arXiv:1107.4503v3 [] 16 Dec 2012

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS
DANG HOP NGUYEN
Abstract. Toric face rings is a generalization of the concepts of affine monoid
rings and Stanley-Reisner rings. We consider several properties which imply
Koszulness for toric face rings over a field k. Generalizing works of Laudal, Sletsjøe
and Herzog et al., graded Betti numbers of k over the toric face rings are computed,
and a characterization of Koszul toric face rings is provided. We investigate a conjecture suggested by Römer about the sufficient condition for the Koszul property.
The conjecture is inspired by Fröberg’s theorem on the Koszulness of quadratic
squarefree monomial ideals. Finally, it is proved that initially Koszul toric face
rings are affine monoid rings.

1. Introduction
Let k be a fixed field. Let Σ be a rational pointed fan in Rd (d ≥ 1), i.e. Σ is a
collection of rational pointed cones in Rd satisfying two conditions. Firstly, if C ∈ Σ
and D is a face of C, then D ∈ Σ. Secondly, if C, C ′ ∈ Σ then C ∩ C ′ is either empty
or a common face of C and C ′ .
A monoidal complex M supported on Σ is a collection of affine monoids MC indexed by elements C of Σ, such that MC generates C and the following compatibility
condition is fulfilled: if D ⊆ C ∈ Σ, then MD = MC ∩ D.
Given Σ and M as above, define the toric face ring k[M] of M over k as follow.
The free k-module k[M] has a basis {ta : a ∈ ∪C∈Σ MC }. The multiplication is
defined for basis elements by the rule
ta · tb = ta+b
if a, b are contained in MC for a cone C ∈ Σ. Otherwise, we let ta ·tb = 0. Expanding
the multiplication rule linearly, we get the multiplication of k[M].
Toric face rings include affine monoid rings and Stanley-Reisner rings as special
cases. Indeed, if the fan Σ is the face poset of one cone, then k[M] is an affine
monoid ring. On the other hand, if all the cones of Σ are simplicial, and MC = C ∩Zd
is generated by exactly dim C elements for every C ∈ Σ, then k[M] is a StanleyReisner ring. Starting with the work of Stanley [24], several authors have considered
toric face rings [5], [8], [18], [22]. For an algebraic treatment of affine monoid rings
and Stanley-Reisner rings, see Bruns- Herzog [7] or Bruns-Gubeladze [5]; for a more
combinatorial treatment of Stanley-Reisner rings see Stanley [25].
The author has been supported by the graduate school “Combinatorial Structures in Algebra
and Topology” at the University of Osnabrück, Germany and (partly) by the Vigoni project for a
stay in Genoa in March 2011.

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DANG HOP NGUYEN

Let R be a homogeneous affine k-algebra. We say that R is a Koszul algebra over
k if k has a R-linear resolution. This is equivalent to the condition that the Betti
R
numbers βi,j
(k) = 0 for all i 6= j. Assume that R is a Koszul k-algebra. Then R is
defined by quadratic relations over some polynomial ring. Fröberg [12] proves that
if R is a Stanley-Reisner ring defined by quadratic monomial relations, then R is
Koszul.
Closely related to the Koszul property is the G-quadratic property. Let R = S/I
be a presentation of R, where S is a standard graded polynomial ring over k, and
I is a homogeneous ideal of S. Then R is said to be G-quadratic if its defining
ideal I has a quadratic Gröbner basis with respect to some term order of S. It is
well-known that G-quadratic algebras over k are Koszul.
Herzog, Hibi and Restuccia [16] defined strongly Koszul algebras. A homogeneous
k-algebra is strongly Koszul if its irrelevant ideal admits a system of generators of
degree 1, namely a1 , . . . , an , such that for all increasing sequence 1 ≤ i1 < · · · < ij ≤
n, the ideal (ai1 , . . . , aij−1 ) : aij is generated by a subset of a1 , . . . , an . Motivated by
the notion of strongly Koszul algebras in [16], Koszul filtrations were introduced in
[10] as an useful tool to deduce Koszulness.
Definition 1.1 (Conca, Trung, Valla). A family F of ideals of R is said to be a
Koszul filtration of R if:
(i) every ideal of F is generated by linear forms;
(ii) the ideal 0 and the graded maximal ideal belong to F ;
(iii) for every I ∈ F different from 0, there exists J ∈ F strictly contained in I
and a linear form x ∈ I such that J + (x) = I and J : I ∈ F .
Note that a Koszul filtration, if it exists, does not contain the unit ideal. It is
known that a ring which has a Koszul filtration must be Koszul. This gives another
proof to the above result of Fröberg, because in this case, the family of those ideals,
each of which is generated by some variables, form a Koszul filtration. Moreover,
a general set of at most 2n points in general linear position has a Koszul filtration
[10, Thm. 2.1].
R is initially Koszul (abbreviated i-Koszul) with respect to a sequence a1 , . . . , an ∈
R1 , if the family of ideals F = {(a1 , . . . , ai ) : i = 0, . . . , n} is a Koszul filtration of
R. Algebras which are i-Koszul must also be Koszul, and in fact they even have the
stronger property of being G-quadratic; see [2], [9] for details. See also the survey
article [13].
Our main concern in this paper is to find simple criteria for the various Koszul-like
properties of the toric face ring k[M]. To illustrate, let us look at the special cases of
Stanley-Reisner rings and affine monoid rings. In the case of Stanley-Reisner rings,
the answers are simple. A Stanley-Reisner ring is Koszul if and only if its defining
ideal is quadratic (see [12]). Moreover, in this case the ring is strongly Koszul [16,
Cor. 2.2]. A Stanley-Reisner ring is i-Koszul only if the simplicial complex is a full
simplex, see [2, Prop. 2.3].
In the case of affine monoid rings R, we have formulae for the Betti numbers of k
over R by the bar resolution [17]. For results and problems about the Koszul property of polytopal algebras, see [5, Chap. 7] and [6]. One can see that the formulae of

