arXiv:1610.00260v2 [] 12 Oct 2016

NON-KOSZUL QUADRATIC GORENSTEIN TORIC RINGS
KAZUNORI MATSUDA
Abstract. Koszulness of Gorenstein quadratic algebras of small socle degree is
studied. In this note, we construct non-Koszul Gorenstein quadratic toric ring
such that its socle degree is more than 3 by using stable set polytopes.

Introduction
Let K be a field and S = K[x1 , . . . , xn ] a polynomial ring over K. Let R = S/I be
a standard graded K-algebra with respect to the grading deg xi = 1 for all 1 ≤ i ≤ n,
where I is a homogeneous ideal of S. Let R+ denote the homogeneous maximal ideal
of R. For an R-module M, we denote βijR (M) by the (i, j)-th graded betti number
of M as an R-module.
The Koszul algebra was originally introduced by Priddy [29]. A standard graded
K-algebra R is said to be Koszul if the residue field K = R/R+ has a linear RR
free resolution as an R-module, that is, βijR (K) = 0 if i 6= j. Since β2j
(K) = 0
for all j > 2, hence Koszul algebras are quadratic, where R = S/I is said to be
quadratic if I is generated by homogeneous elements of degree 2. Every quadratic
complete intersection is Koszul by Tate’s theorem [35]. Moreover, R = S/I is
Koszul if I has a quadratic Gröbner bases by Fröberg’s theorem [10] and the fact that
′
βijR (K) ≤ βijR (K) for all i, j and for all monomial order < on S, where R′ = S/in< (I).
The notion of Koszul algebra has played an important role in the research on graded
K-algebras, and various Koszul-like algebras have been introduced, e.g., universally
Koszul [5], strongly Koszul [12], initially Koszul [2], sequentially Koszul [1], etc.
Koszulness of toric rings of integral convex polytopes is studied. Let P ⊂ Rn
be an integral convex polytope, i.e., a convex polytope each of whose vertices belongs to Zn , and let P ∩ Zn = {a1 , . . . , am }. Assume that Za1 + · · · + Zam = Zn .
−1
Let K[X ±1 , t] := K[x1 , x−1
1 , . . . , xn , xn , t] be the Laurent polynomial ring in n + 1
variables over K. Given an integer vector a = (a1 , . . . , an ) ∈ Zn , we put X a t =
xa11 · · · xann t ∈ K[X ±1 , t]. The toric ring of P, denoted by K[P], is the subalgebra of
K[X ±1 , t] generated by {X a1 t, . . . , X am t} over K. Note that K[P] can be regarded
as a standard graded K-algebra by setting deg X ai t = 1. The toric ideal IP is the
kernel of a surjective ring homomorphism π : K[Y ] = K[y1 , . . . , ym] → K[P] defined
2010 Mathematics Subject Classification. Primary 13P10, Secondary 52B20.
Key words and phrases. Koszul algebra, quadratic algebra, Gröbner bases, stable set polytope.
1

by π(yi ) = X ai t for 1 ≤ i ≤ m. Then K[P] ∼
= K[Y ]/IP . It is known that IP is
generated by homogeneous binomials.
Note that the following implications hold:

✎

quadratic
C. I.
⇓ [35]

Koszul
algebra

✍

quadratic
Gorenstein

⇒

✍

✎

✎

☞
✌

✍

⇐

✌

✍

⇑ [1]

⇑

☞

⇐
✌

✎

☞

✍

✌

quadratic
Cohen-Macaulay

✎

⇑ [33] and [21]

IP has a quadratic
squarefree initial ideal

✍

☞
✌

initially
Koszul [2]

⇐
✌

✍

☞

C.I. = Complete Intersection
✌

GB = Gröbner bases

✎

☞

✎

☞

✍

✌

✍

✌

strongly
Koszul [12]

✎

⇑ [7]
☞

⇒

✍

✌

IP has a
quadratic GB

[10]

sequentially
Koszul [1]

⇒

✎

☞

✎

☞

universally
Koszul [5]

In addition, it is known the following:

