ as $[x,y]$ if there is no danger of confusion. $(x,y]_{Q}$, $[x,y)_{Q}$ and $(x,y)_{Q}$ for $x$, $y\in Q$ with $x<y$ are defined similarly. If $x$, $y\in Q$, $x<y$ and $(x,y)_{Q}=\emptyset$, we say that $y$ covers $x$ and denote $x\mathrel{<\!\!\!\cdot}y$ or $y\mathrel{\cdot\!\!\!>}x$. For $x\in Q$, we define ${\mathrm{star}}_{Q}(x)\mathrel{:=}\{y\in Q\mid y\leq x$ or $y\geq x\}$. For a poset $Q$, let $\infty$ be a new element which is not contained in $Q$. We define the poset $Q^{+}$ whose base set is $Q\cup\{\infty\}$ and $x<y$ if and only if $x$, $y\in Q$ and $x<y$ in $Q$ or $x\in Q$ and $y=\infty$.

Let $H$ be a finite distributive lattice with unique minimal element $\{x_{0}\}$, $P$ the set of join-irreducible elements of $H$, i.e., $P=\{\alpha\in H\mid\alpha=\beta\vee\gamma\Rightarrow\alpha=\beta$ or $\alpha=\gamma\}$. Note that we treat $x_{0}$ as a join-irreducible element. It is known that $H$ is isomorphic to the set of non-empty poset ideals of $P$ ordered by inclusion.

Let $\{T_{x}\}_{x\in P}$ be a family of indeterminates indexed by $P$.

Definition 2.6** ([Hib])**

${\mathcal{R}}_{k}(H)\mathrel{:=}k[\prod_{x\leq\alpha}T_{x}\mid\alpha\in H]$.

${\mathcal{R}}_{k}(H)$ is called the Hibi ring over $k$ on $H$ nowadays. Hibi [Hib, §2 b)] showed that ${\mathcal{R}}_{k}(H)$ is a normal affine semigroup ring and thus is Cohen-Macaulay by the result of Hochster [Hoc]. Further, by setting $\deg T_{x_{0}}=1$ and $\deg T_{x}=0$ for $x\in P\setminus\{x_{0}\}$, ${\mathcal{R}}_{k}(H)$ is a standard graded $k$-algebra. From now on, we set $R={\mathcal{R}}_{k}(H)$.

Here we recall the characterization of Gorenstein property of $R$ by Hibi.

Fact 2.7** ([Hib, §3 d)])**

$R$* is Gorenstein if and only if $P$ is pure.*

Set $\overline{\mathcal{T}}(P)\mathrel{:=}\{\nu\colon P^{+}\to\mathbb{N}\mid\nu(\infty)=0$ and $x<y\Longrightarrow\nu(x)\geq\nu(y)\}$ and ${\mathcal{T}}(P)\mathrel{:=}\{\nu\colon P^{+}\to\mathbb{N}\mid\nu(\infty)=0$ and $x<y\Longrightarrow\nu(x)>\nu(y)\}$. Further, we set $T^{\nu}\mathrel{:=}\prod_{x\in P}T_{x}^{\nu(x)}$ for $\nu\in\overline{\mathcal{T}}(P)$. Note that $\deg T^{\nu}=\nu(x_{0})$. It is easily verified that

[TABLE]

[Hib, §3 b)] and therefore, by the description of the canonical module of a normal affine semigroup ring by Stanley [Sta, p. 82], we see that

[TABLE]

[Hib, §3 b)]. We define the order on ${\mathcal{T}}(P)$ by $\nu\leq\nu^{\prime}\iff\nu^{\prime}-\nu\in\overline{\mathcal{T}}(P)$, where we set $(\nu^{\prime}-\nu)(x)\mathrel{:=}\nu^{\prime}(x)-\nu(x)$. Then for $\nu\in{\mathcal{T}}(P)$, $T^{\nu}$ is a generator of $K_{R}$ if and only if $\nu$ is a minimal element of ${\mathcal{T}}(P)$.