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS

3

Betti numbers of k involve infinitely many complicated simplicial homology groups.
Hence we find sufficient conditions for the Koszul property of k[M] by looking for
situations where the defining ideals of the toric face rings are closed to being quadratic monomial ideals. A reasonable condition is the quadratic condition, which
requires that for some system of generators, the monomial part of the defining ideal
of k[M] is quadratic. We will investigate a question of Römer, asking if the quadratic condition is satisfied and all the k[MC ] is Koszul for all C ∈ Σ then k[M] is
Koszul, see Conjecture 5.6. We are able to prove that the quadratic condition is stable under various natural operations on toric face rings, e.g. taking tensor products,
fiber products, Segre products, Veronese subrings or multigraded algebra retracts.
Furthermore, given the quadratic condition, k[M] is strongly Koszul or G-quadratic
if and only if the same thing is true for the subrings k[MC ] for all C, see Proposition
5.10 and Remark 6.4.
The paper is organized as follows. In Section 2, we recall the basic theory of
toric face rings and introduce homogeneous toric face rings. In the next section,
we prove that the class of homogeneous toric face rings is very natural to consider
ring-theoretically, in the sense that it is closed under basic operations of algebras
over k. In Section 4, we compute certain graded Betti numbers of k over k[M].
From this computation, we get a characterization of the Koszul property of k[M].
In Section 5, we prove that if k[M] is Koszul, then all k[MC ] are Koszul where
C ∈ Σ. We introduce the quadratic condition and prove its stability under many
different constructions involving homogeneous toric face rings. It is expected that
k[M] is Koszul if the quadratic condition is satisfied for some system of generators
and the monoid ring k[MC ] is Koszul for all facet C of Σ, see Conjecture 5.6 of
Römer. Section 6 is devoted to the characterization of strong Koszulness of toric
face rings. Finally, in Section 7, we prove that homogeneous i-Koszul toric face rings
are simply affine monoid rings.
The content of this paper is partially included in the author’s dissertation [21].
2. Notations and background
Let k be a field, d ≥ 1 a natural number. Denote by R+ the non-negative real
numbers.
2.1. Embedded toric face rings. Throughout the paper, we adopt the following
notations, which we explain below: Σ is a rational pointed fan in Rd , M is a monoidal
complex supported on Σ. The toric face ring of M over k is R = k[M]. The set
{a1 , . . . , an } is a homogeneous system of generators of M. The ideal I = IM defines
k[M] as a quotient ring of k[X1 , . . . , Xn ].
A rational pointed fan Σ is a collection of rational pointed cones such that:
(i) if C ∈ Σ and D is a face of C then D ∈ Σ;
(ii) for every C, C ′ ∈ Σ, C ∩ C ′ is either a face of both C and C ′ or empty.
Σ is called simplicial if each of its cones C is generated by linearly independent
vectors in Rd . A maximal element of Σ with respect to (w.r.t.) inclusion is called a
facet of Σ. A one dimensional face of a cone of Σ is called an extremal ray.

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DANG HOP NGUYEN

A monoidal complex M supported on Σ is a collection of affine monoids MC ,
where C varies in Σ such that:
(i) MC ⊆ C ∩ Zd and R+ MC = C;
(ii) for every C, D ∈ Σ with D ⊆ C, MD = MC ∩ D.
For instance, after taking MC = C ∩ Zd for each C we get a monoidal complex
supported on the fan Σ.
The toric face ring of M over k, denoted by k[M] is defined as follows. As a
k-vector space we set
M
k[M] =
kta .
a∈∪C∈Σ MC

The product on basis elements is given by
(
ta+b if for some C ∈ Σ both a and b belongs to MC ;
a
b
t ·t =
0
otherwise.
Sometimes we write a instead of the basis element ta of k[M]. In that case, instead
of ta · tb , we write a · b, hence in k[M]:
(
a + b if for some C ∈ Σ both a and b belongs to MC ;
a·b=
0
otherwise.
It is known that k[M] is a reduced k-algebra with unit 1 = t0 . Moreover, from the
primary decomposition of k[M], we have
dim k[M] = max{dim C : C ∈ Σ},
see [18, Lem. 2.1]. An important aspect is that k[M] inherits the Zd -grading from
the embedding of the monoidal complex. Every Zd -graded component of k[M] has
k-dimension less than or equal to 1.
The two basic examples of toric face rings are Stanley-Reisner rings and affine
monoid rings. We recall these examples for later usage.
Example 2.1. (Stanley-Reisner rings) Let ∆ be a simplicial complex on the vertex
set [n] = {1, . . . , n}. Let e1 , . . . , en be the standard basis vectors of Rn . For each
face F of ∆, consider the cone CF generated by the vectors ei , i ∈ F . It is clear that
the collection Σ = {CF , F ∈ ∆} is a rational pointed fan in Rn . For each F ∈ ∆,
choose MCF = CF ∩ Zn . In this manner, we get a monoidal complex M supported
on Σ.
The ring k[M] is the Stanley-Reisner ring k[∆].
Q By definition, this is k[X1 , . . . , Xn ]
modulo the square-free monomial ideal I∆ = ( j∈G Xj : G ⊆ [n], G ∈
/ ∆).
Example 2.2. (Affine monoid rings) Let M ⊆ Nd be a finitely generated monoid
(d ≥ 1). Choosing Σ to be the face poset of C = R+ M. For each face F of C, let
MF = M ∩ F . The resulting toric face ring is the affine monoid ring k[M].

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS

5

For each cone C ∈ Σ, the affine monoid ring k[MC ] is naturally a subring of R.
We have natural surjections R → k[MC ] defined by:
(
ta if a belongs to MC ;
ta 7−→
0 otherwise.
The composition of the natural inclusion morphism k[MC ] → R with the projection
R → k[MC ] is the identity on k[MC ], in other words k[MC ] is an algebra retract of
R for every C ∈ Σ.
Definition 2.3. The finite set {a1 , . . . , an } is a system of generators of M if ai ∈
∪C∈Σ MC for every i ∈ [n], and the subset {a1 , . . . , an } ∩ MC is a system of generator
of MC for every C ∈ Σ.
This system of generators gives a surjection ϕ : S = k[X1 , . . . , Xn ] → R. Let
I = Ker ϕ. For each cone C of Σ, let SC = k[Xi : ai ∈ MC ], we have a map
ϕC : SC → k[MC ] with kernel IC = Ker ϕC .
Denote by ∆M the following simplicial complex on the set [n]: a subset F ⊆ [n] is a
face of ∆M if and only if there exists some cone C ∈ Σ such that {aj | j ∈ F } ⊆ MC .
Proposition 2.4 ([8], Prop. 2.3). Assume that C1 , . . . , Cr are the facets of Σ. Then
I = AM +

r
X

S · ICi ,

i=1

where AM is generated by square-free monomials

Q

j∈G

Xj , for which G ∈
/ ∆M .

