arXiv:1307.1676v2 [] 5 Jan 2014

ON THE RATIONALITY OF POINCARÉ SERIES
OF GORENSTEIN ALGEBRAS
VIA MACAULAY’S CORRESPONDENCE
GIANFRANCO CASNATI, JOACHIM JELISIEJEW, ROBERTO NOTARI

Abstract. Let A be a local Artinian Gorenstein ring with P
algebraically closed
∞
p
residue field A/M = k of characteristic 0, and let PA (z) := p=0 (TorA
p (k, k))z
be its Poicaré series. We prove that PA (z) is rational if either dimk (M2 /M3 ) ≤ 4
and dimk (A) ≤ 16, or there exist m ≤ 4 and c such that the Hilbert function
HA (n) of A is equal to m for n ∈ [2, c] and equal to 1 for n > c. The results are
obtained thanks to a decomposition of the apolar ideal Ann(F ) when F = G + H
and G and H belong to polynomial rings in different variables.

1. Introduction and notation
Throughout this paper, by ring we mean a Noetherian, associative, commutative
and unitary ring A with maximal ideal M and algebraically closed residue field
k := A/M of characteristic 0.
In [17] the author asked if the Poincaré series of the local ring A, i.e.
PA (z) :=

∞
X

p
dimk (TorA
p (k, k))z ,

p=0

is rational. Moreover he also proved its rationality when A is a regular local ring.
Despite many interesting results showing the rationality of the Poincaré series of
some rings, in [1] the author gave an example of an Artinian local algebra A with
transcendental PA . Later on the existence of an Artinian, Gorenstein, local ring
with M4 = 0 and transcendental PA was proved in [4].
Nevertheless, several results show that large classes of local rings A have rational Poincaré series, e.g. complete intersections rings (see [19]), Gorenstein local rings with dimk (M/M2) ≤ 4 (see [2] and [15]), Gorenstein local rings with
2000 Mathematics Subject Classification. Primary 13D40, Secondary 13H10.
Key words and phrases. Artinian Gorenstein local algebra, rational Poincaré series.
The first and third authors are members of GNSAGA group of INdAM. They are supported by
the framework of PRIN 2010/11 “Geometria delle varietà algebriche”, cofinanced by MIUR.
The second author is supported by the project “Secant varieties, computational complexity, and
toric degenerations” realised within the Homing Plus programme of Foundation for Polish Science,
co-financed from European Union, Regional Development Fund.
This paper is a part of “Computational complexity, generalised Waring type problems and tensor
decompositions” project within “Canaletto”, the executive program for scientific and technological
cooperation between Italy and Poland, 2013-2015.
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G. CASNATI, J. JELISIEJEW, R. NOTARI

dimk (M2 /M3 ) ≤ 2 (see [16], [10]), Gorenstein local rings of multiplicity at most 10
(see [6]), Gorenstein local algebras with dimk (M2 /M3 ) = 4 and M4 = 0 (see [8]).
All the above results are based on the same smart combination of results on the
Poincaré series from [3] and [13] first used in [16] combined with suitable structure
results on Gorenstein rings and algebras. In this last case a fundamental role has
been played by Macaulay’s correspondence.
In Section 2 we give a quick resumé of the main results that we need later on in
the paper about Macaulay’s correspondence. In Section 3 we extend to arbitrary
algebras a very helpful decomposition result already used in a simplified form in
[9] and [8] for algebras with M4 = 0. In Section 4 we explain how to relate the
rationality of the Poincaré series of Gorenstein algebras with their representation in
the setup of Macaulay’s correspondence making use of the aforementioned decomposition result. Finally, in Section 5 we use such relationship in order to prove the
two following results.
Theorem A. Let A be an Artinian, Gorenstein local k–algebra with maximal ideal
M. If there are integers m ≤ 4 and c ≥ 1 such that