Since $\nu(x)\geq\mathrm{rank}[x,\infty]$ for any $x\in P$ and $\nu\in{\mathcal{T}}(P)$, and $\nu_{0}\colon P^{+}\to\mathbb{N}$, $x\mapsto\mathrm{rank}[x,\infty]$ is an element of ${\mathcal{T}}(P)$, we see that

[TABLE]

In particular, $\mathrm{rank}P^{+}=-a$. In order to use notation of [Miy], we define auxiliary notation by $r\mathrel{:=}\mathrm{rank}P^{+}$. We also see by [Hib, §2 a)] that $d=\#P$.

We recall the notion of a sequence with condition N  defined in [Miy].

Definition 2.8

Let $y_{1}$, $x_{1}$, $y_{2}$, $x_{2}$, …, $y_{t}$, $x_{t}$ be a (possibly empty) sequence of elements in $P$. We say the sequence $y_{1}$, $x_{1}$, $y_{2}$, $x_{2}$, …, $y_{t}$, $x_{t}$ satisfies condition N if

1. (1)

$x_{1}\neq x_{0}$. 2. (2)

$y_{1}>x_{1}<y_{2}>x_{2}<\cdots<y_{t}>x_{t}$. 3. (3)

For any $i$, $j$ with $1\leq i<j\leq t$, $y_{i}\not\geq x_{j}$.

Remark 2.9

A sequence with condition N may be an empty sequence, i.e., $t$ may be [math].

For a sequence with condition N, we make the following

Definition 2.10

Let $y_{1}$, $x_{1}$, $y_{2}$, $x_{2}$, …, $y_{t}$, $x_{t}$ be a sequence with condition N. We set

[TABLE]

where we set an empty sum to be 0.

Remark 2.11

For an empty sequence, we set $r()=\mathrm{rank}[x_{0},\infty]=r$.

It is fairly easy to see that there are only finitely many sequences with condition N [Miy, Lemma 3.4] and we state the following

Definition 2.12

We set ${r_{\max}}\mathrel{:=}\max\{r(y_{1},x_{1},\ldots,y_{t},x_{t})\mid y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ is a sequence with condition N$\}$.

By [Miy, Corollary 3.5 and Theorem 3.12], we see the following

Fact 2.13

$[K_{R}/{\mathfrak{m}}K_{R}]_{i}\neq 0\iff r\leq i\leq{r_{\max}}$.

3 A criterion of a Hibi ring to be level, non-Gorenstein and almost Gorenstein

In this section and the next, we state criteria of a Hibi ring to be non-Gorenstein and almost Gorenstein. Recall that we have set $R={\mathcal{R}}_{k}(H)$, $P$ the set of join-irreducible elements of $H$, $r=\mathrm{rank}P^{+}$, $d=\dim R$ and $a=a(R)$. Also recall that $(h_{0},\ldots,h_{s})$, $h_{s}\neq 0$, is the $h$-vector of $R$.

In this section, we treat the level case.

Theorem 3.1

$R$* is level, non-Gorenstein and almost Gorenstein if and only if there exists $z\in P$ with the following conditions.*

1. (1)

$P^{+}\setminus\{z\}$* is a chain of length $r$.* 2. (2)

$\mathrm{rank}[x_{0},z]+\mathrm{rank}[z,\infty]<r$.

• Proof

We first prove the “only if” part. Since $R$ is not Gorenstein, it is not regular. Therefore $s\geq 1$. Take an exact sequence

[TABLE]

such that $\mu(C)=e(C)$ (since $R$ is not Gorenstein, $C\neq 0$).

Since $R$ is level, we see that $[K_{R}(-a)/{\mathfrak{m}}K_{R}(-a)]_{i}=0$ for $i>0$. Therefore, we see by Lemma 2.3 that $[C/{\mathfrak{m}}C]_{i}=0$ for $1\leq i\leq s-1$. If $s\geq 2$, then we see by Corollary 2.5 that $\dim_{k}[C/{\mathfrak{m}}C]_{0}=\dim_{k}[C/{\mathfrak{m}}C]_{s-1}=0$. This contradicts to $C\neq 0$. Therefore, we see that $s=1$.