We call a binomial in ICi , i = 1, . . . , r a pure binomial. A binomial X a − X b
belongs to I if and only if either X a , X b ∈ AM or X a − X b is a pure binomial.
In the following, we call the ideal AM the monomial part of I. Denote by BM the
ideal of S generated by the pure binomials in I.
It is the appropriate place to give a simple example of a genuine toric face ring.
Example 2.5. Consider the points in R3 with the following coordinates
O = (0, 0, 0), A1 = (2, 0, 0), A2 = (0, 2, 0), A3 = (0, 0, 2), A4 = (1, 1, 0).
Consider the fan Σ in R3 with the maximal cones OA1 A2 , OA1 A3 , and OA2 A3 .
Let M be the monoidal complex supported on Σ with the three maximal monoids
generated by {A1 , A2 , A4 }, {A1 , A3 } and {A2 , A3 }.
The defining ideal of the toric face ring k[M] is
IM = (X3 X4 , X1 X2 X3 ) + (X1 X2 − X42 ) = (X1 X2 − X42 , X3 X4 ).
Thus the toric face ring of M is
k[M] = k[X1 , X2 , X3 , X4 ]/(X1 X2 − X42 , X3 X4 ).

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DANG HOP NGUYEN

A

tP
♣♣ 1
A
♣P
♣ ♣ PP
t 4
PP
♣♣
Pt A2
♣♣
♣♣
❇
♣♣
❇
♣♣
❇
♣♣
♣♣
♣♣ ❇
♣♣ ❇
♣ ♣❇
♣ ♣ A3
❇t

O

Clearly k[M] is not a domain, so it is not isomorphic to any monoid ring.
Now assume that k[M] is isomorphic to a Stanley-Reisner ring. Observe that
(X3 ) is a minimal prime ideal of k[M], see [18, Lem. 2.1]. Thus we have k[M]/(X3 )
is isomorphic to a polynomial ring.
Let M be the monoid generated by {A1 , A2 , A4 }. We will show that
k[M]/(X3 ) ∼
= k[M]
is not a polynomial ring. In fact, k[M] is a regular ring only if M ∼
= N2 (see,
e.g., [7, Exercise 6.1.11]). The last isomorphism is not possible. Thus k[M] is not
isomorphic to any Stanley-Reisner ring.
Definition 2.6 (Homogeneous system of generators for M). The set {a1 , . . . , an }
is said to be a homogeneous system of generators of M if for every C ∈ Σ, the
k-algebra k[MC ] is minimally generated in degree 1 by {a1 , . . . , an } ∩ MC . We call
k[M] a homogeneous toric face ring if M has a homogeneous system of generators.
Given a homogeneous system of generators of M, the Z-gradings on the rings
k[MC ] where C ∈ Σ induce a Z-grading on k[M]. We do not require the Z-grading
to be compatible with the existing Zd -grading, as we can use the two gradings
separately.
2.2. Koszul algebras. A homogeneous k-algebra R is a Koszul algebra if and only
R
R
if βi,j
(k) = 0 for all i 6= j, where βi,j
(k) = dimk TorR
i (k, k)j are the Betti numbers
of k as a graded R-module. In other words, R is Koszul if the module k = R/R+
has a linear resolution as an R-module.
Let R = S/I be a presentation of R, where S = k[x1 , . . . , xn ] be a standard graded
polynomial ring and I a homogeneous ideal of S. The following result is well-known;
see, e.g., [3, Prop. 3.13].
Theorem 2.7. If I has a quadratic Gröbner basis with respect to some term order
on S then R = S/I is a Koszul algebra.
When I has a quadratic Gröbner basis with respect to some term order of S, R is
said to be G-quadratic (w.r.t. that term order), where G stands for Gröbner. Hence
G-quadraticity implies Koszulness.
Recall that given an inclusion of rings R ֒→ S, R is called an algebra retract of
S if there is a ring morphism ϕ : S → R (the retraction map) such that ϕ restricts
to the identity on R. If the rings are Zd -graded then we also require the ring

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS

7

homomorphisms to preserve the gradings. Koszul algebras are very well-behaved
with respect to graded algebra retracts.
Proposition 2.8 ([15, Prop. 1.4]). If R ֒→ S is an algebra retract of homogeneous
k-algebras, then S is Koszul if and only if R is Koszul and considered as an S-module
via the retraction map, R has a linear free resolution.
For further information and results on Koszul algebras, the reader may consult
[1], [13].
2.3. Multigraded algebra retracts.
Definition 2.9. Let Γ be a subfan of Σ and MΓ the induced monoidal subcomplex
supported on Γ. We say that MΓ is a restricted subcomplex of M if for every finite
set of elements z1 , . . . , zn in |MΓ | such that there is a cone C ∈ Σ with the property
z1 , . . . , zn ∈ MC , we can also find a cone D ∈ Γ such that z1 , . . . , zn ∈ MD .
For example, given any C ∈ Σ, the monoid MC is a restricted subcomplex of M.
We can classify all Zd -graded algebra retracts of k[M] as follow; for completeness,
we give a full proof below.
Proposition 2.10 ([11, Prop. 4.4]). The ring A is a Zd -graded algebra retract of
k[M] if and only if there is a restricted subcomplex MΓ of M and a Zd -graded
isomorphism A ∼
= k[MΓ ].
Proof. The “if” direction: we have to show that k[MΓ ] is a Zd -graded algebra retract
of k[M] for each restricted subcomplex MΓ . For this, we observe that there is a
natural inclusion k[MΓ ] ֒→ k[M] mapping ta to itself for each a ∈ |MΓ |. To check
that this is indeed an inclusion, we use Proposition 2.4. The binomial relations of
k[MΓ ] clearly map to zero. The monomial relations also map to zero, since any
sequence of elements of MΓ which is not contained in any face of Σ also is not
contained in any face of Γ. The composition of k[MΓ ] ֒→ k[M] with the natural
projection k[M] → k[MΣ ] is the identity of k[MΓ ].
The “only if” direction: if there is an Zd -graded algebra retract A → k[M] with
multigraded inverse ϕ : k[M] → A, then A ∼
= k[M]/L where L = ker ϕ is a Zd graded ideal of k[M]. Now A is reduced since k[M] is so, therefore L is a radical
ideal. From [18, Lem. 2.1], L is generated by elements of |M| which are not in a
subfan Γ, and this implies A ∼
= k[MΓ ]. Similar argument as above shows that MΓ
is a restricted subcomplex of M. This is our desired claim.

