ARTINIAN LEVEL ALGEBRAS OF SOCLE DEGREE 4

arXiv:1701.03180v1 [] 11 Jan 2017

SHREEDEVI K. MASUTI AND M. E. ROSSI
A BSTRACT. In this paper we study the O-sequences of the local (or graded) K-algebras of socle degree 4. More precisely, we prove that an O-sequence h = (1, 3, h2 , h3 , h4 ), where h4 ≥ 2, is the h-vector
of a local level K-algebra if and only if h3 ≤ 3h4 . We also prove that h = (1, 3, h2 , h3 , 1) is the h-vector
of a local Gorenstein K-algebra if and only if h3 ≤ 3 and h2 ≤ (h32+1) + (3 − h3 ). In each of these cases
we give an effective method to construct a local level K-algebra with a given h-vector. Moreover we
refine a result by Elias and Rossi by showing that if h = (1, h1 , h2 , h3 , 1) is an unimodal Gorenstein
O-sequence, then h forces the corresponding Gorenstein K-algebra to be canonically graded if and
only if h1 = h3 and h2 = (h12+1), that is the h-vector is maximal.

1. I NTRODUCTION
Let ( A, m) be an Artinian local or graded K-algebra where K is an algebraically closed field of
characteristic zero. Let Soc( A) = (0 : m) be the socle of A. We denote by s the socle degree of
A, that is the maximum integer j such that m j 6= 0. The type of A is τ := dimK Soc( A). Recall
that A is said to be level of type τ if Soc( A) = ms and dimK ms = τ. If A has type 1, equivalently dimK Soc( A) = 1, then A is Gorenstein. In the literature local rings with low socle degree,
also called short local rings, have emerged as a testing ground for properties of infinite free resolutions (see [AIS08], [AHS13], [HS11], [Sjo79], [Roos05], [CRV01]). They have been also extensively studied in problems related to the irreducibility and the smoothness of the punctual Hilbert
scheme Hilbd (P nK ) parameterizing zero-dimensional subschemes P nK of degree d, see among others [Poo08], [EV08], [CEVV09], [CENR13]. In this paper we study the structure of level K-algebras
of socle degree 4, hence m5 = 0. One of the most significant information on the structure is given
by the Hilbert function.
By definition, the Hilbert function of A, hi = hi ( A) = dimK mi /mi+1 , is the Hilbert function of
the associated graded ring grm ( A) = ⊕i≥0 mi /mi+1 . We also say that h = (h0 , h1 , . . . , hs ) is the hvector of A. In [Mac27] Macaulay characterized the possible sequences of positive integers hi that
can occur as the Hilbert function of A. Since then there has been a great interest in commutative
algebra in determining the h-vectors that can occur as the Hilbert function of A with additional
properties (for example, complete intersection, Gorenstein, level, etc). A sequence of positive
h ii
integers h = (h0 , h1 , . . . , hs ) satisfying Macaulay’s criterion, that is h0 = 1 and hi+1 ≤ hi for
i = 1, . . . , s − 1, is called an O-sequence. A sequence h = (1, h1 , . . . , hs ) is said to be a level (resp.
Gorenstein) O-sequence if h is the Hilbert function of some Artinian level (resp. Gorenstein) Kalgebra A. Remark that h1 is the embedding dimension and, if A is level, hs is the type of A.
Notice that a level O-sequence is not necessarily the Hilbert function of an Artinian level graded
Date: January 13, 2017.
2010 Mathematics Subject Classification. Primary: 13H10; Secondary: 13H15, 14C05.
Key words and phrases. Macaulay’s inverse system, Hilbert functions, Artinian Gorenstein and level algebras, canonically graded algebras.
The first author was supported by INdAM COFOUND Fellowships cofounded by Marie Curie actions, Italy. The
second author was partially supported by PRIN 2014 “Geometria delle varieta’ algebriche”.
1

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MASUTI AND ROSSI

K-algebra. This is because the Hilbert function of the level ring ( A, m) is the Hilbert funtion of
grm ( A) which is not necessarily level. From now on we say that h is a graded level O-sequence if h
is the Hilbert function of a level graded standard K-algebra. For instance it is well known that the
h-vector of a Gorenstein graded K-algebra is symmetric, but this is no longer true for a Gorenstein
local ring. Characterize level O-sequences is a wide open problem in commutative algebra. The
problem is difficult and very few is known even in the graded case as evidenced by [GHMY07].
In the following table we give a summary of known results:
Characterization of
Graded
Local
Gorenstein O-sequences with h1 = 2
[Sta78, Theorem 4.2]
[Ber09, Theorem 2.6]
level O-sequences with h1 = 2
[Iar84, Theorem 4.6A], [Iar04] [Ber09, Theorem 2.6]
Gorenstein O-sequences with h1 = 3
[Sta78, Theorem 4.2]
Open
Open. In [GHMY07] authors
level O-sequences with h1 = 3
gave a complete list with
and τ ≥ 2
s ≤ 5 or s = 6 and τ = 2
Open
level O-sequences with s ≤ 3
[Ste14], [GHMY07]
[Ste14, Theorem 4.3]
In this paper we fill the above table by characterizing the Gorenstein and level O-sequences with
a particular attention to socle degree 4 and embedding dimension h1 = 3.
In our setting we can write A = R/I where R = K [[ x1 , . . . , xr ]] (resp. K [ x1 , . . . , xr ]) is the formal
power series ring (resp. the polynomial ring with standard grading) and I an ideal of R. We
say that A is graded when it can be presented as R/I where I is an homogeneous ideal in R =
K [ x1 , . . . , xr ]. Without loss of generality we assume that h1 = dimK m/m2 = r.
Recall that the socle type of A = R/I is the sequence E = (0, e1 , . . . , es ), where
ei := dimK ((0 : m) ∩ mi )/(0 : m) ∩ mi+1 ).
It is known that for all i ≥ 0,
(1.1)