Take a maximal chain

[TABLE]

of length $r$ in $P^{+}$. Since $1=s=a+d=d-r=\#P-\mathrm{rank}P^{+}$, we see that

[TABLE]

Set $\{z\}=P^{+}\setminus\{x_{0},x_{1},\ldots,x_{r}\}$. We show that $z$ satisfies (1) and (2). (1) is clear. Further, $\mathrm{rank}[x_{0},z]+\mathrm{rank}[z,\infty]<r$ since $P^{+}$ is not pure by Fact 2.7. Thus, $z$ satisfies (2).

Next we prove the “if” part. By (1) and (2), we see that $P^{+}$ is not pure and

[TABLE]

**Therefore, $R$ is not Gorenstein, level and almost Gorenstein by Fact 2.7 and [GTT, Proposition 10.1 and Theorem 10.4]. (Note that in [GTT, §10], they assumed that the base field is an infinite field. However, [GTT, Lemma 10.1 and Theorem 10.4] also hold true in our situation. First, if $s=1$, then $R$ has a 2-linear resolution as a module over a polynomial ring over $k$ with $\dim_{k}R_{1}$ variables. Therefore, $R$ is level. Thus, [GTT, Lemma 10.1] also valid in our situation. As for [GTT, Lemma 10.3], which is used in the proof of [GTT, Theorem 10.4] is clearly valid in our situation, since $R$ is a domain and $K_{R}$ is an ideal of $R$. Further, we can use [GTT, Theorem 10.4] in our case by the argument of base field extension, since $R\otimes_{k}k^{\prime}$ is a domain for any extension field $k^{\prime}$ of $k$. Level property of $R$ may also be deduced from [Miy, Theorem 3.9] and almost Gorenstein property of $R$ is also proved directly by the same argument as in the proof of Theorem 4.1.)  **

Remark 3.2

If $R$ satisfies the conditions in Theorem 3.1, then $R$ is a polynomial ring over the ring considered in [GTT, Example 10.5].

4 A criterion of a Hibi ring to be non-level and almost Gorenstein

In this section, we state a criterion of a Hibi ring to be non-level and almost Gorenstein.

Theorem 4.1

$R$* is not level and almost Gorenstein if and only if there are non-negative integers $m$, $n$ and $n^{\prime}$ with $m\geq 3$, $z_{1}$, …, $z_{m}$, $z^{\prime}_{1}$, …, $z^{\prime}_{m}$, $w_{0}$, …, $w_{n}$, $w^{\prime}_{0}$, …, $w^{\prime}_{n^{\prime}}\in P^{+}$ with the following conditions.*

1. (1)

$P^{+}=\{z_{1}$, …, $z_{m}$, $z^{\prime}_{1}$, …, $z^{\prime}_{m}$, $w_{0}$, …, $w_{n}$, $w^{\prime}_{0}$, …, $w^{\prime}_{n^{\prime}}\}$. 2. (2)

$x_{0}=w_{0}\mathrel{<\!\!\!\cdot}w_{1}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}w_{n}$,

$w^{\prime}_{0}\mathrel{<\!\!\!\cdot}w^{\prime}_{1}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}w^{\prime}_{n^{\prime}}=\infty$,

$w_{n}\mathrel{<\!\!\!\cdot}z_{1}\mathrel{<\!\!\!\cdot}z_{2}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}z_{m}\mathrel{<\!\!\!\cdot}w^{\prime}_{0}$,

$w_{n}\mathrel{<\!\!\!\cdot}z^{\prime}_{1}\mathrel{<\!\!\!\cdot}z^{\prime}_{2}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}z^{\prime}_{m}\mathrel{<\!\!\!\cdot}w^{\prime}_{0}$* and*

$z^{\prime}_{1}\mathrel{<\!\!\!\cdot}z_{m}$. 3. (3)

Other covering relations in $P^{+}$ are one of the followings.

1. (i)

$z_{1}\mathrel{<\!\!\!\cdot}z^{\prime}_{m}$* or* 2. (ii)

there are integers $i_{1}$, …, $i_{p}$ with $1\leq i_{1}<\cdots<i_{p}\leq m-1$ such that

[TABLE]

($p$* may be [math] in this case, i.e., there is no covering relation other than stated in (2).)*

In order to prove this theorem, we first show the following fact.

