Artinian level algebras of socle degree 4

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by  Shreedevi K. Masuti

ARTINIAN LEVEL ALGEBRAS OF SOCLE DEGREE 4
SHREEDEVIK.MASUTIANDM.E.ROSSI
7
1 ABSTRACT. InthispaperwestudytheO-sequencesofthelocal(orgraded)K-algebrasofsoclede-
0 gree4.Moreprecisely,weprovethatanO-sequenceh=(1,3,h2,h3,h4),whereh4 ≥2,istheh-vector
2 ofalocallevelK-algebraifandonlyifh3 ≤3h4.Wealsoprovethath=(1,3,h2,h3,1)istheh-vector
ofalocalGorensteinK-algebraifandonlyifh ≤3andh ≤(h3+1)+(3−h ).Ineachofthesecases
n 3 2 2 3
a wegiveaneffectivemethodtoconstructalocallevelK-algebrawithagivenh-vector. Moreoverwe
J refinearesultbyEliasandRossibyshowingthatifh = (1,h ,h ,h ,1)isanunimodalGorenstein
1 2 3

1 O-sequence, then h forcesthe correspondingGorensteinK-algebrato becanonically gradedif and
1 onlyifh =h andh =(h1+1),thatistheh-vectorismaximal.
1 3 2 2

]
C
A
1. INTRODUCTION
.
h
t Let (A,m) be an Artinian local or graded K-algebra where K is an algebraically closed field of
a characteristic zero. Let Soc(A) = (0 : m) be the socle of A. We denote by s the socle degree of
m
A, that is the maximum integer j such that mj 6= 0. The type of A is τ := dim Soc(A). Recall
[ K
that A is said to be level of type τ if Soc(A) = ms and dim ms = τ. If A has type 1, equiva-
K
1 lently dim Soc(A) = 1, then A is Gorenstein. In the literature local rings with low socle degree,
v K
also called short local rings, have emerged as a testing ground for properties of infinite free res-
0
8 olutions (see [AIS08], [AHS13], [HS11], [Sjo79], [Roos05], [CRV01]). They have been also exten-
1 sivelystudiedinproblemsrelatedtotheirreducibilityandthesmoothnessofthepunctualHilbert
3 scheme Hilb (Pn) parameterizing zero-dimensionalsubschemesPn ofdegreed, seeamong oth-
0 d K K
. ers[Poo08],[EV08],[CEVV09],[CENR13]. InthispaperwestudythestructureoflevelK-algebras
1
ofsocledegree4, hencem5 = 0. Oneofthemostsignificant information onthestructureisgiven
0
7 bytheHilbertfunction.
1 Bydefinition, theHilbert function of A, h = h (A) = dim mi/mi+1, is the Hilbert function of
: i i K
v the associated graded ring grm(A) = ⊕i≥0mi/mi+1. We also say that h = (h0,h1,...,hs) is the h-
i
X vectorof A.In[Mac27]Macaulaycharacterizedthepossiblesequencesofpositiveintegershi that
r can occur as the Hilbert function of A. Since then there has been a great interest in commutative
a
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
hii
integers h = (h ,h ,...,h ) satisfying Macaulay’s criterion, that is h = 1 and h ≤ h for
0 1 s 0 i+1 i
i = 1,...,s−1, is called an O-sequence. A sequence h = (1,h ,...,h ) is said to be a level (resp.
1 s
Gorenstein) O-sequence if h is the Hilbert function of some Artinian level (resp. Gorenstein) K-
algebra A. Remark that h is the embedding dimension and, if A is level, h is the type of A.
1 s
Notice that a level O-sequenceis not necessarilythe Hilbert function of an Artinian levelgraded
Date:January13,2017.
2010MathematicsSubjectClassification. Primary:13H10;Secondary:13H15,14C05.
Keywordsandphrases. Macaulay’sinversesystem,Hilbertfunctions,ArtinianGorensteinandlevelalgebras,canon-
icallygradedalgebras.
ThefirstauthorwassupportedbyINdAMCOFOUNDFellowshipscofoundedbyMarieCurieactions, Italy. The
secondauthorwaspartiallysupportedbyPRIN2014“Geometriadellevarieta’algebriche”.