(1) Conca-De Negri-Rossi posed a conjecture that the defining ideal of a strongly
Koszul algebra has a quadratic Gröbner bases [6, Question 13 (1)]. This conjecture is true for the toric ring of edge polytope [16], order polytope [12],
stable set polytope [23] and cut polytope [31].
(2) A squarefree strongly Koszul toric ringpis compressed [24, Theorem 2.1],
where K[P] is said to be compressed if in< (IP ) = in< (IP ) for any reverse
lexicographic order < on K[Y ]. In particular, a squarefree strongly Koszul
toric ring is quadratic Cohen-Macaulay.
(3) Many of toric rings associated with integral convex polytopes whose toric
ideals has a quadratic Gröbner bases are constructed (e.g., [3], [13], [15],
[17], [18], [19]). In other words, many of Koszul toric rings associated with
2

integral convex polytopes are constructed.
(4) Quadratic algebra is not always Koszul (see [27, Example 2.1], [30, Example
3]). Note that both of these examples are Cohen-Macaulay but are not
Gorenstein.
On the other hand, Koszulness of Gorenstein quadratic algebras is studied.
For a standard graded K-algebra R = ⊕i≥0 Ri with dim R = d, we denote by
HR (t) =

X
i≥0

h0 + h1 t + · · · + hs ts
dimK Ri t =
(1 − t)d
i

the Hilbert series of R, where hs 6= 0, and we say that h(R) := (h0 , h1 , . . . , hs )
is the h-vector of R and the index s is the socle degree of R. It is known
that h0 = 1 and if R is Gorenstein then hi = hs−i for all 0 ≤ i ≤ ⌊s/2⌋ ([32,
Theorem 4.4]). Conca-Rossi-Valla proved that if R is a quadratic Gorenstein
with h(R) = (1, n, 1) (in this case n ≥ 2 since R is quadratic) then R is
Koszul [7, Proposition 2.12].
The case for s = 2 is also studied. Let R be a quadratic Gorenstein with
h(R) = (1, n, n, 1) (in this case n ≥ 3 since R is quadratic). If n = 3, then
R is quadratic complete intersection, hence R is Koszul. Conca-Rossi-Valla
proved that R is Koszul if n = 4 [7, Theorem 6.15] and Caviglia proved that
R is Koszul if n = 5 in his unpublished master thesis. The case for n ≥ 6 is
still open.
In this note, we focus on (4). In Section 1, we remark about known result of toric
rings and toric ideals of stable set polytopes, and construct non-Koszul quadratic
Gorenstein toric rings by using stable set polytopes. In Section 2, we present some
questions.
Remark 0.1. In this note, we use Macaulay2 [11] to check to be not Koszul. About
checking of non-Koszulness by using Macaulay2, see [34, p. 289].
1. Stable set polytope and
non-Koszul quadratic Gorenstein toric ring
The stable set polytope is an integral convex polytope associated with stable sets
of a simple graph.
Let G be a finite simple graph on the vertex set [n] = {1, 2, . . . , n} and let E(G)
denote the set of edges of G. Recall that a finite graph is simple if it possesses no
loops or multiple edges. We denote by G the complement graph of G.
P
Given a subset W ⊂ [n], we define the (0, 1)-vector ρ(W ) = i∈W ei ∈ Rn , where
ei is the i-th unit coordinate vector of Rn . In particular, ρ(∅) is the origin of Rn .
3