Lemma 4.2

Let $\nu$ be a minimal element of ${\mathcal{T}}(P)$. Then there exists a sequence $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ of elements in $P$ with the following conditions, where we set $y_{t+1}=\infty$.

1. (1)

$y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ satisfies condition N. 2. (2)

$\nu(x_{i})-\nu(y_{i+1})=\mathrm{rank}[x_{i},y_{i+1}]$* for $0\leq i\leq t$.* 3. (3)

For any $i$ with $1\leq i\leq t$, $\mathrm{rank}[x_{i},z]+\mathrm{rank}[z,y_{i+1}]<\mathrm{rank}[x_{i},y_{i+1}]$ for any $z\in(x_{i},y_{i}]\cap(x_{i},y_{i+1}]$ and $\mathrm{rank}[x_{i-1},z]+\mathrm{rank}[z,y_{i}]<\mathrm{rank}[x_{i-1},y_{i}]$ for any $z\in[x_{i-1},y_{i})\cap[x_{i},y_{i})$.

• Proof

By [Miy, Lemma 3.3], we see that there is a sequence $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ of elements in $P$ with condition N such that $\nu(x_{i})-\nu(y_{i+1})=\mathrm{rank}[x_{i},y_{i+1}]$ for $0\leq i\leq t$.

If there exist $i$ with $1\leq i\leq t$ and $z\in(x_{i},y_{i}]\cap(x_{i},y_{i+1}]$ such that

[TABLE]

then we can replace $x_{i}$ with $z$. Similarly, if there exist $i$ with $1\leq i\leq t$ and $z\in[x_{i-1},y_{i})\cap[x_{i},y_{i})$ such that

[TABLE]

then we can replace $y_{i}$ with $z$.

**It is clear that after finite steps of replacements, we get a sequence which satisfies (1), (2) and (3).  **

We also note the following fact, whose proof is straightforward.

Lemma 4.3

Let $\nu$ be an element of ${\mathcal{T}}(P)$ and $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ a sequence of elements in $P$ which satisfies (1) and (2) of Lemma 4.2. Then

[TABLE]

Now we state

Proof of Theorem 4.1 We first prove the “only if” part. Take an exact sequence

[TABLE]

such that $\mu(C)=e(C)$ (since $R$ is not Gorenstein, $C\neq 0$). Since $R$ is not level, we see by [Miy, Theorem 3.9] that ${r_{\max}}>r$. Therefore, by Fact 2.13 and Lemma 2.3, we see that

[TABLE]

Thus, we see by Fact 2.4 and Lemma 2.1 that $s\geq 2$. Further, since

[TABLE]

by Corollary 2.5, we see by Fact 2.13 that

[TABLE]

Let $\nu$ be a minimal element of ${\mathcal{T}}(P)$ with $\nu(x_{0})={r_{\max}}$. Take a sequence $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ of elements in $P$ which satisfies (1), (2) and (3) of Lemma 4.2. Further take $\alpha_{ij}$ and $\beta_{ij}$ such that $x_{i}=\alpha_{i0}\mathrel{<\!\!\!\cdot}\alpha_{i1}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}\alpha_{iu_{i}}=y_{i+1}$ for $0\leq i\leq t$, where we set $y_{t+1}=\infty$ and $u_{i}=\mathrm{rank}[x_{i},y_{i+1}]$ and $x_{i}=\beta_{i0}\mathrel{<\!\!\!\cdot}\beta_{i1}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}\beta_{iv_{i}}=y_{i}$ for $1\leq i\leq t$, where we set $v_{i}=\mathrm{rank}[x_{i},y_{i}]$. Then $\alpha_{00}$, …, $\alpha_{0,u_{0}}$, $\beta_{11}$, …, $\beta_{1,v_{1}-1}$, $\alpha_{10}$, …, $\alpha_{1,u_{1}}$, $\beta_{21}$, …, $\beta_{2,v_{1}-1}$, …, $\alpha_{t-1,0}$, …, $\alpha_{t-1,u_{t-1}}$, $\beta_{t,1}$, …, $\beta_{t,v_{t}-1}$, $\alpha_{t,0}$, …, $\alpha_{t,u_{t}}$ are distinct elements of $P^{+}$, since the sequence $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ satisfies (1), (2) and (3) of Lemma 4.2. Since