hi ≤ min{dimK Ri , ei dimK R0 + ei+1 dimK R1 + · · · + es dimK Rs−i }

(see [Iar84]). Hence a necessary condition for h to be a level O-sequence is that hs−1 ≤ h1 hs being
es = hs and ei = 0 otherwise. In the following theorem we prove that this condition is also
sufficient for h = (1, 3, h2 , h3 , h4 ) to be a level O-sequence, provided h4 ≥ 2. However, if h4 = 1,
we need an additional assumption for h to be a Gorenstein O-sequence. We remark that the result
can not be extended to h1 > 3 (see Example 3.7).
Theorem 1. Let h = (1, 3, h2 , h3 , h4 ) be an O-sequence.
(a) Let h4 = 1. Then h is a Gorenstein O-sequence if and only if h3 ≤ 3 and h2 ≤ (h32+1) + (3 −
h3 ) .
(b) Let h4 ≥ 2. Then h is a level O-sequence if and only if h3 ≤ 3h4 .
The proof of the above result is effective in the sense that in each case we construct a local level
K-algebra with a given h-vector verifying the necessary conditions (see Theorem 3.6).
Combining Theorem 1(b) and results in [GHMY07] we show that there are level O-sequences
which are not admissible in the graded case (see Section 3, Table 1). A similar behavior was
observed in [Ste14] for socle degree 3. Theorem 1(a) is a consequence of the following more general
result which holds for any embedding dimension:

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Theorem 2.
(1.2)

(a) If (1, h1 , . . . , hs−2 , hs−1 , 1) is a Gorenstein O-sequence, then


hs −1 + 1
hs−1 ≤ h1 and hs−2 ≤
+ ( h1 − h s − 1 ) .
2

(b) If h = (1, h1 , h2 , h3 , 1) is an unimodal O-sequence satisfying (1.2), then h is a Gorenstein
O-sequence.
The bound (1.2) improves the bound for hs−2 given in (1.1).
If A is a Gorenstein local K-algebra with symmetric h-vector, then grm ( A) is Gorenstein, see
[Iar94]. It is a natural question to ask, in this case, whether A is analytically isomorphic to grm ( A).
Accordingly with the definition given in [Ems78, Page 408] and in [ER12], recall that an Artinian
local K-algebra ( A, m) is said to be canonically graded if there exists a K-algebra isomorphism between A and its associated graded ring grm ( A).
For instance J. Elias and M. E. Rossi in [ER12] proved that every Gorenstein K-algebra with symmetric h-vector and m4 = 0 (s ≤ 3) is canonically graded. A local K-algebra A of socle type E is said
to be compressed if equality holds in (1.1) for all 1 ≤ i ≤ s, equivalently the h-vector is maximal (see
[Iar84, Definition 2.3]). In [ER15, Theorem 3.1] Elias and Rossi proved that if A is any compressed
Gorenstein local K-algebra of socle degree s ≤ 4, then A is canonically graded. In Section 4 we
prove that if the socle degree is 4, then the assumption can not be relaxed. More precisely only the
maximal h-vector forces every corresponding Gorenstein K-algebra to be canonically graded. We
prove that if h is unimodal and not maximal, then there exists a Gorenstein K-algebra with Hilbert
function h which is not canonically graded. To prove that a local K-algebra is not canonically
graded is in general a very difficult task. See also [Jel16] for interesting discussions.
Theorem 3. Let h = (1, h1 , h2 , h3 , 1) be a Gorenstein O-sequence with h2 ≥ h1 . Then every local
Gorenstein K-algebra with Hilbert function h is necessarily canonically graded if and only if h1 =
h3 and h2 = (h12+1).
The main tool of the paper is Macaulay’s inverse system [Mac1916] which gives a one-to-one
correspondence between ideals I ⊆ R such that R/I is an Artinian local ring and finitely generated
R-submodules of a polynomial ring. In Section 2 we gather preliminary results needed for this
purpose. We prove Theorems 1 and 2 in Section 3, and Theorem 3 in Section 4.
We have used Singular [Singular], [Eli15] and CoCoA [CoCoA] for various computations and
examples.
A CKNOWLEDGEMENTS
We thank Juan Elias for providing us the updated version of I NVERSE - SYST. LIB for computations.
2.