A subset W ⊂ [n] is said to be stable if {i, j} 6∈ E(G) for all i, j ∈ W with i 6= j.
Note that the empty set and each single-element subset of [n] are stable. Let S(G)
denote the set of all stable sets of G. The stable set polytope of a simple graph G,
denoted by QG , is the convex hull of {ρ(W ) | W ∈ S(G)}. By definition, QG is a
Q
(0, 1)-polytope and K[QG ] = K[t · i∈W xi | W ∈ S(G)] ⊂ K[x1 , . . . , xn , t]. Note
that dim K[QG ] = n + 1. Let K[Y ] = K[yW | W ∈ S(G)] be the polynomial ring
over K. Now we define a surjective ring homomorphism π : K[Y ] → K[QG ] by
Q
π(yW ) = t · i∈W xi and let IQG = ker π.
To state known results of the toric ring K[QG ] and the toric ideal IQG of the
stable set polytope QG of a simple graph G, we introduce some classes of graphs.
About terminologies for the graph theory, see [8].
A cycle graph with length n, denoted by Cn , is a connected graph which satisfies
E(Cn ) = {{1, 2}, {2, 3}, . . . , {n − 1, n}, {1, n}}. An odd cycle is a cycle such that its
length is odd.
A graph G is said to be perfect if the chromatic number of every induced subgraph
of G is equal to the size of the largest clique of that subgraph. A graph G is perfect
if and only if both G and G are (C2n+3 , n ≥ 1)-free [4].
The comparability graph G(P ) of a partially ordered set P = ([n], <P ) is the graph
such that V (G(P )) = [n] and {i, j} ∈ E(G(P )) if and only if i <P j or j <P i. A
graph G is said to be comparability if G is the comparability graph of some partially
ordered set. Forbidden induced subgraphs of comparability graphs are known (see
[22, p.13]).
A graph G is said to be bipartite if there exist V1 , V2 with V1 ∪ V2 = V (G) and
V1 ∩ V2 = ∅ such that if {i, j} ∈ E(G) then either i ∈ V1 and j ∈ V2 or i ∈ V2 and
j ∈ V1 . It is known that a graph G is bipartite if and only if G is (C2n+1 , n ≥ 1)-free.
A graph G is said to be almost bipartite (see [9, p.87]) if there exists a vertex v
such that the induced subgraph G[n]\v is bipartite.
Remark 1.1. It is known that
(1) Let G be a perfect graph. Then K[QG ] is Gorenstein if and only if all maximal cliques of G have the same cardinality [28, Theorem 2.1(b)].
(2) Let G(P ) be the comparability graph of a partially ordered set P . Then
K[QG(P ) ] is Koszul since QG(P ) is equal to the chain polytope of P and the
toric ideal of a chain polytope has a squarefree quadratic initial ideal (see
[14, Corollary 3.1]).
(3) If G is almost bipartite, then K[QG ] is Koszul since its toric ideal IQG has a
squarefree quadratic initial ideal (see [9, Theorem 8.1]).
4

(4) Let G be a graph such that G is bipartite. Then K[QG ] is quadratic if and
only of it is Koszul [25, Corollary 3.4].
Hence, if K[QG ] is quadratic but not Koszul, then G is neither comparability nor
almost bipartite, and G is not bipartite. From this fact and the classifications of
these graphs, we have
Proposition 1.2. Let G be a graph on [n]. If K[QG ] is non-Koszul quadratic
Gorenstein, then n ≥ 7.
Proof. First, we assume that n ≤ 5. Then G is a comparability graph if G is not
C5 . Since C5 is almost bipartite, we have that K[QG ] is Koszul if n ≤ 5.
Next, we assume that n = 6. If G is not connected, then G is a comparability
graph if G is not C5 ∪ K1 . Since C5 ∪ K1 is almost bipartite, we have that K[QG(P ) ]
is Koszul.
Assume that G is connected. From the classifications of comparability and almost
bipartite graphs, G is one of the following (see [23, p.10]):
•❏❏ t•✴
✎✎ t❏t❏t❏ ✴✴
✎✎t✎ttt ❏❏❏✴❏✴✴
•
•✴✎t✴
✴✴
✎✎
✴✴
✎✎✎
✎