[TABLE]

by Lemma 4.3, we see that $t=1$, $v_{1}=1$ and ${r_{\max}}=u_{0}-1+u_{1}=\#P-2=d-2$. In particular, $P^{+}=\{\alpha_{00}$, $\alpha_{01}$, …, $\alpha_{0u_{0}}$, $\alpha_{10}$, $\alpha_{11}$, …, $\alpha_{1u_{1}}\}$ and $x_{0}=\alpha_{00}\mathrel{<\!\!\!\cdot}\alpha_{01}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}\alpha_{0u_{0}}$, $\alpha_{10}\mathrel{<\!\!\!\cdot}\alpha_{11}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}\alpha_{1u_{1}}=\infty$ and $\alpha_{10}\mathrel{<\!\!\!\cdot}\alpha_{0u_{0}}$.

Set

[TABLE]

[TABLE]

Set also

[TABLE]

Then (1) and (2) are satisfied. Further, since $w_{n}\mathrel{<\!\!\!\cdot}z^{\prime}_{1}\mathrel{<\!\!\!\cdot}z_{m}\mathrel{<\!\!\!\cdot}w^{\prime}_{0}$, $z^{\prime}_{m^{\prime}}\mathrel{<\!\!\!\cdot}w^{\prime}_{0}$ and $w_{n}\mathrel{<\!\!\!\cdot}z_{1}$, we see that $z_{1}\neq z_{m}$ and $z^{\prime}_{1}\neq z^{\prime}_{m^{\prime}}$, i.e., $m$, $m^{\prime}\geq 2$, by the definition of covering relation.

Next we show that

[TABLE]

for any $x\in P^{+}$. Assume the contrary. Then by [Miy, Lemma 4.1], we see that $\dim_{k}[K_{R}(-a)/{\mathfrak{m}}K_{R}(-a)]_{0}\geq 2$. Therefore, we see by Corollary 2.5 that $\dim[K_{R}(-a)/{\mathfrak{m}}K_{R}(-a)]_{s-1}>0$. It follows that

[TABLE]

by Fact 2.13, contradicts to the fact that ${r_{\max}}=d-2$.

Since $\mathrm{rank}[x_{0},z_{m}]=u_{0}=n+m$, $\mathrm{rank}[z_{m},\infty]=n^{\prime}+1$, $\mathrm{rank}[x_{0},z^{\prime}_{1}]=n+1$ and $\mathrm{rank}[z^{\prime}_{1},\infty]=u_{1}=m^{\prime}+n^{\prime}$, we see that $n+m+n^{\prime}+1=n+1+n^{\prime}+m^{\prime}=r$. In particular, $m=m^{\prime}$. Further, we see that $d=u_{0}+u_{1}+1=n+2m+n^{\prime}+1$ and $s=d-r=m$. Moreover, since $n+m+n^{\prime}+1=r<{r_{\max}}=d-2=n+2m+n^{\prime}-1$, we see that $s=m\geq 3$.

Consider the possible covering relations in $P^{+}$ other than listed in (2). By the definition of covering relation, $z^{\prime}_{i}\not\mathrel{<\!\!\!\cdot}z_{j}$ if $(i,j)\neq(1,m)$. Therefore, we consider what kind of covering relations of the form $z_{i}\mathrel{<\!\!\!\cdot}z^{\prime}_{j}$ possible to exist.

If $i\geq j$ and $z_{i}\mathrel{<\!\!\!\cdot}z^{\prime}_{j}$, then

[TABLE]

contradicting to the fact that $r=\mathrm{rank}P^{+}$.