PRELIMINARIES

2.1. Macaulay’s Inverse System. In this subsection we recall some results on Macaulay’s inverse
system which we will use in the subsequent sections. This theory is well-known in the literature,
especially in the graded setting (see for example [Mac1916, Chapter IV] and [IK99]). However, the
local case is not so well explored. We refer the reader to [Ems78], [Iar94] for an extended treatment.

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MASUTI AND ROSSI

It is known that the injective hull of K as R-module is isomorphic to a divided power ring
P = K [ X1 , . . . , Xr ] which has a structure of R-module by means of the following action

◦ : R × P −→
( f , g) −→

P
f ◦ g = f (∂X1 , . . . , ∂Xr )( g)

where ∂Xi denotes the partial derivative with respect to Xi . For the sake of simplicity from now
on we will use xi instead of the capital letters Xi . If { f 1 , . . . , f t } ⊆ P is a set of polynomials, we
will denote by h f 1 , . . . , f t i R the R-submodule of P generated by f1 , . . . , f t , i.e., the K-vector space
generated by f 1 , . . . , f t and by the corresponding derivatives of all orders. We consider the exact
pairing of K-vector spaces:
h , i : R × P −→
K
( f , g) −→ ( f ◦ g)(0).
For any ideal I ⊂ R we define the following R-submodule of P called Macaulay’s inverse system:
I ⊥ : = { g ∈ P | h f , g i = 0 ∀ f ∈ I }.
Conversely, for every R-submodule M of P we define
AnnR ( M ) := { g ∈ R | h g, f i = 0 ∀ f ∈ M }
which is an ideal of R. If M is generated by polynomials f := f1 , . . . , f t , with f i ∈ P, then we will
write AnnR ( M ) = AnnR (f) and Af = R/ AnnR (f).
By using Matlis duality one proves that there exists a one-to-one correspondence between ideals
I ⊆ R such that R/I is an Artinian local ring and R-submodules M of P which are finitely generated. More precisely, Emsalem in [Ems78, Proposition 2] and Iarrobino in [Iar94, Lemma 1.2]
proved that:
Proposition 2.1. There is a one-to-one correspondence between ideals I such that R/I is a level local ring
of socle degree s and type τ and R-submodules of P generated by τ polynomials of degree s having linearly
independent forms of degree s. The correspondence is defined as follows:




 I ⊆ R such that R/I 
 M ⊆ P submodule generated by 
1− 1
is a level local ring of
τ polynomials of degree
←→




socle degree s and type τ
s with l.i. forms of degree s
I
AnnR ( M )

−→
←−

I⊥
M

By [Ems78, Proposition 2(a)], the action h , i induces the following isomorphism of K-vector
spaces:
(2.2)

( R/I )∗ ≃ I ⊥ ,

(where ( R/I )∗ denotes the dual with respect to the pairing h , i induced on R/I). Hence dimK R/I =
dimK I ⊥ . As in the graded case, it is possible to compute the Hilbert function of A = R/I via the
inverse system. We define the following K-vector space:

( I ⊥ )i :=

I ⊥ ∩ P≤i + P<i
.
P<i

Then, by (2.2), it is known that
(2.3)

hi ( R/I ) = dimK ( I ⊥ )i .

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2.2. Q-decomposition. It is well-known that the Hilbert function of an Artinian graded Gorenstein K-algebra is symmetric, which is not true in the local case. The problem comes from the fact
that, in general, the associated algebra G := grm ( A) of a Gorenstein local algebra A is no longer
Gorenstein. However, in [Iar94] Iarrobino proved that the Hilbert function of a Gorenstein local
K-algebra A admits a “symmetric” decomposition. To be more precise, consider a filtration of G
by a descending sequence of ideals:
G = C (0) ⊇ C (1) ⊇ · · · ⊇ C (s) = 0,
where
C ( a) i : =

( 0 : m s + 1− a − i ) ∩ m i
.
( 0 : m s + 1− a − i ) ∩ m i + 1

Let
Q( a) = C ( a)/C ( a + 1).
Then

{Q( a) : a = 0, . . . , s − 1}
is called Q-decomposition of the associated graded ring G. We have
s −1

hi ( A) = dimK Gi =

∑ dimK Q(a)i .
a=0

Iarrobino [Iar94, Theorem 1.5] proved that if A = R/I is a Gorenstein local ring then for all
a = 0, . . . , s − 1, Q( a) is a reflexive graded G-module, up to a shift in degree: HomK ( Q( a)i , K ) ∼
=
Q( a)s− a−i . Hence the Hilbert function of Q( a) is symmetric about s−2 a . Moreover, he showed that
Q(0) = G/C (1) is the unique (up isomorphism) socle degree s graded Gorenstein quotient of G.
Let f = f [s] + lower degree terms... be a polynomial in P of degree s where f [s] is the homogeneous part of degree s and consider A f the corresponding Gorenstein local K-algebra. Then,
Q ( 0) ∼
= R/ AnnR ( f [s]) (see [Ems78, Proposition 7] and [Iar94, Lemma 1.10]).
3. C HARACTERIZATION