1
2 MASUTIANDROSSI
K-algebra. This is because the Hilbert function of the level ring (A,m) is the Hilbert funtion of
grm(A)whichisnotnecessarilylevel. FromnowonwesaythathisagradedlevelO-sequenceifh
istheHilbertfunctionofalevelgradedstandardK-algebra. Forinstanceitiswellknownthatthe
h-vectorofaGorensteingradedK-algebraissymmetric,butthisisnolongertrueforaGorenstein
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].
Inthefollowingtablewegiveasummaryofknownresults:
Characterization of Graded Local
GorensteinO-sequenceswithh = 2 [Sta78,Theorem4.2] [Ber09,Theorem2.6]
1
levelO-sequenceswithh = 2 [Iar84,Theorem4.6A],[Iar04] [Ber09,Theorem2.6]
1
GorensteinO-sequenceswithh = 3 [Sta78,Theorem4.2] Open
1
Open. In[GHMY07]authors
levelO-sequenceswithh = 3 gaveacompletelistwith
1
andτ ≥ 2 s ≤ 5ors = 6andτ = 2 Open
levelO-sequenceswiths ≤ 3 [Ste14],[GHMY07] [Ste14,Theorem4.3]
Inthispaperwefilltheabove table bycharacterizing theGorensteinandlevelO-sequenceswith
aparticularattentiontosocledegree4andembeddingdimensionh = 3.
1
In our setting we can write A = R/I where R = K[[x ,...,x ]] (resp. K[x ,...,x ]) is the formal
1 r 1 r
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[x ,...,x ].Withoutlossofgeneralityweassumethath = dim m/m2 = r.
1 r 1 K
Recallthatthesocletypeof A = R/I isthesequenceE = (0,e ,...,e ),where
1 s
e := dim ((0 : m)∩mi)/(0 : m)∩mi+1).
i K
Itisknownthatforalli ≥ 0,
(1.1) h ≤ min{dim R ,e dim R +e dim R +···+e dim R }
i K i i K 0 i+1 K 1 s K s−i
(see[Iar84]). Henceanecessaryconditionfor h tobealevelO-sequenceisthat h ≤ h h being
s−1 1 s
e = h and e = 0 otherwise. In the following theorem we prove that this condition is also
s s i
sufficient for h = (1,3,h ,h ,h ) to be a level O-sequence, provided h ≥ 2. However, if h = 1,
2 3 4 4 4
weneedanadditionalassumptionforhtobeaGorensteinO-sequence. Weremarkthattheresult
>
cannotbeextendedtoh 3(seeExample3.7).
1
Theorem1. Leth = (1,3,h ,h ,h )beanO-sequence.
2 3 4
(a) Leth = 1.Thenh isaGorensteinO-sequenceifandonlyif h ≤ 3and h ≤ (h3+1)+(3−
4 3 2 2
h ).
3
(b) Leth ≥ 2. ThenhisalevelO-sequenceifandonlyifh ≤ 3h .
4 3 4
Theproofoftheaboveresultiseffectiveinthesensethatineachcaseweconstructalocallevel
K-algebrawithagivenh-vectorverifyingthenecessaryconditions(seeTheorem3.6).
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
observedin[Ste14]forsocledegree3.Theorem1(a)isaconsequenceofthefollowingmoregeneral
resultwhichholdsforanyembeddingdimension:
3
Theorem2. (a) If(1,h ,...,h ,h ,1)isaGorensteinO-sequence,then
1 s−2 s−1
h +1
(1.2) h ≤ h andh ≤ s−1 +(h −h ).
s−1 1 s−2 1 s−1
2
(cid:18) (cid:19)
(b) If h = (1,h ,h ,h ,1) is an unimodal O-sequence satisfying (1.2), then h is a Gorenstein
1 2 3
O-sequence.
Thebound(1.2)improvestheboundforh givenin(1.1).
s−2
If A is a Gorenstein local K-algebra with symmetric h-vector, then grm(A) is Gorenstein, see
[Iar94]. Itisanaturalquestiontoask,inthiscase,whether Aisanalyticallyisomorphictogrm(A).
Accordingly with thedefinition 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 be-
tween Aanditsassociatedgradedring grm(A).