m
if t = 2, . . . , c,
t
t+1
dimk (M /M ) =
1
if t = c + 1,
then PA is rational.
Theorem B. Let A be an Artinian, Gorenstein local k–algebra with maximal ideal
M. If dimk (M2 /M3 ) ≤ 4 and dimk (A) ≤ 16, then PA is rational.
The above theorems generalize the quoted results on stretched, almost–stretched
and short algebras (see [16], [10], [6], [8]).
1.1. Notation. In what follows k is an algebraically closed field of characteristic
0. A k–algebra is an associative, commutative and unitary algebra over k. For
each N ∈ N we set S[N] := k[[x1 , . . . , xN ]] and P [N] := k[y1 , . . . , yN ]. We denote
by S[N]q (resp. P [N]q ) the homogeneous
component of degree
Lqq of such a graded
Lq
k–algebra, and we set S[N]≤q := i=1 S[N]i (resp. P [n]≤q := i=1 P [n]i ). Finally,
we set S[n]+ := (x1 , . . . , xn ) ⊆ S[n]. The ideal S[n]+ is the unique maximal ideal of
S[N].
A local ring R is Gorenstein if its injective dimension as R–module is finite.
If γ := (γ1 , . . . , γN ) ∈ NN is a multi–index, then we set tγ := tγ11 . . . tγNN ∈
k[t1 , . . . , tN ].
For all the other notations and results we refer to [12].
2. Preliminary results
In this section we list the main results on algebras we need in next sections. Let A
be a local, Artinian k–algebra with maximal ideal M. We denote by HA the Hilbert
function of the graded associated algebra
gr(A) :=

+∞
M
t=0

Mt /Mt+1 .

RATIONALITY OF THE POINCARÉ SERIES

3

We know that

A∼
= S[n]/J
2
for a suitable ideal J ⊆ S[n]+ ⊆ S[n], where n = emdim(A) := HA (1). Recall that
the socle degree sdeg(A) of A is the greatest integer s such that Ms 6= 0.
We have an action of S[n] over P [n] given by partial derivation defined by identifying xi with ∂/∂yi . Hence


α! αβ y β−α
if β ≥ α,
α
β
x ◦ y :=
0
if β 6≥ α.
Such an action endows P [n] with a structure of module over S[n]. If J ⊆ S[n] is an
ideal and M ⊆ P [n] is a S[n]–submodule we set
J ⊥ := { F ∈ P [n] | g ◦ F = 0, ∀g ∈ J },
Ann(M) := { g ∈ S[n] | g ◦ F = 0, ∀F ∈ M }.
For the following results see e.g. [11], [14] and the references therein. Macaulay’s
theory of inverse system is based on the fact that constructions J 7→ J ⊥ and M 7→
Ann(M) give rise to a inclusion–reversing bijection between ideals J ⊆ S[n] such
that S[n]/J is a local Artinian k–algebra and finitely generated S[n]–submodules
M ⊆ P [n]. In this bijection Gorenstein algebras A with sdeg(A) = s correspond
to cyclic S[n]–submodules hF iS[n] ⊆ P [n] generated by a polynomial F of degree s.
We simply write Ann(F ) instead of Ann(hF iS[n]).
On the one hand, given a S[n]–module M, we define
M ∩ P [n]≤q + P [n]≤q−1
P [n]≤q−1
L∞
Lq
where P [n]≤q :=
q=0 tdf(M)q . The module tdf(M)
i=0 P [n]i , and tdf(M) :=
can be interpreted as the S[n]–submodule of P [n] generated by the top degree forms
of all polynomials in M.
On the other hand, for each f ∈ S[n], the lowest degree of monomials appearing
with non–zero coefficient in the minimal
P representation of f is called the order of f
and it is denoted by ord(f ). If f = ∞
i=ord(f ) fi , fi ∈ S[n]i then ford(f ) is called the
lower degree form of f . It will be denoted in what follows with ldf(f ).
If f ∈ J, then ord(f ) ≥ 2. The lower degree form ideal ldf(J) associated to J is
tdf(M)q :=

ldf(J) := (ldf(f )|f ∈ J) ⊆ S[n].
We have ldf(Ann(M)) = Ann(tdf(M)) (see [11]: see also [9], Formulas (2) and
(3)) whence
gr(S[n]/Ann(M)) ∼
= S[n]/Ann(tdf(M)).
= S[n]/ldf(Ann(M)) ∼
Thus
(1)

HS[n]/Ann(M ) (q) = dimk (tdf(M)q ).