OF LEVEL

O- SEQUENCES

In this section we characterize Gorenstein and level O-sequences of socle degree 4 and embedding dimension 3. First if h = (1, h1 , . . . , hs ) is a Gorenstein O-sequence (in any embedding
dimension), we give an upper bound on hs−2 in terms of hs−1 which improves the already known
inequality given in (1.1).
Theorem 3.1.
(3.2)

(a) If (1, h1 , . . . , hs−2 , hs−1 , 1) is a Gorenstein O-sequence, then



hs −1 + 1
hs−1 ≤ h1 and hs−2 ≤
+ ( h1 − h s − 1 ) .
2

(b) If h = (1, h1 , h2 , h3 , 1) is an O-sequence such that h2 ≥ h3 and it satisfies (3.2), then h is a
Gorenstein O-sequence.
Proof. (a): From (1.1) it follows that hs−1 ≤ h1 . Let A be a Gorenstein local K-algebra with the
Hilbert function h and {Q( a) : a = 0, . . . , s − 1} be a Q-decomposition of grm ( A). Since Q(i )s−1 =
0 for i > 0, dimK Q(0)s−1 = hs−1 . Hence dimK Q(0)1 = dimK Q(0)s−1 = hs−1 and dimK Q(0)s−2 =
h 1i
dimK Q(0)2 ≤ hs−1 . This in turn implies that dimK Q(1)s−2 = dimK Q(1)1 ≤ h1 − hs−1 . Therefore
hs−2 = dimK Q(0)s−2 + dimK Q(1)s−2


hs −1 + 1
≤
+ ( h1 − h s − 1 ) .
2

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MASUTI AND ROSSI

(b): Suppose h2 ≤ h1 . Set
f = x14 + · · · + x4h3 + x3h3 +1 + · · · + x3h2 + x2h2 +1 + · · · + x2h1 .
Then A f has the Hilbert function h. Now assume h2 > h1 . Denote h3 := n and define monomials
gi ∈ K [ x1 , . . . , xn ] as follows:

2

if 1 ≤ i ≤ n
 xi
gi = xi−n xi−n+1 if n + 1 ≤ i ≤ 2n − 1


x n x1
if i = 2n.

For i = (i1 , . . . , in ) ∈ N n , let xi := x1i1 . . . xnin . Let T be the set of monomials xi of degree 2 in
K [ x1 , . . . , xn ] such that
/ { gi : 1 ≤ i ≤ 2n}.
xi ∈
1
n +1
Then | T | = (n+
2 ) − 2n. We write T = { gi : 2n < i ≤ ( 2 )}. Define

h2 − h1
n


2

x
g
+

∑ x2i gn+i + x3n+1 + · · · + x3h1
∑ i i
i=1
f = i=n 1
h2 − h1 + n
n


 ∑ x2 gi + ∑ x2 gn+i + ∑ g2 + x3n+1 + · · · + x3

i
i
i
h1

i=1

Then h3 = dimK



if h2 − h1 > n.

2n +1

i=1

∂f
∂f
∂x1 , . . . , ∂xn

if h2 − h1 ≤ n



= n and



h2 = dimK { gi : 1 ≤ i ≤ h2 − h1 + n}

Thus A f has the Hilbert function (1, h1 , h2 , h3 , 1).

S
·


{ x2n+1 , . . . , x2h1 } .



Remark 3.3. If h3 = h1 , then f in the proof of Theorem 3.1(b) is homogeneous and hence A f is a
graded Gorenstein K-algebra.
If the socle degree is 4 and h1 ≤ 12, then the converse holds in (a).
Corollary 3.4. An O-sequence h = (1, h1 , h2 , h3 , 1) with h1 ≤ 12 is a Gorenstein O-sequence if and only
if h satisfies (3.2).
Proof. By Theorem 3.1, it suffices to show that h2 ≥ h3 if h1 ≤ 12. Let A be a local Gorenstein Kalgebra with the Hilbert function h. Considering the symmetric Q-decomposition of grm ( A), we
observe that in this case Q(0) has the Hilbert function (1, h3 , h2 − m, h3 , 1), for some non-negative
integer m. Since h1 ≤ 12, by (3.2) h3 ≤ 12. As Q(0) is a graded Gorenstein K-algebra, by [MZ16,
Theorem 3.2] we conclude that h2 − m ≥ h3 since Q(0) has unimodal Hilbert function, hence
h2 ≥ h3 .

Remark 3.5. A Gorenstein O-sequence (1, h1 , h2 , h3 , 1) does not necessarily satisfy h2 ≥ h3 . For
example, consider the the sequence h = (1, 13, 12, 13, 1). By [Sta78, Example 4.3] there exists a
graded Gorenstein K-algebra with h as h-vector.
In the following theorem we characterize the h-vector of local level algebras of socle degree 4
and embedding dimension 3.
Theorem 3.6. Let h = (1, 3, h2 , h3 , h4 ) be an O-sequence.
(a) Let h4 = 1. Then h is a Gorenstein O-sequence if and only if h3 ≤ 3 and h2 ≤ (h32+1) + (3 − h3 ).
(b) Let h4 ≥ 2. Then h is a level O-sequence if and only if h3 ≤ 3h4 .