ForinstanceJ.EliasandM.E.Rossiin[ER12]provedthateveryGorensteinK-algebrawithsym-
metrich-vectorandm4 = 0(s ≤ 3)iscanonicallygraded. AlocalK-algebraAofsocletypeEissaid
tobecompressedifequalityholdsin(1.1)forall1 ≤ i ≤ s,equivalentlytheh-vectorismaximal(see
[Iar84,Definition2.3]). In[ER15,Theorem3.1]EliasandRossiprovedthatif Aisanycompressed
Gorenstein local K-algebra of socle degree s ≤ 4, then A is canonically graded. In Section 4 we
provethatifthesocledegreeis4,thentheassumptioncannotberelaxed. Morepreciselyonlythe
maximal h-vector forceseverycorrespondingGorensteinK-algebratobecanonically graded. We
provethatifhisunimodalandnotmaximal,thenthereexistsaGorensteinK-algebrawithHilbert
function h which is not canonically graded. To prove that a local K-algebra is not canonically
gradedisingeneralaverydifficulttask. Seealso[Jel16]forinterestingdiscussions.
Theorem 3. Let h = (1,h ,h ,h ,1) be a Gorenstein O-sequence with h ≥ h . Then every local
1 2 3 2 1
GorensteinK-algebrawithHilbertfunction hisnecessarilycanonically gradedifandonlyifh =
1
h andh = (h1+1).
3 2 2
The main tool of the paper is Macaulay’s inverse system [Mac1916] which gives a one-to-one
correspondencebetweenidealsI ⊆ RsuchthatR/IisanArtinianlocalringandfinitelygenerated
R-submodules of a polynomial ring. In Section 2 we gather preliminary results needed for this
purpose. WeproveTheorems1and2inSection3,andTheorem3inSection4.
We have used Singular [Singular], [Eli15] and CoCoA [CoCoA] for various computations and
examples.
ACKNOWLEDGEMENTS
We thank Juan Elias for providing us the updated version of INVERSE-SYST.LIB for computa-
tions.
2. PRELIMINARIES
2.1. Macaulay’sInverseSystem. InthissubsectionwerecallsomeresultsonMacaulay’sinverse
systemwhich wewill useinthesubsequentsections. Thistheoryiswell-knownintheliterature,
especiallyinthegradedsetting(seeforexample[Mac1916,ChapterIV]and[IK99]). However,the
localcaseisnotsowellexplored. Wereferthereaderto[Ems78],[Iar94]foranextendedtreatment.
4 MASUTIANDROSSI
It is known that the injective hull of K as R-module is isomorphic to a divided power ring
P = K[X ,...,X ]whichhasastructureofR-modulebymeansofthefollowingaction
1 r
◦ : R×P −→ P
(f,g) −→ f ◦g = f(∂ ,...,∂ )(g)
X1 Xr
where ∂ denotes the partial derivative with respect to X . For the sake of simplicity from now
Xi i
on we will use x instead of the capital letters X . If {f ,..., f } ⊆ P is a set of polynomials, we
i i 1 t
will denote by hf ,..., f i the R-submodule of P generated by f ,..., f , i.e., the K-vector space
1 t R 1 t
generatedby f ,..., f and by the correspondingderivatives of all orders. We consider the exact
1 t
pairingofK-vectorspaces:
h, i : R×P −→ K
(f,g) −→ (f ◦g)(0).
Foranyideal I ⊂ Rwedefinethefollowing R-submoduleof PcalledMacaulay’sinversesystem:
I⊥ := {g ∈ P |hf,gi = 0 ∀f ∈ I }.
Conversely,foreveryR-submodule Mof Pwedefine
Ann (M) := {g ∈ R|hg, fi = 0 ∀f ∈ M}
R
which is an ideal of R. If M is generatedby polynomials f := f ,..., f , with f ∈ P, thenwe will
1 t i
writeAnn (M) = Ann (f)and A = R/Ann (f).