We say that M is non–degenerate if HS[n]/Ann(M ) (1) = dimk (tdf(M)1 ) = n, i.e. if
and only if the classes of y1 , . . . , yn are in tdf(M). If M = hF iS[n], then we write
tdf(F ) instead of tdf(M).

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G. CASNATI, J. JELISIEJEW, R. NOTARI

Let A be Gorenstein with s := sdeg(A),
so that Soc(A) = Ms ∼
= k. In particular
P
s
∼
), where F :=
A = S[n]/Ann(F
i=0 Fi , Fi ∈ P [n]i . For each h ≥ 0 we set
P
F≥h := si=h Fi (hence Fs = F≥s ). We have that tdf(F≥h )i ⊆ tdf(F )i and equality
obviously holds if i ≥ h − 1 (see Lemma 2.1 of [7]).
Trivially, if s ≥ 1, we can always assume that the homogeneous part of F of degree
0 vanishes, i.e. F = F≥1 . Moreover, thanks to Lemma 2.2 of [7] we know that, if
s ≥ 2 and Ann(F ) ⊆ S[n]2+ , then we can also assume F1 = 0, i.e. F = F≥2 : we will
always make such an assumption in what follows.
We have a filtration with proper ideals (see [14]) of gr(A) ∼
= S[n]/ldf(Ann(F ))
CA (0) := gr(A) ⊃ CA (1) ⊇ CA (2) ⊇ · · · ⊇ CA (s − 2) ⊇ CA (s − 1) := 0.
Via the epimorphism S[n] ։ gr(A) we obtain an induced filtration
bA (0) := S[n] ⊃ C
bA (1) ⊇ C
bA (2) ⊇ · · · ⊇ C
bA (s − 2) ⊇ C
bA (s − 1) := ldf(Ann(F )).
C

bA (a)/C
bA (a + 1) are reflexive graded
The quotients QA (a) := CA (a)/CA (a + 1) ∼
=C
gr(A)–modules whose Hilbert function is symmetric around (s − a)/2. In general
gr(A) is no more Gorenstein, but the first quotient
(2)
G(A) := QA (0) ∼
= S[n]/Ann(Fs )
is characterized by the property of being the unique (up to isomorphism) graded
Gorenstein quotient k–algebra of gr(A) with the same socle degree. Moreover, the
Hilbert function of A satisfies
s−2
X
(3)
HA (i) = Hgr(A) (i) =
HQA(a) (i),
i ≥ 0.
a=0

Since HA (0) = HG(A) (0) = 1, it follows that if a ≥ 1, then QA (a)0 = 0, whence
QA (a)i = 0 when i ≥ s − a (see [14]) for the same values of a.
Moreover
a
X
Hgr(A)/CA (a+1) (i) = HS[n]/CbA(a+1) (i) =
HQA(α) (i),
i ≥ 0.
α=0

We set
fh :=

s−h
X

HQA (α) (1) = HS[n]/CbA(s−h+1) (1) = Hgr(A)/CA (s−h+1) (1)

α=0

(so that n = HA (1) = f2 ).
Finally we introduce the following new invariant.
Definition 2.1. Let A be a local, Artinian k–algebra with maximal ideal M and
s := sdeg(A). The capital degree, cdeg(A), of A is defined as the maximum integer
i, if any, such that HA (i) > 1, 0 otherwise. If c = cdeg(A) we also say that A is a
c–stretched algebra (for short, stretched if c ≤ 1).
By definition cdeg(A) ≥ 0 and cdeg(A) ≤ sdeg(A): if A is Gorenstein, then we
also have cdeg(A) < sdeg(A).