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Proof. (a): Follows from Corollary 3.4.
(b): The “only if” part follows from (1.1). The converse is constructive and we prove it by inductive
steps on h4 . First we consider the cases h4 = 2, 3, 4 and then h4 ≥ 5. In each case we define the
polynomials f := f1 , . . . , f h4 ∈ P = K [ x1 , x2 , x3 ] of degree 4 such that Af has the Hilbert function h.
We set g1′ = x33 , g2′ = x22 x3 , g3′ = x12 x2 , g4′ = x1 x32 .
For short, in this proof we use the following notation: m := h2 and n := h3 .
Case 1: h4 = 2.
In this case n ≤ 6 as h3 ≤ 3h4 by assumption. Suppose m = 2. Then h is an O-sequence implies
that n = 2. In this case, let f1 = x14 + x32 and f2 = x24 . Then Af has the Hilbert function (1, 3, 2, 2, 2).
Now assume that m ≥ 3.

′

 x4−i gi if 1 ≤ i ≤ n − 2
Subase 1: m ≥ n. We set gi = gi′
if n − 1 ≤ i ≤ m − 2


0
if m − 1 ≤ i ≤ 4.
(Here x0 = x3 ). Define
f1 = x14 + g1 + g2 and f 2 = x24 + g3 + g4 .

Then
h3 = dimK { x13 , x23 , g1′ , . . . , gn′ −2 } = n and
g′
h2 = dimK { x12 , x22 , i : 1 ≤ i ≤ m − 2} = m
x4 − i
and hence Af has the required Hilbert function h.
Subcase 2: m < n. The only possible ordered tuples (m, n) with m < n ≤ 6 such that h is an Osequence are {(3, 4), (4, 5), (5, 6)}. For each 2-tuple (m, n) we define f1 , f2 as:
a.(m, n) = (3, 4) : f1 = x14 + x12 x22 + x32 ; f2 = x24 + x12 x22 .
b.(m, n) = (4, 5) : f1 = x14 + x12 x22 + x34 ; f2 = x24 + x12 x22 .
c.(m, n) = (5, 6) : f1 = x14 + x12 x22 + x34 ; f2 = x24 + x12 x22 + x23 x3 .
Case 2: h4 = 3.
In this case n ≤ 9. We consider the following subcases:
Subcase 1: n ≤ 6. Let f′ = f 1 , f 2 be polynomials defined as in Case 1 such that Af′ has the Hilbert
(
x34
if m ≥ n
function (1, 3, m, n, 2). Now define f3 =
2
2
x1 x2 if m < n.
Then Af has the required Hilbert function h.
Subcase 2: 7 ≤ n ≤ 9. Let f′ = f1 , f 2 be polynomials defined as in Case 1 such that Af′ has the
2 2
2 2
2 2
Hilbert function (1, 3, m, 6, 2). We set p1 = x(
2 x3 , p2 = x1 x2 and p3 = x1 x3 . Since h is an O-sequence
n −6
∑i=1 pi if m = 6
and n ≥ 7, we get m ≥ 5. Now define f 3 =
if m = 5.
x22 x32
Then Af has the required Hilbert function h.

Case 3: h4 = 4.

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MASUTI AND ROSSI

Since h is an O-sequence, n ≤ 10. We consider the following subcases:
Subcase 1: n ≤ 9. Let f′ = f1 , f 2 , f 3 be polynomials defined as in Case 2 such that Af′ has the
Hilbert function (1, 3, m, n, 3). Define
(
x23 x3 if {m ≥ n and n ≤ 6} OR {n ≥ 7 and m = 6}
f4 =
x13 x2 if {m < n ≤ 6} OR {(m, n) = (5, 7)}.
Then Af has the Hilbert function (1, 3, m, n, 4).
Subcase 2: n = 10. As h is an O-sequence, we conclude that m = 6. Let f′ = f 1 , f2 , f 3 be polynomials defined as in Case 2 such that Af′ has the Hilbert function (1, 3, 6, 9, 3). Define f4 = x12 x2 x3 .
Then Af has the Hilbert function (1, 3, 6, 10, 4).