3. Basic operations on toric face rings
In this section, we show that the class of homogeneous toric face rings over k is
closed under taking Veronese subrings, tensor products, fiber products and Segre
products over k. These results show that toric face rings are natural objects to
consider, even if we are mainly interested in Stanley-Reisner rings. For example, if
we take Veronese subrings of a Stanley-Reisner ring, then we do not get StanleyReisner rings but toric face rings.
The following result is implicit in [4]; we give the details for completeness.

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DANG HOP NGUYEN

Proposition 3.1. For each m ≥ 1, the Veronese subring k[M](m) of a homogeneous
toric face ring k[M] is also a toric face ring.
Proof. Consider the Veronese M(m) of the monoidal complex M defined as follows.
(m)
For each C ∈ Σ, we let MC = mMC := {ma : a ∈ MC }. It is easy to check that
(m)
the affine monoids MC , C ∈ Σ form a monoidal complex M(m) supported on Σ.

Moreover k[M(m) ] ∼
= k[M](m) . Hence k[M](m) is also a toric face ring.
Proposition 3.2. For two monoidal complexes M, N supported on the fans Σ ⊆ Rd ,
Γ ⊆ Re , the tensor product k[M] ⊗k k[N ] is also a toric face ring.
Proof. Assume that the affine monoids of M are MC where C ∈ Σ, and the monoids
of N are ND where D ∈ Γ.
We will construct a join monoidal complex J(M, N ) of M and N . This construction is the same as building a pyramid after given a base polytope and an apex
when N = N ⊆ R. Embed Rd and Re into Rd+e as vector subspaces such that they
intersect only at the origin.
For each C ∈ Σ, D ∈ Γ, let J(MC , ND ) be the submonoid of Zd+e generated by
MC and ND , considered as subsets of Zd+e . Moreover, let J(C, D) be the convex
hull of C and D. Then J(C, D) is a cone in Rd+e .
We observe that the cones J(C, D) form a fan J(Σ, Γ) in Rd+e . To see this, choose
arbitrary C, C ′ ∈ Σ, D, D ′ ∈ Γ. Since Rd ∩ Re = {0}, we have Rd ∩ J(C, D) = C and
Re ∩ J(C, D) = D. So J(C, D) is a face of the cone J(C ′ , D ′ ) if and only if C ⊆ C ′
and D ⊆ D ′ .
Moreover, the monoids J(MC , ND ) (where C ∈ Σ, D ∈ Γ) form a monoidal complex J(M, N ) supported on J(Σ, Γ). In fact, for each C, C ′ ∈ Σ, D, D ′ ∈ Γ with
C ⊆ C ′ , D ⊆ D ′ , we have
J(MC ′ , ND′ ) ∩ J(C, D) = J(MC , ND ).
Finally, we will show that k[M] ⊗k k[N ] ∼
= k[J(M, N )]. Indeed, the isomorphism
is given by
tm ⊗ tn 7→ t(m,n) , m ∈ |M|, n ∈ |N |.
Hence k[M] ⊗k k[N ] is a toric face ring.

Corollary 3.3. Polynomial extensions of a toric face ring are also toric face rings.
Proof. Apply Proposition 3.2 for M and N = N ⊆ R.



Proposition 3.4. The fiber product over k of two toric face rings is a toric face
ring.
Proof. Recall that if A = k[x1 , . . . , xn ]/I and B = k[y1 , . . . , ym ] are k-algebras where
I ⊆ k[x1 , . . . , xn ], J ⊆ k[y1 , . . . , ym ] are ideals of polynomial rings, then the fiber
product of A and B is the pull-back of the diagram A → k ← B. Concretely, we
have
A ×k B ∼
= k[x1 , . . . , xn , y1 , . . . , ym]/(I + J + (xi yj : 1 ≤ i ≤ n, 1 ≤ j ≤ m)).
Let M, N be monoidal complexes supported on the fans Σ ⊆ Rd , Γ ⊆ Re respectively. Embed Rd and Re as vector subspaces into Rd+e such that Rd ∩ Re = {0}.

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS

9

We consider the union Σ ∪ Γ as a fan in Rd+e . Consider the union M ∪ N as a
monoidal complex supported on Σ ∪ Γ. Then we have
k[M ∪ N ] ∼
= k[M] ×k k[N ].
Hence k[M] ×k k[N ] is a toric face ring.



Proposition 3.5. The Segre product of two homogeneous toric face rings over k is
again a toric face ring.
Proof. Firstly, we note that the Segre product of two homogeneous affine monoid
rings over k is again an affine monoid ring over k. Concretely, let M and N are two
homogeneous affine monoids, then the Segre product k[M] ∗ k[N] = k[M ∗ N], where
M ∗ N is the submonoid of the direct sum M × N consisting of elements (u, v) with
deg(u) = deg(v).
We can generalize this construction for any two homogeneous toric face rings
k[M], k[N ]. For any two cones C ∈ Σ, D ∈ Γ, let C ∗ D be the cone generated by
MC ∗ ND . Let Σ ∗ Γ be the collection of cones C ∗ D and M ∗ N be the collection
of affine monoids MC ∗ ND for such cones C, D. Then M ∗ N is a homogeneous

monoidal complex supported on the fan Σ ∗ Γ, and k[M ∗ N ] ∼
= k[M] ∗ k[N ].
4. Betti numbers of the residue field
Laudal and Sletsjøe [19, Prop. 1.3] computed Betti numbers of affine monoid rings.
The result was later reproved and extended in [23, (1.2)] and [17, Thm. 2.1]. In this
section, using the method employed in [23] and [17], we find the Betti numbers of k
as a k[M]-module in the natural monoid grading (to be defined below).
Recall that I is the defining ideal of k[M] and BM is the ideal generated by pure
binomials in I.
Definition 4.1 (The associated monoid). Define a relation ∼ in Nn : a ∼ b if and
only if X a − X b ∈ BM . This relation is compatible with vector sum: a ∼ b implies
a + c ∼ b + c for a, b, c ∈ Nn . Let H be the monoid whose elements are equivalent
classes of Nn / ∼ and the addition inherited from that of Nn . Then we call H the
monoid associated with M and the system of generators a1 , . . . , an .
In general, H depends both on M and on the choice of the system of generators.
Remark 4.2. In Example 2.2, the monoidal complex M is a positive affine monoid
M, and we can choose {a1 , . . . , an } to be the minimal set of generators of M. Here
H = M and BM is the toric ideal defining k[M]. In other words, the H-grading is
the monoid grading induced by M. In Example 2.1, we have BM = 0 and H = Nn ;
the H-grading is simply the Zn -grading.
We say that a monoid is positive if for elements λ, µ of this monoid with λ+µ = 0,
we must have λ = µ = 0. We say that a monoid is cancellative with respect to 0 if
an equation λ + µ = λ in the monoid implies that µ = 0. It is not hard to see that
H is a commutative positive monoid.