R R f R
ByusingMatlisdualityoneprovesthatthereexistsaone-to-onecorrespondencebetweenideals
I ⊆ R such that R/I is an Artinian local ring and R-submodules M of P which are finitely gen-
erated. More precisely, Emsalem in [Ems78, Proposition 2] and Iarrobino in [Iar94, Lemma 1.2]
provedthat:
Proposition2.1. Thereisaone-to-onecorrespondence betweenideals I suchthat R/I isalevel localring
ofsocle degree s andtype τ and R-submodulesof Pgenerated byτ polynomials ofdegree s havinglinearly
independentformsofdegree s.Thecorrespondence isdefinedasfollows:
I ⊆ Rsuchthat R/I M ⊆ Psubmodulegeneratedby
1−1
isalevellocalringof ←→ τ polynomialsofdegree
   
socledegree sandtypeτ swithl.i. formsofdegree s
   
I −→ I⊥
   
Ann (M) ←− M
R
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)∗denotesthedualwithrespecttothepairingh, iinducedonR/I). Hencedim R/I =
K
dim I⊥. Asin thegradedcase, it is possibletocomputetheHilbert function of A = R/I via the
K
inversesystem. WedefinethefollowingK-vectorspace:
(I⊥) := I⊥∩P≤i+P<i.
i
P<i
Then,by(2.2),itisknownthat
(2.3) h (R/I) = dim (I⊥) .
i K i
5
2.2. Q-decomposition. It is well-known that the Hilbert function of an Artinian graded Goren-
steinK-algebraissymmetric,whichisnottrueinthelocalcase. Theproblemcomesfromthefact
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
byadescendingsequenceofideals:
G = C(0) ⊇ C(1) ⊇ ··· ⊇ C(s) = 0,
where
(0 : ms+1−a−i)∩mi
C(a) := .
i (0 : ms+1−a−i)∩mi+1
Let
Q(a) = C(a)/C(a+1).
Then
{Q(a) : a = 0,...,s−1}
iscalledQ-decomposition oftheassociatedgradedring G.Wehave
s−1
h (A) = dim G = ∑ dim Q(a) .
i K i K 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 toa shift in degree: Hom (Q(a) ,K) =
K i
Q(a) .HencetheHilbertfunction of Q(a) issymmetricabout s−a. Moreover,heshowedthat
s−a−i 2
Q(0) = G/C(1) is the unique (up isomorphism)socle degrees gradedGorensteinquotientof G.
Let f = f[s] + lowerdegreeterms... be a polynomial in P of degree s where f[s] is the homo-
geneous part of degree s and consider A the corresponding Gorenstein local K-algebra. Then,
f
∼
Q(0) = R/Ann (f[s]) (see[Ems78,Proposition7]and[Iar94,Lemma1.10]).
R
3. CHARACTERIZATION OF LEVEL O-SEQUENCES
In this section we characterize Gorenstein and level O-sequences of socle degree 4 and em-
bedding dimension 3. First if h = (1,h ,...,h ) is a Gorenstein O-sequence (in any embedding
1 s
dimension),wegiveanupperboundonh intermsofh whichimprovesthealreadyknown
s−2 s−1
inequalitygivenin(1.1).
Theorem3.1. (a) If(1,h ,...,h ,h ,1)isaGorensteinO-sequence,then
1 s−2 s−1
h +1
(3.2) h ≤ h andh ≤ s−1 +(h −h ) .
s−1 1 s−2 1 s−1
2
(cid:18)(cid:18) (cid:19) (cid:19)
(b) If h = (1,h ,h ,h ,1) is an O-sequence such that h ≥ h and it satisfies (3.2), then h is a
1 2 3 2 3
GorensteinO-sequence.
Proof. (a): From (1.1) it follows that h ≤ h . Let A be a Gorenstein local K-algebra with the
s−1 1
Hilbertfunctionhand{Q(a) : a = 0,...,s−1}bea Q-decompositionofgrm(A).Since Q(i)s−1 =
0fori > 0,dim Q(0) = h .Hencedim Q(0) = dim Q(0) = h anddim Q(0) =
K s−1 s−1 K 1 K s−1 s−1 K s−2
dim Q(0) ≤ hh1i .Thisinturnimpliesthatdim Q(1) = dim Q(1) ≤ h −h .Therefore
K 2 s−1 K s−2 K 1 1 s−1
h = dim Q(0) +dim Q(1)
s−2 K s−2 K s−2
h +1
≤ s−1 +(h −h ).
1 s−1
2
(cid:18) (cid:19)
6 MASUTIANDROSSI
(b): Supposeh ≤ h .Set
2 1
f = x4+···+x4 +x3 +···+x3 +x2 +···+x2 .