Next assume that there are $i$ and $j$ such that $j\geq i+2$, $(i,j)\neq(1,m)$ and $z_{i}\mathrel{<\!\!\!\cdot}z^{\prime}_{j}$. Then, by the definition of covering relation, we see that $z_{1}\not\mathrel{<\!\!\!\cdot}z^{\prime}_{m}$. Let $\nu$ be a minimal element of ${\mathcal{T}}(P)$ such that $\nu(x_{0})={r_{\max}}=d-2$. Take a sequence $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ of elements in $P$ which satisfies (1), (2) and (3) of Lemma 4.2. Then by the same argument as above, we see that $t=1$, $x_{1}\mathrel{<\!\!\!\cdot}y_{1}$ and $\mathrm{rank}[x_{0},y_{1}]+\mathrm{rank}[x_{1},\infty]=d-1$. Therefore, we see $y_{1}=z_{m}$ and $x_{1}=z^{\prime}_{1}$ since $z_{1}\not\mathrel{<\!\!\!\cdot}z^{\prime}_{m}$. Since $\nu(x_{0})-\nu(z_{m})=\nu(x_{0})-\nu(y_{1})=\mathrm{rank}[x_{0},y_{1}]=\mathrm{rank}[x_{0},z_{m}]$ and $\nu(z^{\prime}_{1})=\nu(x_{1})-\nu(\infty)=\mathrm{rank}[x_{1},\infty]=\mathrm{rank}[z^{\prime}_{1},\infty]$, we see that

[TABLE]

In particular, there is exactly one minimal element $\nu$ of ${\mathcal{T}}(P)$ with $\nu(x_{0})=d-2$. Therefore,

[TABLE]

On the other hand, if we define $\nu_{1}$, $\nu_{2}\colon P^{+}\to\mathbb{N}$ by

[TABLE]

then $\nu_{1}$ and $\nu_{2}$ are minimal elements of ${\mathcal{T}}(P)$ by [Miy, Lemma 3.2] and $\nu_{1}\neq\nu_{2}$ since $\nu_{1}(z_{m})\neq\nu_{2}(z_{m})$. Further $\nu_{1}(x_{0})=\nu_{2}(x_{0})=r+1$. Thus, we see that

[TABLE]

This contradicts to Corollary 2.5. Therefore, we see that there is no covering relation $z_{i}\mathrel{<\!\!\!\cdot}z^{\prime}_{j}$ with $j\geq i+2$ and $(i,j)\neq(1,m)$.

Summing up, possible covering relations of the form $z_{i}\mathrel{<\!\!\!\cdot}z^{\prime}_{j}$ are

• •

$z_{1}\mathrel{<\!\!\!\cdot}z^{\prime}_{m}$ and

• •

$z_{i}\mathrel{<\!\!\!\cdot}z^{\prime}_{i+1}$ with $1\leq i\leq m-1$.

If $z_{1}\mathrel{<\!\!\!\cdot}z^{\prime}_{m}$, then there is no other covering relations of the form $z_{i}\mathrel{<\!\!\!\cdot}z^{\prime}_{j}$ by the definition of covering relation. Therefore, we see (3) of Theorem 4.1 is satisfied by $z_{1}$, …, $z_{m}$, $z^{\prime}_{1}$, …, $z^{\prime}_{m}$, $w_{0}$, …, $w_{n}$, $w^{\prime}_{0}$, …, $w^{\prime}_{n^{\prime}}$. This proves the “only if” part of Theorem 4.1.

Next we prove the “if” part of Theorem 4.1. We first assume that there are $z_{1}$, …, $z_{m}$, $z^{\prime}_{1}$, …, $z^{\prime}_{m}$, $w_{0}$, …, $w_{n}$, $w^{\prime}_{0}$, …, $w^{\prime}_{n^{\prime}}\in P^{+}$ which satisfy (1), (2) and (3)(ii). First note that for any $\alpha_{1}$, $\alpha_{2}\in P^{+}$ with $\alpha_{1}\leq\alpha_{2}$,

[TABLE]

if and only if $(\alpha_{1},\alpha_{2})=(z^{\prime}_{1},z_{m})$. In particular, $\mathrm{rank}[x_{0},x]+\mathrm{rank}[x,\infty]=r$ for any $x\in P^{+}$