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DANG HOP NGUYEN

Lemma 4.3 ([8], Lem. 4.4). Denote the class of a ∈ Nn in H by a. We have:
(i) If a + c = b + c for a, b, c ∈ Nn then X a − X b ∈ I.
(ii) H is cancellative w.r.t. 0.
(iii) If X a − X b ∈ I and X a , X b ∈
/ I then a = b in H.
It is easy to see that S = k[X1 , . . . , Xn ] and R = k[M] are H-graded. Note that
S/BM is exactly the monoid algebra k[H] of H.
Denote by J the ideal I/BM of k[H]. Let eg , g = 1, . . . , n be the standard basis
vectors in Rn . Then J is a monomial ideal of k[H] in the sense that it is generated
P
Q
by elements g∈G eg in H, where G ⊆ [n] such that g∈G ag = 0. From Proposition
2.4, we get k[M] = k[H]/J.
For elements λ, µ ∈ H, we say that λ < µ if λ 6= µ and µ − λ ∈ H. Then (H, <)
is a partially ordered set. It follows from 4.3(ii) that < is a strict order.
Definition 4.4 (The order complexes of divisor posets). For each λ ∈ H, denote by
∆λ the set of chains α1 < · · · < αi ∈ H such that 0 = α0 < α1 and αi < λ = αi+1 .
Let ∆λ,J be the subset of ∆λ consisting of chains 0 = α0 < α1 < · · · < αi < λ =
αi+1 in ∆λ such that for some 0 ≤ j ≤ i, the element X αj+1 −αj , as an element of the
group ring k[H], belongs to J.
e ℓ (∆λ , ∆λ,J ; k)
For each λ ∈ H, the sets ∆λ , ∆λ,J are simplicial complexes. Let H
be the ℓ-th reduced simplicial homology with coefficients in k of the pair (∆λ , ∆λ,J ).
Since {a1 , . . . , an } is a homogeneous system of generators, there is a function
| · | : H → Z mapping λ = a ∈ H to |λ|, which is the sum of coordinates of a. This
follows from the fact that k[MC ] is standard graded for every C ∈ Σ.
Finally, we get the formula for graded Betti numbers of toric face rings, and a
characterization of the Koszul property.
R
Theorem 4.5. With the above notations, the bi-graded Betti number βi,λ
(k) =
R
dimk Tori (k, k)λ of k as an H-graded module is given by
R
e i−2 (∆λ , ∆λ,J ; k),
βi,λ
(k) = dimk H

for every λ ∈ H and i > 0.
In particular, the following statements are equivalent:

(i) k[M] is a Koszul algebra;
e i−2 (∆λ , ∆λ,J ; k) = 0 for every i > 0 and λ ∈ H such that |λ| > i.
(ii) H

Proof. Similar to the proof of [17, Thm. 2.1], we will exploit the bar resolution.
Consider the following free resolution of k over R = k[H]/J, the so-called bar
resolution (see Maclane [20, Chap. 10, §2]):
B : · · · → Bi → Bi−1 → · · · → B0 = k.
For each i, the module Bi is the free R-module with basis elements [λ1 | · · · |λi ], where
λj ∈ (H \ J) − {0} for all j. The differential map di : Bi → Bi−1 is defined on basis

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS

11

elements by the formula:
di [λ1 | · · · |λi ] = X λ1 [λ2 | · · · |λi ] +

i−1
X

(−1)j [λ1 | · · · |λj + λj+1 | · · · |λi ].

j=1

Here we will understand the symbol [λ1 | · · · |λi ] to be 0 if one of the λj is in J.
By definition, the Betti numbers TorR
i (k, k) are computed by the homology of
B ⊗ k. We can write the latter complex as follows:
B ⊗ k : · · · → Bi ⊗ k → Bi−1 ⊗ k → · · · → B0 ⊗ k = k.
The differential of this complex is given by:
X
(di ⊗ k)[λ1 | · · · |λi ] =
(−1)j [λ1 | · · · |λj + λj+1 | · · · |λi ].
1≤j≤i−1

The important observation is that the differential map di ⊗ k preserves the sum
λ = λ1 + · · · + λi . Hence B ⊗ k is H-graded and the homology modules of (B ⊗ k)λ
compute TorR
i (k, k)λ .
Since < is a strict order on H, we can identify [λ1 | · · · |λi ] with the chain λ1 <
λ1 + λ2 < · · · < λ1 + λ2 + · · · + λi−1 in the open interval (0, λ). Then the differential
of the complex (B ⊗ k)λ is nothing but the boundary map of the reduced chain
e. −2 (∆λ , ∆λ,J ; k). Thus
complex of pair C
e i−2 (∆λ , ∆λ,J ; k),
β R (k) = dimk H
i,λ

for all i > 0 and λ ∈ H, as claimed.
The remaining characterization is obvious.



5. The quadratic condition
We want to obtain necessary and sufficient conditions on M so that R = k[M]
a Koszul algebra. Theorem 4.5 is not easy to use, even for Stanley-Reisner ring,
see [17]. Instead of looking directly at relative homology of simplicial complexes, we
turn our attention to the defining ideal of k[M] and the monoid rings k[MC ], C ∈ Σ.
Firstly, we will relate the Koszul property of R with the Koszul property of the
involved affine monoid rings k[MC ] where C ∈ Σ. As a corollary to Proposition 2.8
we get:
Proposition 5.1. If k[M] is Koszul then for any C ∈ Σ, the monoid ring k[MC ] is
Koszul.
Now we come to an useful condition for the Koszul property of toric face rings.
Definition 5.2. (Quadratic condition) Given a homogeneous system of generators
a1 , . . . , an , M is said to satisfies the quadratic condition if the monomial ideal AM
is generated in degree 2. In that case, we also say that k[M] satisfies the quadratic
condition.
We immediately have the following interpretation of the quadratic condition.