1 h3 h3+1 h2 h2+1 h1
Then A has theHilbert function h. Nowassume h > h .Denote h := n and definemonomials
f 2 1 3
g ∈ K[x ,...,x ]asfollows:
i 1 n
x2 if1 ≤ i ≤ n
i
g = x x ifn+1 ≤ i ≤ 2n−1
i  i−n i−n+1
x x ifi = 2n.
n 1
For i = (i1,...,in) ∈ Nn, letxi := x1i1...xnin. Let T be the set of monomials xi of degree 2 in
K[x ,...,x ]suchthat
1 n
xi ∈/ {g : 1 ≤ i ≤ 2n}.
i
Then|T| = (n+1)−2n.WewriteT = {g : 2n < i ≤ (n+1)}.Define
2 i 2
n h2−h1
∑x2g + ∑ x2g +x3 +···+x3 ifh −h ≤ n
 i i i n+i n+1 h1 2 1
f = i=1 i=1
∑n x2g +∑n x2g +h2−∑h1+ng2+x3 +···+x3 ifh −h > n.
i i i n+i i n+1 h1 2 1
i=1 i=1 2n+1


Thenh3 = dimK ∂∂xf1,..., ∂∂xfn = nand
(cid:16) (cid:17)
h2 = dimK {gi : 1 ≤ i ≤ h2−h1+n} · {x2n+1,...,x2h1} .
Thus Af hastheHilbertfunc(cid:16)tion(1,h1,h2,h3,1). S (cid:17) (cid:3)
Remark 3.3. If h = h , then f in the proofof Theorem3.1(b) is homogeneousand hence A is a
3 1 f
gradedGorensteinK-algebra.
Ifthesocledegreeis4andh ≤ 12,thentheconverseholdsin(a).
1
Corollary3.4. AnO-sequence h = (1,h ,h ,h ,1) withh ≤ 12isaGorenstein O-sequenceifandonly
1 2 3 1
ifhsatisfies(3.2).
Proof. By Theorem 3.1, it suffices to show that h ≥ h if h ≤ 12. Let A be a local Gorenstein K-
2 3 1
algebra with the Hilbert function h. Considering the symmetric Q-decomposition of grm(A), we
observethatinthiscase Q(0) hastheHilbertfunction(1,h ,h −m,h ,1),forsomenon-negative
3 2 3
integer m. Since h ≤ 12, by (3.2) h ≤ 12. As Q(0) is a graded Gorenstein K-algebra, by [MZ16,
1 3
Theorem 3.2] we conclude that h −m ≥ h since Q(0) has unimodal Hilbert function, hence
2 3
h ≥ h . (cid:3)
2 3
Remark 3.5. A Gorenstein O-sequence (1,h ,h ,h ,1) does not necessarily satisfy h ≥ h . For
1 2 3 2 3
example, consider the the sequence h = (1,13,12,13,1). By [Sta78, Example 4.3] there exists a
gradedGorensteinK-algebrawithhash-vector.
In the following theorem we characterize the h-vector of local level algebras of socle degree 4
andembeddingdimension3.
Theorem3.6. Leth = (1,3,h ,h ,h )beanO-sequence.
2 3 4
(a) Leth = 1.ThenhisaGorensteinO-sequenceifandonlyifh ≤ 3andh ≤ (h3+1)+(3−h ).
4 3 2 2 3
(b) Leth ≥ 2. ThenhisalevelO-sequenceifandonlyifh ≤ 3h .
4 3 4
7
Proof. (a): FollowsfromCorollary3.4.
(b): The“onlyif”partfollowsfrom(1.1). Theconverseisconstructiveandweproveitbyinductive
steps on h . First we consider the cases h = 2,3,4 and then h ≥ 5. In each case we define the
4 4 4
polynomialsf := f ,..., f ∈ P = K[x ,x ,x ]ofdegree4suchthat A hastheHilbertfunctionh.
1 h4 1 2 3 f
Wesetg′ = x3,g′ = x2x ,g′ = x2x ,g′ = x x2.
1 3 2 2 3 3 1 2 4 1 3
Forshort,inthisproofweusethefollowingnotation: m := h andn := h .
2 3
Case1: h = 2.
4
In this case n ≤ 6 as h ≤ 3h by assumption. Suppose m = 2. Then h is an O-sequence implies
3 4
thatn = 2.Inthiscase,let f = x4+x2 and f = x4.Then A hastheHilbertfunction(1,3,2,2,2).