For $1\leq h\leq m-1$, set $\nu_{h}\colon P^{+}\to\mathbb{N}$ by

[TABLE]

Then by [Miy, Lemma 3.2], we see that $\nu_{h}$ is a minimal element of ${\mathcal{T}}(P)$, since $\nu_{h}(x_{0})-\nu_{h}(z_{m})=\mathrm{rank}[x_{0},z_{m}]$ and $\nu_{h}(z^{\prime}_{1})=\mathrm{rank}[z^{\prime}_{1},\infty]$. Further, $\nu_{h}(z^{\prime}_{1})-\nu_{h}(z_{m})=h$ for $1\leq h\leq m-1$.

Let $\nu$ be an arbitrary minimal element of ${\mathcal{T}}(P)$. Then by [Miy, Lemma 3.3], we see that there is a sequence $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ with condition N such that

[TABLE]

for $1\leq i\leq t+1$, where we set $y_{t+1}=\infty$. Take $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ as short as possible. If $t=0$, then $\nu(x_{0})=r$. Thus, $\nu(x)=\mathrm{rank}[x,\infty]$ for any $x\in P$, since $\mathrm{rank}[x_{0},x]+\mathrm{rank}[x,\infty]=r$. Therefore, $\nu=\nu_{m-1}$. Suppose $t>0$. Then $\nu(x_{0})-\nu(y_{2})>\mathrm{rank}[x_{0},y_{2}]$ since we took $y_{1}$, $x_{1}$, …, $y_{t}$, $x_{t}$ as short as possible. Thus,

[TABLE]

and therefore,

[TABLE]

Thus, $x_{1}=z^{\prime}_{1}$ and $y_{1}=z_{m}$. We see that $x_{t}=z^{\prime}_{1}$ and $y_{t}=z_{m}$ by the same way. Therefore $t=1$ since $y_{1}\not\geq x_{i}$ for any $i\geq 2$ by the definition of condition N. Further, we see that $\nu=\nu_{h}$, where $h=\nu(z^{\prime}_{1})-\nu(z_{m})$. Thus, we see that there are exactly $m-1$ minimal elements $\nu_{1}$, …, $\nu_{m-1}$ of ${\mathcal{T}}(P)$.

Let $\varphi\colon R\to K_{R}(-a)$ be the $R$-homomorphism such that $\varphi(1)=T^{\nu_{m-1}}$. Since

[TABLE]

as a $k$-vector space and

[TABLE]

we see that

[TABLE]

for any $i\in\mathbb{Z}$. Further, we see that $\mu({\mathrm{Cok}}\varphi)=m-2$, since $K_{R}$ is generated by $T^{\nu_{1}}$, …, $T^{\nu_{m-1}}$. Set

[TABLE]

Then $R^{\prime}$ is a $k$-subalgebra of $R$. Further, $R^{\prime}$ is isomorphic to a polynomial ring over $k$ with $d-1$ variables, since $R^{\prime}$ is isomorphic to the Hibi ring over $k$ on a chain of length $d-2$. Moreover, we see that $\bigoplus_{\nu\in{\mathcal{T}}(P)\atop\nu(z^{\prime}_{1})-\nu(z_{m})=h}kT^{\nu}$ is a rank 1 free module over $R^{\prime}$ generated by $T^{\nu_{h}}$ for $1\leq h\leq m-2$. Since multiplicity is defined by the Hilbert series, we see that $e({\mathrm{Cok}}\varphi)=m-2$. Therefore, we see that $R$ is almost Gorenstein.

Next, we assume that there are $z_{1}$, …, $z_{m}$, $z^{\prime}_{1}$, …, $z^{\prime}_{m}$, $w_{0}$, …, $w_{n}$, $w^{\prime}_{0}$, …, $w^{\prime}_{n^{\prime}}\in P^{+}$ which satisfy (1), (2) and (3)(i). For $1\leq h\leq m-1$, set $\nu_{h}\colon P^{+}\to\mathbb{N}$ by

[TABLE]

and for $1\leq h\leq m-2$, set $\nu^{\prime}_{h}\colon P^{+}\to\mathbb{N}$ by

[TABLE]

Then we see by the same argument as above, that there are exactly $2m-3$ minimal elements $\nu_{1}$, …, $\nu_{m-1}$, $\nu^{\prime}_{1}$, …, $\nu^{\prime}_{m-2}$ of ${\mathcal{T}}(P)$ and

[TABLE]

where $\varphi$ is the $R$-homomorphism $R\to K_{R}(-a)$ such that $\varphi(1)=T^{\nu_{m-1}}$.