12

DANG HOP NGUYEN

Proposition 5.3. The following statements are equivalent:
(i) The monoidal complex M satisfies the quadratic condition w.r.t. the homogeneous system of generators a1 , . . . , an .
(ii) For any set I ⊆ {1, 2, . . . , n} such that the elements {ai : i ∈ I} do not
belong to a common cone of Σ, there are two different indices i, j ∈ I such
that ai and aj do not belong to a common cone of Σ.
Clearly a monoidal complex which gives rise to a Stanley-Reisner rings defined by
quadrics satisfies the quadratic condition. Moreover, the correspongding StanleyReisner ring is Koszul by Fröberg’s theorem.
Remark 5.4. Without the quadratic condition, even when the ring k[MC ] is a
Koszul algebra for each facet C ∈ Σ and the defining ideal I is generated by quadrics,
k[M] can be non-Koszul. See the example below.
Example 5.5. Take k = Q. Consider the points in R4 with the following coordinates
A1 = (2, 0, 0, 0), A2 = (0, 2, 0, 0), A3 = (0, 0, 2, 0),
A4 = (1, 1, 0, 0), A5 = (0, 1, 1, 0).
Hence these points live in the affine subspace R3 = {x4 = 0}. The monoid ring
generated by those 5 points is k[X1 , . . . , X5 ]/I1 where I1 = (X1 X2 −X42 , X2 X3 −X52 ).
Let O = (0, 0, 0, 0) be the origin of R4 .
Take the point A6 = (0, 0, 0, 2). Consider the fan in R4 with the following facets,
which are simplicial cones: OA1A2 A3 , OA1 A3 A6 , OA2 A6 .

A

♣♣ 1
tP
A
♣P
♣ ♣ PP
t 4
PP
♣♣
Pt A2
♣♣
♣♣
❇
♣♣
❇
♣♣
❇ A5
♣♣
t
♣♣
♣♣ ❇
♣♣ ❇
♣ ♣❇
♣ ♣ A3
❇t
✟
✟ O
✟✟
✟
t✟
✟✟A6
✟✟

The (homogeneous) toric face ring we get is R = k[X1 , . . . , X6 ]/I with I = I1 +
(X4 X6 , X5 X6 ).
The affine monoid rings supported on the 3 maximal cones are Koszul. In fact, in
the revlex order with X1 < X2 < · · · < X5 , the polynomials {X1 X2 −X42 , X2 X3 −X52 }
form a Gröbner basis for I1 . So k[X1 , . . . , X5 ]/I1 is Koszul. The other two monoid
rings are polynomial rings.
However, we can check by Macaulay2 [14] that R is not a Koszul algebra, because
k has a non-linear second syzygy. The Betti table we get is the following.

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS

total:
0:
1:
2:

0
1
1
.
.

1
6
6
.
.

2
19
19
.
.

3
46
45
1
.

4
101
92
9
.

5
217
173
44
.

6
468
309
158
1

13

7
1016
534
470
12

In this example, AM = (X4 X6 , X5 X6 , X1 X2 X6 , X2 X3 X6 ) is not quadratic, so M
does not satisfy the quadratic condition w.r.t. the generators A1 , . . . , A6 .
The following conjecture was suggested by T. Römer.
Conjecture 5.6 (Römer). Assume that M satisfies the quadratic condition (with
respect to a homogeneous system of generators), and for every cone C ∈ Σ, k[MC ]
is a Koszul algebra. Then k[M] is Koszul.
The quadratic condition behaves well under various constructions on toric face
rings.
Proposition 5.7. Let k[M], k[N ] be monoidal complexes satisfying the quadratic
condition. Then the following toric face rings also satisfy the quadratic condition
with respect to suitable system of generators:
(i) tensor product over k of k[M] and k[N ],
(ii) fiber product over k of k[M] and k[N ],
(iii) Segre product over k of k[M] and k[N ],
(iv) Veronese subrings of k[M],
(v) multigraded algebra retracts of k[M].
Proof. From the hypothesis M and N satisfy the quadratic condition w.r.t. some
system of generators. Assume that a1 , . . . , an is that homogeneous system of generators of M and b1 , . . . , bm is that of N . Let k[X1 , . . . , Xn ]/IM be the presentation
of k[M], k[Y1 , . . . , Ym ]/IN the presentation of k[N ].
(i) The monomial part of the defining ideal of k[M] ⊗k k[N ] is AM + AN , which
is quadratic.
(ii) The monomial part of the defining ideal of k[M ∪ N ] is AM + AN + (Xi Yj :
1 ≤ i ≤ n, 1 ≤ j ≤ m), which is also quadratic.
(iii) Recall from the proof of Proposition 3.5 that k[M]∗k[N ] = k[M∗N ]. An element in the support of M∗N has the form (a, b) where a, b belongs to the support of
M and N respectively, and deg(a) = deg(b). Now assume that (a1 , b1 ), . . . , (am , bm )
is a sequence of elements of |M ∗ N | such that (a1 , b1 ) · · · (am , bm ) = 0. This implies
that either a1 · · · am = 0 or b1 · · · bm = 0. We may assume that the first equality
holds. Since M satisfies the quadratic condition, there exists some i 6= j such that
ai aj = 0. But then (ai , bi ) · (aj , bj ) = 0. Hence M ∗ N also satisfies the quadratic
condition.
(m)
(iv) Let M(m) be the mth
satisfies the
PnVeronesePofn M. We show that M
quadratic condition w.r.t. { i=1 ti ai :
i=1 ti = m, ti ≥ 0}. In fact, we will show
that if b1 · · · bp = 0 where bi ∈ |M(m) | then bi bj = 0 for some i 6= j.
We call bi extremal element if it is bi = maj for some j. We prove by induction
on the number of non-extremal elements among b1 , . . . , bp .

14

DANG HOP NGUYEN

If there is no non-extremal element, we are done by hypothesis on M.
Assume that P
b1 is a non-extremal.
P Choose a cone C ∈ Σ such that b1 ∈ int C. We
can write b1 = si=1 ti ai where si=1 ti = m, ti > 0 and a1 , . . . , as ∈ C.
Clearly (ma1 ) · · · (mas )b2 · · · bp = 0, hence by induction hypothesis on the number
of non-extremal elements, there exist 2 ≤ i < j ≤ p such that bi bj = 0 (in this case
we are done) or mat bi = 0 for 1 ≤ t ≤ s and 2 ≤ i ≤ p. In the second case bi does
not belong to any facet containing C, hence b1 bi = 0, hence we are also done.
(v) From Proposition 2.10, we know that any multigraded retract of k[M] is
isomorphic to some k[MΓ ], where MΓ is a restricted subcomplex of M. It is not
hard to check that the monomial part of the defining ideal of k[MΓ ] is
AMΓ = AM ∩ k[Xi : ai ∈ |MΓ |].
Since AM is quadratic in S = k[Xi : ai ∈ |M|], we have AMΓ is quadratic in
k[Xi : ai ∈ |MΓ |].