1 1 3 2 2 f
Nowassumethatm ≥ 3.
x g′ if1 ≤ i ≤ n−2
4−i i
Subase1: m ≥ n.Wesetg = g′ ifn−1 ≤ i ≤ m−2
i  i
0 ifm−1 ≤ i ≤ 4.
(Here x = x ). Define
0 3 

f = x4+g +g and f = x4+g +g .
1 1 1 2 2 2 3 4
Then
h = dim {x3,x3,g′,...,g′ } = nand
3 K 1 2 1 n−2
g′
h = dim {x2,x2, i : 1 ≤ i ≤ m−2} = m
2 K 1 2 x
4−i
andhence A hastherequiredHilbertfunctionh.
f
Subcase2: m < n. The only possible ordered tuples (m,n) with m < n ≤ 6 such that h is an O-
sequenceare{(3,4),(4,5),(5,6)}. Foreach2-tuple(m,n)wedefine f , f as:
1 2
a.(m,n) = (3,4) : f = x4+x2x2+x2; f = x4+x2x2.
1 1 1 2 3 2 2 1 2
b.(m,n) = (4,5) : f = x4+x2x2+x4; f = x4+x2x2.
1 1 1 2 3 2 2 1 2
c.(m,n) = (5,6) : f = x4+x2x2+x4; f = x4+x2x2+x3x .
1 1 1 2 3 2 2 1 2 2 3
Case2: h = 3.
4
Inthiscasen ≤ 9.Weconsiderthefollowingsubcases:
Subcase1: n ≤ 6.Letf′ = f1, f2 bepolynomialsdefinedasinCase 1suchthat Af′ hastheHilbert
x4 ifm ≥ n
function(1,3,m,n,2).Nowdefine f = 3
3 (x2x2 ifm < n.
1 2
Then A hastherequiredHilbertfunctionh.
f
Subcase2: 7 ≤ n ≤ 9. Let f′ = f1, f2 be polynomials defined as in Case 1 such that Af′ has the
Hilbertfunction(1,3,m,6,2).Weset p = x2x2,p = x2x2and p = x2x2.SincehisanO-sequence
1 2 3 2 1 2 3 1 3
∑n−6p ifm = 6
andn ≥ 7,wegetm ≥ 5.Nowdefine f = i=1 i
3 (x2x2 ifm = 5.
2 3
Then A hastherequiredHilbertfunctionh.
f
Case3: h = 4.
4
8 MASUTIANDROSSI
SincehisanO-sequence,n ≤ 10.Weconsiderthefollowingsubcases:
Subcase1: n ≤ 9. Let f′ = f1, f2, f3 be polynomials defined as in Case 2 such that Af′ has the
Hilbertfunction(1,3,m,n,3).Define
x3x if{m ≥ nandn ≤ 6}OR{n ≥ 7andm = 6}
f = 2 3
4 (x13x2 if{m < n ≤ 6}OR{(m,n) = (5,7)}.
Then A hastheHilbertfunction(1,3,m,n,4).
f
Subcase2: n = 10. As h is an O-sequence, we conclude that m = 6. Let f′ = f , f , f be polyno-
1 2 3
mials definedas in Case 2such that Af′ has theHilbert function (1,3,6,9,3). Define f4 = x12x2x3.
Then A hastheHilbertfunction(1,3,6,10,4).
f
Case4: h ≥ 5.
4
SincehisanO-sequence,n ≤ 10andh ≤ 15
4
Subcase1: n ≥ h4 ORh4 ≥ 11. Let f′ = f1, f2, f3, f4 be defined as in Case 3 such that Af′ has the
Hilbertfunction(1,3,m,n,4).For5 ≤ i ≤ 15,define f asfollows:
i
x3x if{m ≥ nandn ≤ 6}OR{n ≥ 7andm = 6}
f = 1 2
5 (x1x23 if{m < n ≤ 6}OR{(m,n) = (5,7)},
x x3 if{m ≥ nandn ≤ 6}OR{n ≥ 7andm = 6}
f = 1 3
6 (x4 if{m < n ≤ 6}OR{(m,n) = (5,7)},
3
x4 ifn ≥ 7andm = 6
f = 3
7 (x23x3 if{(m,n) = (5,7)}.