Remark 4.4

$Q_{m}$ in [Hig, Example 5.7] is the poset which satisfies (1), (2) and (3)(i) with $n=n^{\prime}=0$ and $P_{m}$ in [Hig, Theorem 5.3] is the poset which satisfies (1), (2) and (3)(ii) with $n=n^{\prime}=p=0$.

5 Almost Gorenstein property of ladder determinantal rings defined by 2-minors

In this section, we state a criterion of a ladder determinantal ring defined by 2-minors to be a non-Gorenstein almost Gorenstein ring. Note that a (ladder) determinantal ring defined by 2-minors is a Hibi ring.

We use the notation of [Con, §1]. Let $m$ and $n$ be integers with $2\leq m\leq n$ and $X$ an $m\times n$ matrix of indeterminates. Let $P_{0}$ be a poset which is a disjoint union of two chains $\alpha_{1}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}\alpha_{m-1}$ and $\beta_{1}\mathrel{<\!\!\!\cdot}\cdots\mathrel{<\!\!\!\cdot}\beta_{n-1}$. Then by the correspondence

[TABLE]

we see the set of poset ideals (including the empty set) $J(P_{0})$ of $P_{0}$ ordered by inclusion is isomorphic to $X$ ordered by $\preceq$ of [Con, p. 121]. Let $P_{1}$ be a poset whose base set is $P_{0}$ and adding a covering relation $\alpha_{s}\mathrel{<\!\!\!\cdot}\beta_{t}$. Then a poset ideal of $P_{1}$ is a poset ideal of $P_{0}$, i.e., $J(P_{1})\subset J(P_{0})$. Further, it is easily verified that by the above correspondence, $J(P_{1})$ corresponds to

[TABLE]

i.e., $J(P_{1})$ corresponds to a ladder with inside upper corner $(m-s,t)$ in the terminology of [Con, p. 122].

By repeating this argument, one sees that any ladder satisfying (a) and (b) of [Con, p. 121] can be obtained by adding several covering relations of form $\alpha_{s}\mathrel{<\!\!\!\cdot}\beta_{t}$ or $\alpha_{s^{\prime}}\mathrel{\cdot\!\!\!>}\beta_{t^{\prime}}$ to $P_{0}$. Therefore, by the results of previous sections, we see the following fact.

Theorem 5.1

Let $m$, $n$ be integers with $2\leq m\leq n$, $X$ an $m\times n$ matrix of indeterminates, $Y$ a ladder of $X$ which satisfies (a) and (b) of [Con, p. 121]. Assume that all the indeterminates of $Y$ are involved in some 2-minors of $Y$. Then $R_{2}(Y)$ is non-Gorenstein and almost Gorenstein if and only if one of the following conditions is satisfied.

1. (1)

$m=2$, $n\geq 3$ and $Y=X$. 2. (2)

$m=n\geq 4$, inside lower corner of $Y$ is $(2,2)$ and inside upper corner is just $(m-1,m-1)$ or lie on the line of equation $x+y=m+1$. (We allow one sided ladder, i.e., there may be no inside upper corner.) 3. (3)

$m=n\geq 4$, inside upper corner of $Y$ is $(m-1,m-1)$ and inside lower corner is just $(2,2)$ or lie on the line of equation $x+y=m+1$. (We allow one sided ladder, i.e., there may be no inside lower corner.)

Remark 5.2

Suppose $t\geq 2$. Taniguchi [Tan] showed that $R_{t}(X)$ is non-Gorenstein and almost Gorenstein if and only if $m=2$, $n\geq 3$ and $t=2$.