These observations suggest the feasibility of the above conjecture.
We will provide in the following some evidences to support this conjecture, see
5.10, 6.2, 6.4 and 7.2. As a special case, if k[M] is a Stanley-Reisner ring, this
conjecture is confirmed by Fröberg’s theorem. In the case of affine monoid rings
(where Σ is the face poset of a cone), there is nothing to do.
Remark 5.8. On the other hand, it is not true that the ideal AM is generated by
quadratic monomials given R being Koszul. See the following example 5.9.
Example 5.9. Continue with Example 2.5. The toric face ring
k[M] = k[X1 , X2 , X3 , X4 ]/(X1 X2 − X42 , X3 X4 )
is Koszul. Indeed, in the lex order, the set {X1 X2 − X42 , X3 X4 } is a quadratic
Gröbner basis for I. However, the ideal AM = (X3 X4 , X1 X2 X3 ) is not generated by
quadrics.
The next result is similar to Bruns, Koch and Römer [8, Prop. 3.2]. We give a
proof for the convenience of the reader.
Proposition 5.10. Let < be a monomial order on S = k[X1 , . . . , Xn ] and <i the
induced order on SCi = k[Xj |aj ∈ MCi ], where C1 , . . . , Cr are the facets of Σ. Then:
(i) Assume that k[M] is G-quadratic w.r.t. a monomial order < on S. Then
for i = 1, . . . , r, k[MCi ] is G-quadratic w.r.t. <i .
(ii) Assume that M satisfies the quadratic condition. Additionally, assume that
< is a monomial order on S such that the ideal k[MCi ] is G-quadratic
w.r.t. <i for i = 1, . . . , r. Then k[M] is G-quadratic w.r.t. <.
We will use the following simple lemma.
Lemma 5.11. Let I be an ideal of S which is generated by monomials and binomials
and < a monomial order on S. Then with respect to <, the ideal I has a Gröbner
basis consisting of monomials and binomials.

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS

15

Proof. We compute a Gröbner basis for I starting with the monomials and binomials
generating I. By using the Buchberger algorithm, at most we add new monomials
or new binomials to the Gröbner basis of I. Hence the conclusion of the lemma
follows.

Proof of Proposition 5.10. The two statements follow from the formula
in< (I) = AM +

r
X

S · in<i (ICi ),

i=1

and
in<i (ICi ) = SCi ∩ in< (I),
for each i = 1, . . . , r. We will prove these formulae by using Lemma 5.11.
For the first formula, clearly the right-hand side is contained in the left-hand side.
For the other inclusion, note that from Proposition 2.4, I is generated by monomials
in and some binomials.
Consider a Gröbner basis consisting of monomials and binomials of I. Clearly the
initial forms of the monomials in question belong to AM . Assume that b is a proper
binomial belongs to the Gröbner basis of I. Then either both monomials of b are
in AM , which gives us nothing to do, or b is in ICi for some i ∈ {1, . . . , r}. In the
second case, again in(b) belongs to the right-hand side of the first formula.
The second formula is proven in the same way.

Example 2.5 shows that the converse of Proposition 5.10.(ii) is not true. In fact,
in this case the ring k[M] has a quadratic Gröbner basis w.r.t. the lex order, but
AM is not generated by quadrics.
6. Strongly Koszul property
Recall the notion of strongly Koszul algebras.
Definition 6.1 (Herzog, Hibi, Restuccia [16]). Let R be a homogeneous k-algebra,
m = R+ . Suppose that a1 , . . . , an ∈ R1 and minimally generate m. Then R is called
strongly Koszul with respect to the sequence a1 , . . . , an if for every 1 ≤ i1 < · · · <
ij ≤ n, the ideal (ai1 , . . . , aij−1 ) : aij is generated by a subset of {a1 , . . . , an }.
The main result of this section characterizes strongly Koszul toric face rings.
Theorem 6.2. The following statements are equivalent:
(i) k[M] is strongly Koszul w.r.t. the sequence {a1 , . . . , an };
(ii) (a) for each i = 1, . . . , n, we have 0 :k[M] ai = (ai1 , . . . , aij ), for some
elements ai1 , . . . , aij in {a1 , . . . , an },
(b) for each facet C of Σ, the ring k[MC ] is strongly Koszul w.r.t. the
sequence {a1 , . . . , an } ∩ MC .
Firstly, we have a simple lemma.

16

DANG HOP NGUYEN

Lemma 6.3. Let I = (a1 , . . . , ai−1 ) :R ai . Let C1 , . . . , Cr be the facets of Σ. For
each C ∈ Σ, consider the following ideal of k[MC ] :
(
(aj |j < i, aj ∈ MC ) :k[MC ] ai if ai ∈ MC ;
IC =
0
if ai ∈
/ MC .
Then:
(i) For each C ∈ Σ, we have (a1 , . . . , ai ) ∩ k[MC ] = (aj |j ≤ i, aj ∈ MC ).
(ii) For each C ∈ ΣP
such that ai ∈ MC , we have I ∩ k[MC ] = IC .
(iii) I = (0 :R ai ) + rl=1 R · ICl .

Proof. This is a standard utilization of the Zd -grading. For (i), let y ∈ (a1 , . . . , ai ) ∩
k[MC ] be homogeneous. If aj divides y then because y ∈ MC , we have aj ∈ MC .
Thus (i) is proved.
For (ii), we see that IC ⊆ I ∩ k[MC ]. Let z ∈ I ∩ k[MC ] be homogeneous. Since
zai ∈ (a1 , . . . , ai−1 ), we have zai = waj where j < i. But waj ∈ MC , so w, aj ∈ MC .
Using the projection from R to k[MC ], we get z ∈ IC .
The last part follows from (ii). We leave it as an exercise to the reader.

Proof of Theorem 6.2. With Lemma 6.3.(iii), it is immediate that (ii) ⇒ (i).
Assuming that we have (i). Then (ii)(a) is clear.
Consider a facet C of Σ, and a subsequence of {a1 , . . . , an } ∩ MC . Without loss
of generality we can assume this subsequence to be a1 , . . . , ai . By (i), we have the
equality (a1 , . . . , ai−1 ) :k[M] ai = (ai1 , . . . , aik ).
Using Lemma 6.3(i), the ideal (a1 , . . . , ai−1 ) :k[MC ] ai = (ail |ail ∈ MC ). This
implies (ii)(b).