Case 4: h4 ≥ 5.
Since h is an O-sequence, n ≤ 10 and h4 ≤ 15
Subcase 1: n ≥ h4 OR h4 ≥ 11. Let f′ = f 1 , f 2 , f 3 , f 4 be defined as in Case 3 such that Af′ has the
Hilbert function (1, 3, m, n, 4). For 5 ≤ i ≤ 15, define f i as follows:
(
x13 x2 if {m ≥ n and n ≤ 6} OR {n ≥ 7 and m = 6}
f5 =
x1 x23 if {m < n ≤ 6} OR {(m, n) = (5, 7)},
(
x1 x33 if {m ≥ n and n ≤ 6} OR {n ≥ 7 and m = 6}
f6 =
x34
if {m < n ≤ 6} OR {(m, n) = (5, 7)},
(
if n ≥ 7 and m = 6
x34
f7 =
3
x2 x3 if {(m, n) = (5, 7)}.
(Note that in the last case, h4 ≥ 7 implies that n ≥ 7). If h4 ≥ 8, then n ≥ 8 which implies that
m = 6. We set
f8 = x12 x22 , f 9 = x12 x32 , f 10 = x2 x33 , f 11 = x1 x23 , f 12 = x13 x3 , f 13 = x23 x3 , f 14 = x1 x22 x3 , f 15 = x1 x2 x32 .
Now Af has the Hilbert function (1, 3, m, n, h4 ).
Subcase 2: n < h4 ≤ 10. The smallest ordered tuple (n, h4 ) such that h is an O-sequence and n <
h4 is (4, 5). (Here smallest ordered tuple means smallest with respect to the order ≤ defined as:
(n1 , n2 ) ≤ (m1 , m2 ) if and only if n1 ≤ m1 and n2 ≤ m2 ). Let
(
(
(
x32 if m = 3
0
if m < 5
0
if m < 6
q1 =
q2 =
and q3 =
x33 if m ≥ 4,
x22 x3 if m ≥ 5
x1 x32 if m = 6.
We define
f1 = x14 + q1 + q2 , f 2 = x24 + q3 , f3 = x13 x2 , f 4 = x1 x23 , f5 = x12 x22 .
Then Af has the Hilbert function (1, 3, m, 4, 5).
Let h4 ≥ 6. We set
(
f6 =

x34 , f 7

=

x23 x3 , f 8

=

x2 x33 , f 9

=

x22 x32
x1 x33

Then Af has the Hilbert function (1, 3, m, n, h4 ).

if n = 7
f 10 =
if n ≥ 8,

(

x22 x32
x13 x3

if n = 8
if n = 9.



9

Using [GHMY07, Appendix D] and Theorem 3.6(b) we list in the following table all the Osequences which are realizable for local level K-algebras with h4 ≥ 2, but not for graded level
K-algebras.
TABLE 1

(1, 3, 2, 2, 2)
(1, 3, 6, 3, 2)
(1, 3, 6, 4, 3)
(1, 3, 5, 4, 5)
(1, 3, 6, 7, 9)

(1, 3, 3, 2, 2)
(1, 3, 3, 4, 2)
(1, 3, 3, 4, 4)
(1, 3, 6, 4, 5)

(1, 3, 4, 2, 2)
(1, 3, 4, 3, 3)
(1, 3, 5, 4, 4)
(1, 3, 6, 5, 5)

(1, 3, 5, 2, 2)
(1, 3, 5, 3, 3)
(1, 3, 6, 4, 4)
(1, 3, 5, 5, 6)

(1, 3, 6, 2, 2)
(1, 3, 6, 3, 3)
(1, 3, 3, 4, 5)
(1, 3, 6, 5, 6)

(1, 3, 5, 3, 2)
(1, 3, 3, 4, 3)
(1, 3, 4, 4, 5)
(1, 3, 6, 6, 7)

Analogously, by Theorem 3.6(a), the following O-sequences are Gorenstein sequences, but they
are not graded Gorenstein sequences since they are not symmetric. If an O-sequence h is symmetric with h1 = 3 then h is also a graded Gorenstein O-sequence by Remark 3.3.
TABLE 2

(1, 3, 1, 1, 1) (1, 3, 2, 1, 1) (1, 3, 3, 1, 1) (1, 3, 2, 2, 1) (1, 3, 3, 2, 1) (1, 3, 4, 2, 1)
The following example shows that Theorem 3.6(b) can not be extended to h1 ≥ 4 because the
necessary condition h3 ≤ h1 hs is not longer sufficient for characterizing level O-sequences of socle
degree 4.
Example 3.7. The O-sequence h = (1, 4, 9, 2, 2) is not a level O-sequence.
Proof. Let A = R/I be a local level K-algebra with the Hilbert function h. The lex-ideal L ∈ P =
K [ x1 , . . . , x4 ] with the Hilbert function h is
L = ( x12 , x1 x22 , x1 x2 x3 , x1 x2 x4 , x1 x32 , x1 x3 x4 , x1 x42 , x23 , x22 x3 , x22 x4 , x2 x32 , x2 x3 x4 , x2 x42 , x33 , x32 x4 , x3 x44 , x45 ).
A minimal graded P-free resolution of P/L is:
0 −→ P(−6)7 ⊕ P(−8)2 −→ P(−5)26 ⊕ P(−7)6 −→ P(−4)33 ⊕ P(−6)6 −→ P(−2) ⊕ P(−3)14 ⊕ P(−5)2.