•
t•✴
✎✎ ttt ✴✴✴
✴✴
✎✎t✎ttt
t
✴✎✴❏
•
•❏
✴✴❏❏❏
✎✎
✴✴ ❏❏❏ ✎✎✎
❏•✎
•

G1

G2

•

•

•
✎✎
✎✎ ✎
•✴✎✴
✴✴
✴✴
•

•✴✴

✴✴
✴✴
•
✎✎
✎✎✎
•✎

G3

•✴
t•✴
✎✎ ✴✴ ttt ✴✴✴
✴✴
✎✎t✎ttt✴✴
✴✴
•✴✎t✴
✴✴ ✎✎•
✴✴
✴✴ ✎✎✎
✴✴
✎
•

•

G4

•✴✴
•✴
✴✴
✎✎ ✴✴
✴✴
✎✎ ✎ ✴✴
✴✎✴❏ ✴✴
•
•❏
t
✴✴❏❏❏ ✴✴ttt✎✎✎
✴✴ ❏t❏t❏t✴✴ ✎✎
tt ❏✎
•

•

G5

Then we can see that
•
•
•
•
•

K[QG1 ] is not Gorenstein since h(K[QG1 ]) = (1, 7, 10, 3).
K[QG2 ] is Koszul since IQG2 has a quadratic Gröbner bases.
G3 is C6 , hence bipartite.
K[QG4 ] is not Gorenstein since h(K[QG4 ]) = (1, 6, 8, 2).
K[QG5 ] is Koszul since IQG5 = IQC5 and IQC5 has a quadratic Gröbner bases.

Therefore we have the desired conclusion.



For each integer k ≥ 3, the complement of a odd cycle C2k+1 , denoted by C2k+1 ,
is neither comparability nor almost bipartite. Note that C2k+1 is not perfect and
S(C2k+1 ) = {∅, {1}, {2}, . . . , {2k + 1}, {1, 2}, {2, 3}, . . . , {2k, 2k + 1}, {1, 2k + 1}}.
Let K[Y ] = K[y∅ , y{1} , y{2} , . . . , y{2k+1} , y{1,2} , y{2,3} , . . . , y{2k,2k+1}, y{1,2k+1} ]. Now
we study the toric ring
K[Y ]
K[QC2k+1 ] ∼
.
=
IQ C
2k+1

Proposition 1.3. We have the following:
5

(1)
(2)
(3)
(4)
Proof.

K[QC2k+1 ] is quadratic Cohen-Macaulay for all k ≥ 3.
K[QC2k+1 ] is not Gorenstein for all k ≥ 4.
K[QC7 ] is Gorenstein.
possesses no quadratic Gröbner bases for all k ≥ 3.
IQ C
2k+1

(1) First, by [25, Theorem 2.1], we have that K[QC2k+1 ] is normal. Hence
K[QC2k+1 ] is Cohen-Macaulay. Moreover, by [25, Theorem 3.2], we have that
is generated by the following 4k + 2 binomials:
the toric ideal IQC
2k+1

•
•
•
•

y{i} y{i+1} − y∅ y{i,i+1} (1 ≤ i ≤ 2k);
y{1} y{2k+1} − y∅ y{1,2k+1} ;
y{i} y{i+1,i+2} − y{i+2} y{i,i+1} (1 ≤ i ≤ 2k − 1);
y{2k} y{1,2k+1} − y{1} y{2k,2k+1} , y{2k+1} y{1,2} − y{2} y{1,2k+1} .

Hence K[QC2k+1 ] is quadratic. Therefore K[QC2k+1 ] is quadratic CohenMacaulay.
is Cohen-Macaulay with dim K[QC2k+1 ] =
(2) By (1), K[QC2k+1 ] ∼
= K[Y ]/IQC
2k+1
2k + 2. We note that y = y∅ , y{1} − y{2,3} , y{2} − y{3,4} , . . . , y{2k−1} − y{2k,2k+1} ,
. Then
y{2k} − y{1,2k+1} , y{2k+1} − y{1,2} is a regular sequence of K[Y ]/IQC
2k+1
we have that
K[y{1} , y{2} , . . . , y{2k+1} ]
K[Y ]
∼
=
+ (y)
IQ C
I2k+1
2k+1

is a artinian quadratic Cohen-Macaulay ring, where I2k+1 is generated by the
followings:
•
•
•
•

y{i} y{i+1} (1 ≤ i ≤ 2k);
y{1} y{2k+1} ;
2
y{i}
− y{i−1} y{i+2} (2 ≤ i ≤ 2k − 1);
2
2
2
y{1} − y{3} y{2k+1} , y{2k}
− y{1} y{2k−1} , y{2k+1}
− y{2} y{2k} .