(Note that in the last case, h ≥ 7 implies that n ≥ 7). If h ≥ 8, then n ≥ 8 which implies that
4 4
m = 6.Weset
f = x2x2, f = x2x2, f = x x3, f = x x3, f = x3x , f = x3x , f = x x2x , f = x x x2.
8 1 2 9 1 3 10 2 3 11 1 2 12 1 3 13 2 3 14 1 2 3 15 1 2 3
Now A hastheHilbertfunction(1,3,m,n,h ).
f 4
Subcase2: n < h ≤ 10. Thesmallestorderedtuple (n,h ) suchthat h isan O-sequenceand n <
4 4
h is (4,5). (Here smallest ordered tuple means smallest with respect to the order ≤ defined as:
4
(n ,n ) ≤ (m ,m )ifandonlyifn ≤ m andn ≤ m ). Let
1 2 1 2 1 1 2 2
x2 ifm = 3 0 ifm < 5 0 ifm < 6
q = 3 q = andq =
1 (x33 ifm ≥ 4, 2 (x22x3 ifm ≥ 5 3 (x1x32 ifm = 6.
Wedefine
f = x4+q +q , f = x4+q , f = x3x , f = x x3, f = x2x2.
1 1 1 2 2 2 3 3 1 2 4 1 2 5 1 2
Then A hastheHilbertfunction(1,3,m,4,5).
f
Leth ≥ 6.Weset
4
x2x2 ifn = 7 x2x2 ifn = 8
f = x4, f = x3x , f = x x3, f = 2 3 f = 2 3
6 3 7 2 3 8 2 3 9 (x1x33 ifn ≥ 8, 10 (x13x3 ifn = 9.
Then A hastheHilbertfunction(1,3,m,n,h ). (cid:3)
f 4
9
Using [GHMY07, Appendix D] and Theorem 3.6(b) we list in the following table all the O-
sequences which are realizable for local level K-algebras with h ≥ 2, but not for graded level
4
K-algebras.
TABLE 1
(1,3,2,2,2) (1,3,3,2,2) (1,3,4,2,2) (1,3,5,2,2) (1,3,6,2,2) (1,3,5,3,2)
(1,3,6,3,2) (1,3,3,4,2) (1,3,4,3,3) (1,3,5,3,3) (1,3,6,3,3) (1,3,3,4,3)
(1,3,6,4,3) (1,3,3,4,4) (1,3,5,4,4) (1,3,6,4,4) (1,3,3,4,5) (1,3,4,4,5)
(1,3,5,4,5) (1,3,6,4,5) (1,3,6,5,5) (1,3,5,5,6) (1,3,6,5,6) (1,3,6,6,7)
(1,3,6,7,9)
Analogously,byTheorem3.6(a),thefollowingO-sequencesareGorensteinsequences,butthey
arenotgradedGorensteinsequencessincetheyarenotsymmetric. IfanO-sequencehissymmet-
ricwithh = 3thenhisalsoagradedGorensteinO-sequencebyRemark3.3.
1
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 h ≥ 4 because the
1
necessaryconditionh ≤ h h isnotlongersufficientforcharacterizinglevelO-sequencesofsocle
3 1 s
degree4.
Example3.7. TheO-sequenceh = (1,4,9,2,2) isnotalevelO-sequence.
Proof. Let A = R/I be a local level K-algebra with the Hilbert function h. The lex-ideal L ∈ P =
K[x ,...,x ]withtheHilbertfunctionhis
1 4
L = (x2,x x2,x x x ,x x x ,x x2,x x x ,x x2,x3,x2x ,x2x ,x x2,x x x ,x x2,x3,x2x ,x x4,x5).
1 1 2 1 2 3 1 2 4 1 3 1 3 4 1 4 2 2 3 2 4 2 3 2 3 4 2 4 3 3 4 3 4 4
Aminimalgraded P-freeresolutionofP/Lis:
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,Theorem4.1]theBettinumbersof AcanbeobtainedfromtheBettinumbersofP/Lby
asequenceofnegativeandzeroconsecutivecancellations. Thisimpliesthat β (A) ≥ 3andhence
4
(cid:3)
Ahastypeatleast3,whichleadstoacontradiction.