By [RS10, Theorem 4.1] the Betti numbers of A can be obtained from the Betti numbers of P/L by
a sequence of negative and zero consecutive cancellations. This implies that β4 ( A) ≥ 3 and hence
A has type at least 3, which leads to a contradiction.

4. C ANONICALLY

GRADED ALGEBRAS WITH

m5 = 0.

It is clear that a necessary condition for a Gorenstein local K-algebra A being canonically graded
is that the Hilbert function of A must be symmetric. Hence we investigate whether a Gorenstein Kalgebra A with the Hilbert function (1, h1 , h2 , h1 , 1) is necessarily canonically graded. If h2 = (h12+1)
(equiv. A is compressed), then by [ER15, Theorem 3.1] A is canonically graded. In this section we
prove that if h = (1, h1 , h2 , h1 , 1) is an O-sequence with h1 ≤ h2 < (h12+1), then there exists a
polynomial F of degree 4 such that A F has the Hilbert function h and it is not canonically graded.
Theorem 4.1. Let h = (1, h1 , h2 , h3 , 1) be a Gorenstein O-sequence with h2 ≥ h1 . Then every local
Gorenstein K-algebra with Hilbert function h is necessarily canonically graded if and only if h1 = h3 and
h2 = (h12+1).

10

MASUTI AND ROSSI

Proof. The assertion is clear for h1 = 1. Hence we assume h1 > 1. The “if” part of the theorem
follows from [ER15, Theorem 3.1]. We prove the converse, that is we show that if h2 < (h12+1),
then there exists a polynomial G of degree 4 such that A G has the Hilbert function h and it is not
canonically graded. We may assume h3 = h1 , otherwise the result is clear. For simplicity in the
notation we put h1 := n and h2 := m.
First we prove the assertion for n ≤ 3. We define
 3
x1 x2



 x4 + x4 + x3 x
2 3
2
1
F=
4 + x4 + x3 x + x3 x

x

2
2 3
1 2

 14
x1 + x24 + x23 x3 + x13 x2 + x1 x22 x3

if n
if n
if n
if n

ϕ ( x n ) = u 1 x1 + · · · + u n x n +

∑

=m=2
= 3 and m = 3
= 3 and m = 4
= 3 and m = 5.

Let G = F + x3n . It is easy to check that A G has the Hilbert function h. We claim that A G is not
canonically graded. Suppose that A G is canonically graded. Then A F ∼
= grm ( A) ∼
= A G . Let
2
2
ϕ : A F −→ A G be a K-algebra automorphism. Since xn ◦ F = 0, xn ∈ AnnR ( F ). This implies that
ϕ( xn )2 ∈ AnnR ( G ) and hence ϕ( xn )2 ◦ G = 0. For i = (i1 , . . . , in ) ∈ N n , let |i | = i1 + · · · + in .
Suppose

i ∈N n ,

ai x i .

|i|≥2

Comparing the coefficients of the monomials of degree ≤ 2 in ϕ( xn )2 ◦ G = 0, it is easy to verify that u1 = · · · = un = 0. This implies that ϕ( xn ) has no linear terms and thus ϕ is not an
automorphisms, a contradiction.
Suppose n > 3. First we define a homogeneous polynomial F ∈ P of degree 4 such that A F
has the Hilbert function h and x2n does not divide any monomial in F (in other words, if xi is a
monomial that occurs in F with nonzero coefficient, then in ≤ 1).
Let T be a monomials basis of P2 . We split the set T \ { x2n } into a disjoint union of monomials as
follows. We set
 2
xi



x x
2 n
pi =

x
i − n x i + 1− n



x1 x n

for 1 ≤ i ≤ n − 1
for i = n
for n + 1 ≤ i < 2n
for i = 2n.

Let E = { pi : 1 ≤ i ≤ n}, B = { pi : n + 1 ≤ i ≤ 2n}, C = { xi x j : 1 ≤ i < j < n such that j − i > 1}
and D := { xi xn : 3 ≤ i ≤ n − 2}. Then
T \ { x2n } = E

S S S
·

B

·

C

·

D.

1
Denote by | · | the cardinality, then |C | = (n+
2 ) − 2n − ( n − 4) − 1 and | D | = n − 4. Hence we write
1
n +1
n +1
C = { pi : 2n < i ≤ (n+
2 ) − ( n − 4) − 1} and D = { pi : ( 2 ) − ( n − 4) − 1 < i ≤ ( 2 ) − 1}. We

11

set

Define

 4
xi





x23 xn



2


 xi− n pi
gi = x22 x32



x1 x22 xn





p2


 x2i 2
xn pi

for 1 ≤ i ≤ n − 1
for i = n
for n + 1 ≤ i < 2n and i 6= n + 2
for i = n + 2
for i = 2n
1
for 2n < i ≤ (n+
2 ) − ( n − 4) − 1
1
n +1
for (n+
2 ) − ( n − 4) − 1 < i < ( 2 ) .
m