Qk−1
2
Assume k ≥ 4. Then both y{2k+1}
· i=1
y{2i} and
 2k+1

3
 Y



y{3i}
(k ≡ 1 mod 3),



i=1


2k−1


3
Y
y{2k+1} ·
y{3i} (k ≡ 2 mod 3),



i=1


2k−3

3

Y


2

y{3i}
(k ≡ 0 mod 3),

 y{2k} ·
i=1

6

are socle elements of K[y{1} , y{2} , . . . , y{2k+1} ]/I2k+1 , hence it is not Gorenstein. Therefore K[QC2k+1 ] is not Gorenstein for all k ≥ 4.
(3) By the proof of (2), we have
K[Y ] ∼ K[y{1} , y{2} , . . . , y{7} ]
.
=
IQC + (y)
I7
7

Let <rev be the reverse lexicographic order on K[y{1} , y{2} , . . . , y{7} ] induced by the ordering y{1} < y{2} < · · · < y{7} . Then the initial ideal
in<rev (I7 ) is generated by the following monomials:
(y{1} y{2} , y{2} y{3} , y{3} y{4} , y{4} y{5} , y{5} y{6} , y{6} y{7} , y{1} y{7} ,
3
2
2
2
2
2
2
2
2
y{1}
, y{2}
, y{3}
, y{4}
, y{5}
, y{6}
, y{7}
, y{3} y{7} , y{1}
y{4} , y{1}
y{6} , y{2} y{5} y{7} ).

K[y{1} , y{2} , . . . , y{7} ]
in<rev (I7 )
2
3
4
is 1 + 7t + 14t + 7t + t . Hence h(K[QC7 ]) = (1, 7, 14, 7, 1), therefore it is
Gorenstein.
(4) Assume that there exists a monomial order < on K[Y ] such that the Gröbner
bases of IQC
with respect to < is quadratic.
2k+1
We may assume that y{1} y{2,3} < y{3} y{1,2} . Then y{3} y{4,5} < y{5} y{3,4}
and its initial monomial is
since y{5} y{1,2} y{3,4} − y{1} y{2,3} y{4,5} ∈ IQC
2k+1
and its initial
y{5} y{1,2} y{3,4} . Since y{7} y{3,4} y{5,6} − y{3} y{4,5} y{6,7} ∈ IQC
2k+1
monomial is y{7} y{3,4} y{5,6} , we have y{5} y{6,7} < y{7} y{5,6} . By repeating this
argument, we have
From this, we can compute that the Hilbert series of

y{1} y{2,3} < y{3} y{1,2} ,
y{3} y{4,5} < y{5} y{3,4} ,
..
.
y{2k−1} y{2k,2k+1}
y{2k+1} y{1,2}
y{2} y{3,4}
y{4} y{5,6}

< y{2k+1} y{2k−1,2k} ,
< y{2} y{1,2k+1} ,
< y{4} y{2,3} ,
< y{6} y{4,5} ,
..
.

y{2k−2} y{2k−1,2k} < y{2k} y{2k−2,2k−1} ,
y{2k} y{1,2k+1} < y{1} y{2k,2k+1}.
These inequalities induce a contradiction. Hence we have the desired conclusion.

R
(K) = 1 6= 0 by using
We can check that R = K[QC7 ] is not Koszul to check β34
Macaulay2. Hence we have

Corollary 1.4. The toric ring K[QC7 ] is non-Koszul quadratic Gorenstein.
7

We can construct an infinite family of non-Koszul quadratic Gorenstein toric rings
by using stable set polytopes.
Proposition 1.5. Let k ≥ 1 be an integer. Let G be a graph on [2k + 7] such that
G = C7 ∪ K2 ∪ · · · ∪ K2 and the labeling of vertices is as follows:
1

6

•❏❏
ttt ❏❏❏❏
t
t
❏❏
t
❏•✰
tt
7 ✓•✓t
✰✰ 2
✰✰
✓✓
✰✰
✓✻✓ ✓
•✻
✟•
✻✻
✟✟
✻✻
✟
✻✻
✟✟
✟✟

5

•

•

8

2k + 6

•

•

···
3

4

•

•

9

2k + 7

Then we have
(1) K[QG ] is quadraic Gorenstein such that
HK[QG] (t) = (1 + 7t + 14t2 + 7t3 + t4 )(1 + t)k /(1 − t)2k+8 .
(2) K[QG ] is not Koszul.
Proof.