Remark 6.4. Part (a) of (ii) in the Theorem 6.2 is true if the M satisfies the
quadratic condition. However, the converse is not true as the next example demonstrates.
Example 6.5. In R3 take six points with the following coordinates:
A1 = (2, 0, 0), A2 = (0, 2, 0), A3 = (0, 0, 2), A4 = (0, 1, 1),
A5 = (1, 0, 1), A6 = (1, 1, 0).
Consider the fan in R3 with three maximal cones C1 , C2 , C3 , where C1 is generated
by the points A1 , A2 , A6 , the cone C2 is generated by A3 , A1 , A5 and the cone C3
is generated by A2 , A3 , A4 with the monoid relations A1 A2 − A26 = A2 A3 − A24 =
A3 A1 − A25 = 0.
Take M1 to be generated by A1 , A2 , A6 and in the same way, we have two other
maximal monoids M2 and M3 of a monoidal complex M. The presentation ideal of
k[M] is
I = (X1 X2 − X62 , X2 X3 − X42 , X3 X1 − X52 , X1 X4 , X2 X5 , X3 X6 ,
X4 X5 , X4 X6 , X5 X6 ).

ON THE KOSZUL PROPERTY OF TORIC FACE RINGS

17

A

tP
♣♣ 1
A6
♣P
♣ ♣ PP
♣ ♣ t PP
♣♣
Pt A2
♣♣
❆
♣♣
♣♣
t♣ ♣A5 ❆
❆ A4
♣♣
♣ ♣ ❆t
♣♣
♣♣ ❆
♣♣❆
♣ ♣ A3
❆♣ t

O

Because of Theorem 6.2, the ring k[M] is strongly Koszul w.r.t. the sequence
X1 , X2 , . . . , X6 . Indeed, part (b) of (ii) is true. For example, k[M1 ] is strongly
Koszul w.r.t. the sequence X1 , X2 , X6 .
Part (a) if (ii) is true because of the following two identities:
(i) 0 :k[M] X1 = (X4 )
(ii) 0 :k[M] X4 = (X1 , X5 , X6 )
and four similar identities.
However, the monomial part
AM = (X1 X2 X3 , X1 X4 , X2 X5 , X3 X6 , X4 X5 , X4 X6 , X5 X6 )
is not quadratic.
7. Initially Koszul property
In the followings, we consider initially and universally initially Koszul algebras in
the sense of Blum [2, Definition 1.3], Conca, Rossi and Valla [9, Definition 2.2].
Definition 7.1. Let R be a homogeneous k-algebra, and a1 , . . . , an ∈ R1 minimally
generate R+ . The ring R is called initially Koszul (or i-Koszul) with respect to the
sequence a1 , . . . , an if the set
F = {(a1 , . . . , ai ) : i = 0, . . . , n}
is a Koszul filtration for R in the sense of Definition 1.1. In other words, for i =
1, . . . , n, we have
(a1 , . . . , ai−1 ) : ai ∈ F .
We say that R is universally initially Koszul (or u-i-Koszul) if R is i-Koszul w.r.t. any
k-basis of R1 .
In this section, for a homogeneous quotient ring R = k[X1 , . . . , Xn ]/I, i-Koszulness
means i-Koszulness w.r.t. the sequence of elements X1 , . . . , Xn . By [9, Thm. 2.4] and
[2, Thm. 2.1], i-Koszulness of R implies that I has quadratic Gröbner basis in certain
monomial order.
From [2, Prop. 2.3], we see that a Stanley-Reisner ring k[∆] is i-Koszul if and only
if ∆ is the full simplex.
The following theorem is the main result of this section.

18

DANG HOP NGUYEN

Theorem 7.2. If k[M] is i-Koszul w.r.t. the sequence a1 , . . . , an , then Σ is the face
poset of a cone. In particular, k[M] is an affine monoid ring.
In the proof of the theorem, we need the following lemma.
Lemma 7.3. Under the assumptions of Theorem 7.2, if ai generates an extremal
ray of Σ then (a1 , . . . , ai−1 ) : ai = (a1 , . . . , ai−1 ).
Proof. From the i-Koszulness assumption (a1 , . . . , ai−1 ) : ai = (a1 , . . . , aj ). If j ≥ i,
then ai ∈ (a1 , . . . , ai−1 ) : ai . Thus a2i = bal for some l < i, b ∈ ∪C∈Σ MC . Since
a2i 6= 0, we conclude that ai , b, al belong to a common cone C of Σ. Since ai
generates an extremal ray of C, we must have al belongs to that extremal ray. In
other words, al is a non-zero multiple of ai . This contradicts with the minimality of
{a1 , . . . , an }.

Proof of Theorem 7.2. We only need to prove that all the extremal rays of Σ belong
to the same face of Σ. Assume that this is not the case. If ai1 , . . . , ait generate the
extremal rays of Σ with 1 ≤ i1 < · · · < it ≤ n then ai1 ai2 · · · ait = 0.
Of course ai2 · · · ait ∈ (a1 , . . . , ai1 −1 ) : ai1 . From Lemma 7.3, we have that
ai2 · · · ait ∈ (a1 , . . . , ai1 −1 ). This in turn implies ai3 · · · ait ∈ (a1 , . . . , ai2 −1 ) : ai2 .
So again from Lemma 7.3, we get ai3 · · · ait ∈ (a1 , . . . , ai2 −1 ). Iterate this argument
and in the end we will have ait ∈ (a1 , . . . , ait−1 −1 ). This is a contradiction and hence
we are done.

Corollary 7.4. If k[M] is a homogeneous u-i-Koszul toric face ring then k[M] is
a polynomial ring.
Proof. From Theorem 7.2, we get that k[M] is an affine monoid ring. The Corollary
follows from [2, Prop. 5.5], which says that affine monoid rings which are u-i-Koszul
must be polynomial rings.

Acknowledgment
We are grateful to Tim Römer for generously suggesting problems and many
insightful ideas on the subject of this paper. I am indebted to him for allowing me
to include Conjecture 5.6 in the paper. We want to express our sincere thank to
Ngo Viet Trung and Aldo Conca for their encouragement and inspiring comments.
We thank the anonymous referee for his/her thoughtful comments which considerably helped us to improve the presentation of this paper.
References
[1] J. Backelin and R. Fröberg, Koszul algebras, Veronese subrings and rings with linear resolution.
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