F=

∑ gi .
i=1

Since m ≥ n, dimK (h F i R )i = n for i = 1, 3. Also, dimK (h F i R )2 = dimK { pi : 1 ≤ i ≤ m} = m.
Hence A F has the Hilbert function h.
Let G = F + x3n . We prove that A G is not canonically graded. Suppose that A G is canonically
graded. Then, as before, A G ∼
= A F . Let ϕ : A F −→ A G be a K-algebra automorphism. Since F does
not contain a monomial multiple of x2n , x2n ◦ F = 0 and hence x2n ∈ AnnR ( F ) which implies that
ϕ( xn )2 ∈ AnnR ( G ). Let
ϕ ( x n ) = u 1 x1 + · · · + u n x n +

∑

ai x i .

i∈N n , |i|≥2

We claim that u1 = · · · = un−1 = 0.
Case 1: m = n. Comparing the coefficients of x12 , x2 xn , x32 , . . . , x2n−1 in ϕ( xn )2 ◦ G = 0, we get u1 =
· · · = un−1 = 0.
Case 2: m = n + 1 OR m = n + 2. Compare the coefficients of x1 x2 , x2 xn , x32 , . . . , x2n−1 in ϕ( xn )2 ◦
G = 0, to get u1 = · · · = un−1 = 0.
Case 3: n + 2 < m < 2n. Comparing the coefficients of x1 x2 , x2 xn , x3 x4 , . . . , xm−n xm−n+1 , x2m−n+1,
x2m−n+2 . . . , x2n−1 in ϕ( xn )2 ◦ G = 0, we get u1 = · · · = un−1 = 0.
Case 4: m ≥ 2n. Comparing the coefficients of x1 x2 , x1 xn , x3 x4 , . . . , xn−1 xn in ϕ( xn )2 ◦ G = 0, we
get u1 = · · · = un−1 = 0.
This proves the claim. Now, comparing the coefficients of xn in ϕ( xn )2 ◦ G = 0, we get un = 0
(since F does not contain a monomial divisible by x2n ). This implies that ϕ( xn ) has no linear terms
and hence ϕ is not an automorphism, a contradiction.

We expect that the Theorem 4.1 holds true without the assumption h2 ≥ h1 . The problem is
that, as far as we know, the admissible Gorenstein not unimodal h-vectors are not classified even
if s = 4. However, starting from an example by Stanley, we are able to construct a not canonically
graded Gorenstein K-algebra with (not unimodal) h-vector (1, 13, 12, 13, 1).
Corollary 4.2. Let h = (1, h1 , h2 , h1 , 1) where h1 ≤ 13 be a Gorenstein O-sequence. Then every Gorenstein K-algebra with the Hilbert function h is necessarily canonically graded if and only if h2 = (h12+1).
Proof. If a local Gorenstein K-algebra A has Hilbert function h = (1, h1 , h2 , h1 , 1), then by considering Q-decomposition of grm ( A) we conclude that grm ( A) ∼
= Q(0). This implies that h is also the
Hilbert function of a graded Gorenstein K-algebra. By [MZ16, Theorem 3.2] if h1 ≤ 12, then the
Hilbert function of a graded Gorenstein K-algebra is unimodal. Hence by Theorem 4.1 the result
follows.

12

MASUTI AND ROSSI

If h1 = 13 and h is unimodal, then the assertion follows from Theorem 4.1. Now, by [MZ16, Theorem 3.2] the only nonunimodal graded Gorenstein O-sequence with h1 = 13 is h = (1, 13, 12, 13, 1).
In this case we write P = [ x1 , . . . , x10 , x, y, z]. Let
10

F=

∑ xi µi ,
i=1

3 .
where µ = { x3 , x2 y, x2 z, xy2 , xyz, xz2 , y3 , y2 z, yz2 , z3 } = {µ1 , . . . , µ10 }. Let G = F + x13 + · · · + x10
Then A G has the Hilbert function h. We claim that A G is not canonically graded. Suppose A G is
canonically graded. Then A G ∼
= A F . Let ϕ : A F −→ A G be a K-algebra automorphism. Since
x12 ∈ AnnR ( F ), ϕ( x1 )2 ∈ AnnR ( G ). Let

ϕ( x1 ) = u1 x1 + · · · + u10 x10 + u11 x + u12 y + u13 z +

∑

ai x i .

i∈N n , |i|≥2

Comparing the coefficients of x1 x, x7 y, x10 z in ϕ( x1 )2 ◦ G = 0, we get u11 = u12 = u13 = 0. Now,
comparing the coefficients of x1 , . . . , x10 in ϕ( x1 )2 ◦ G = 0, we get u1 = · · · = u10 = 0. This implies
that ϕ( x1 ) has no linear terms and thus ϕ is not an automorphism, a contradiction.

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D IPARTIMENTO DI M ATEMATICA , U NIVERSIT À DI G ENOVA , V IA D ODECANESO 35, 16146 G ENOVA , I TALY
E-mail address: masuti@dima.unige.it
D IPARTIMENTO DI M ATEMATICA , U NIVERSIT À DI G ENOVA , V IA D ODECANESO 35, 16146 G ENOVA , I TALY
E-mail address: rossim@dima.unige.it