(1) By [25, Theorem 3.2], we have that the toric ideal IQG is generated
by the following binomials:
• y{i} y{i+1} − y∅ y{i,i+1} (1 ≤ i ≤ 6);
• y{1} y{7} − y∅ y{1,7} ;
• y{i} y{i+1,i+2} − y{i+2} y{i,i+1} (1 ≤ i ≤ 5);
• y{6} y{1,7} − y{1} y{6,7} , y{7} y{1,2} − y{2} y{1,7} ;
• y{2i} y{2i+1} − y∅ y{2i,2i+1} (4 ≤ i ≤ k + 3).
Let K[Y ] = K[yW | W ∈ S(G)]. Then K[QG ] ∼
= K[Y ]/IQG . Note that
y = y∅ , y{1} − y{2,3} , y{2} − y{3,4} , . . . , y{5} − y{6,7} , y{6} − y{1,7} , y{7} − y{1,2} ,
y{8} − y{9} , . . . , y{2k+6} − y{2k+7} , y{8,9} , . . . , y{2k+6,2k+7} is a regular sequence
of K[Y ]/IQG . Hence we have
K[y{2i} | 4 ≤ i ≤ k + 3]
K[Y ] ∼ K[y{1} , y{2} , . . . , y{7} ]
⊗K
.
=
2
IQG + (y)
I7
(y{2i}
| 4 ≤ i ≤ k + 3)

Thus the Hilbert series of K[Y ]/IQG + (y) is (1 + 7t + 14t2 + 7t3 + t4 )(1 + t)k .
Therefore we have the desired conclusion.
(2) K[QC7 ] is a conbinatorial pure subring (see [26]) of K[QG ]. Since K[QC7 ] is
not Koszul, hence K[QG ] is not Koszul by [26, Proposition 1.3].

8

2. Questions
As the end of this note, we present some questions.
Recall that the h-vector of K[QC7 ] is (1, 7, 14, 7, 1). Hence the following question
is interesting.
Question 2.1. Does exist a non-Koszul quadratic Gorenstein algebra R such that
h(R) = (1, n1 , n2 , n1 , 1) and n1 ≤ 6 ?
Note that, in this case n1 ≥ 4 since R is quadratic. Since n1 = embdim R − dim R
and embdim K[QG ] = #S(G) = 1+n+#{W ∈ S(G) | #W ≥ 2} and dim K[QG ] =
n + 1, if embdim K[QG ] − dim K[QG ] ≤ 6, then #{W ∈ S(G) | #W ≥ 2} ≤ 6. In
particular, we have α(G) = 2, where α(G) := max{#W | W ∈ S(G)} is the stability
number of G. Since if G is perfect graph with α(G) = 2 then G is bipartite, In this
case G is not perfect.
Let G be a graph on [n] and with E(G) its edge set. The edge ring of G, denoted
by K[G], is defined by
K[G] := K[xi xj | {i, j} ∈ E(G)] ⊂ K[x1 , . . . , xn ].
The second question is
Question 2.2. Does exist a graph G such that the edge ring K[G] is non-Koszul
quadratic Gorenstein ?
In [27, Theorem 1.2], a criterion for the edge ring K[G] of G to be quadratic is
given. Moreover, in [20], a class of graphs with the property that the toric ideal IG
of the edge ring K[G] of G is quadratic but IG possesses no quadratic Gröbner bases
is studied. A graph G is said to be (∗)-minimal if G satisfies the above property and
every induced subgraph H ( G does not satisfy the property. By the computation
by using Macaulay2, we have that if G is (∗)-minimal and the edge ring K[G] is
non-Koszul quadratic Gorenstein, then n ≥ 9.
Acknowledgment. The author wish to thank Professor Takayuki Hibi for his
financial support.
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(Kazunori Matsuda) Department of Pure and Applied Mathematics, Graduate School
of Information Science and Technology, Osaka University, Suita, Osaka 565-0871,
Japan
E-mail address: kaz-matsuda@ist.osaka-u.ac.jp

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