4. CANONICALLY GRADED ALGEBRAS WITH m5 = 0.
ItisclearthatanecessaryconditionforaGorensteinlocalK-algebraAbeingcanonicallygraded
isthattheHilbertfunctionofAmustbesymmetric. HenceweinvestigatewhetheraGorensteinK-
algebra AwiththeHilbertfunction(1,h ,h ,h ,1)isnecessarilycanonicallygraded. Ifh = (h1+1)
1 2 1 2 2
(equiv. Aiscompressed),thenby[ER15,Theorem3.1] Aiscanonically graded. Inthissectionwe
prove that if h = (1,h ,h ,h ,1) is an O-sequence with h ≤ h < (h1+1), then there exists a
1 2 1 1 2 2
polynomial F ofdegree4suchthat A hastheHilbertfunctionhanditisnotcanonically graded.
F
Theorem 4.1. Let h = (1,h ,h ,h ,1) be a Gorenstein O-sequence with h ≥ h . Then every local
1 2 3 2 1
Gorenstein K-algebra with Hilbert function h is necessarily canonically graded if and only if h = h and
1 3
h = (h1+1).
2 2
10 MASUTIANDROSSI
Proof. The assertion is clear for h = 1. Hence we assume h > 1. The “if” part of the theorem
1 1
follows from [ER15, Theorem 3.1]. We prove the converse, that is we show that if h < (h1+1),
2 2
thenthereexistsa polynomial G of degree4 suchthat A has theHilbert function h and it is not
G
canonically graded. We may assume h = h , otherwise the result is clear. For simplicity in the
3 1
notationweputh := nandh := m.
1 2
Firstweprovetheassertionforn ≤ 3.Wedefine
x3x ifn = m = 2
1 2
x4+x4+x3x ifn = 3andm = 3
F =  1 2 2 3
x14+x24+x23x3+x13x2 ifn = 3andm = 4
x4+x4+x3x +x3x +x x2x ifn = 3andm = 5.
1 2 2 3 1 2 1 2 3




Let G = F+ x3. It is easy to check that A has the Hilbert function h. We claim that A is not
n G G
∼ ∼
canonically graded. Suppose that AG is canonically graded. Then AF = grm(A) = AG. Let
ϕ : A −→ A be a K-algebra automorphism. Since x2 ◦F = 0, x2 ∈ Ann (F). This implies that
F G n n R
ϕ(x )2 ∈ Ann (G) and hence ϕ(x )2 ◦G = 0. For i = (i ,...,i ) ∈ Nn, let |i| = i +···+i .
n R n 1 n 1 n
Suppose
ϕ(x ) = u x +···+u x + ∑ a xi.
n 1 1 n n i
i∈Nn,|i|≥2
Comparing the coefficients of the monomials of degree ≤ 2 in ϕ(x )2 ◦G = 0, it is easy to ver-
n
ify that u = ··· = u = 0. This implies that ϕ(x ) has no linear terms and thus ϕ is not an
1 n n
automorphisms,acontradiction.
Suppose n > 3. First we define a homogeneous polynomial F ∈ P of degree 4 such that A
F
has the Hilbert function h and x2 does not divide any monomial in F (in other words, if xi is a
n
monomialthatoccursin Fwithnonzerocoefficient,theni ≤ 1).
n
LetTbeamonomialsbasisofP .WesplitthesetT\{x2}intoadisjointunionofmonomialsas
2 n
follows. Weset
x2 for1 ≤ i ≤ n−1
i
x x fori = n
p =  2 n
i xi−nxi+1−n forn+1 ≤ i < 2n

x x fori = 2n.
1 n




LetE = {p : 1 ≤ i ≤ n},B = {p : n+1 ≤ i ≤ 2n},C = {x x : 1 ≤ i < j < nsuchthat j−i > 1}
i i i j
and D := {x x : 3 ≤ i ≤ n−2}.Then
i n
T\{x2n} = E · B · C · D.
S S S
Denoteby|·|thecardinality,then|C| = (n+1)−2n−(n−4)−1and|D| = n−4.Hencewewrite
2
C = {p : 2n < i ≤ (n+1)−(n−4)−1} and D = {p : (n+1)−(n−4)−1 < i ≤ (n+1)−1}. We
i 2 i 2 2