RATIONALITY OF THE POINCARÉ SERIES

5

The rationality of the Poincaré series PA of every stretched ring A is proved in
[16]. The proof has been generalized to rings with HA (2) = 2 in [10] and to rings
with HA (2) = 3, HA (3) = 1 in [6] . The rationality of PA when A is a 2–stretched
algebra has been studied in [8] with the restriction sdeg(A) = 3.
3. Decomposition of the apolar ideal
In the present section we explain how to decompose the ideal Ann(F ) as the sum
of two simpler ideals. Such a decomposition will be used in the next section in order
to reduce the calculation of the Poincaré series of A to the one of a simpler algebra.
Lemma 3.1. Let m ≤ n, G ∈ P [m], H ∈ k[ym+1 , . . . , yn ] and F = G + H. Let
us denote by Ann(G) and Ann(H) the annihilators of G and H inside S[m] and
k[[xm+1 , . . . , xn ]] respectively. Then
Ann(F ) = Ann(G)S[n] + Ann(H)S[n] + (σG − σH , xi xj )1≤i≤m,

m+1≤j≤n .

where σG ∈ S[m] and σH ∈ k[[xm+1 , . . . , xn ]] are any series of order deg(G) and
deg(H) such that σG ◦ G = σH ◦ H = 1.
Proof. The inclusions Ann(G)S[n], Ann(H)S[n] ⊆ Ann(F ) are completely trivial.
Also the inclusion (σG − σH , xi xj )1≤i≤m, m+1≤j≤n ⊆ Ann(F ) is easy to check. Thus
Ann(G)S[n] + Ann(H)S[n] + (σG − σH , xi xj )1≤i≤m,

m+1≤j≤n

⊆ Ann(F ).

Conversely let p ∈ Ann(F ). Grouping the different monomials in p, we can write
a decomposition p = p≤m + p>m + pmix , where p≤m ∈ S[m], p>m ∈ k[[xm+1 , . . . , xn ]]
and, finally, pmix ∈ (xi xj )1≤i≤m, m+1≤j≤n ⊆ S[n].
It is clear that pmix ∈ Ann(G)S[n] + Ann(H)S[n] + (σG − σH , xi xj )1≤i≤m, m+1≤j≤n ,
hence it suffices to prove that
p≤m + p>m ∈ Ann(G)S[n] + Ann(H)S[n] + (σG − σH , xi xj )1≤i≤m,

m+1≤j≤n .

To this purpose recall that 0 = p ◦ F = p≤m ◦ G + p>m ◦ H, by definition. Hence
p≤m ◦ G = u = −p>m ◦ H. Since p≤m ◦ G ∈ P [m] and p>m ◦ H ∈ k[ym+1 , . . . , yn ], it
follows that u ∈ k. So p≤m − u(σG − σH ) ∈ Ann(G)S[n], whence
p≤m ∈ (σG − σH ) + Ann(G)S[n] ⊆
⊆ Ann(G)S[n] + Ann(H)S[n] + (σG − σH , xi xj )1≤i≤m,

m+1≤j≤n .

A similar argument shows that
p>m ∈ (σG − σH ) + Ann(H)S[n] ⊆
⊆ Ann(G)S[n] + Ann(H)S[n] + (σG − σH , xi xj )1≤i≤m,

m+1≤j≤n ,

and this concludes the proof.
Let P
F be as in the statement above. Then Lemma 3.1 with G :=
H := nj=m+1 yj2 yield the following corollary.


Ps

i=2

Fi and

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G. CASNATI, J. JELISIEJEW, R. NOTARI

Corollary 3.2. Let m ≤ n, G ∈ P [m] non–degenerate and F = G +
Let us denote by Ann(G) the annihilator of G inside S[m]. Then
Ann(F ) = Ann(G)S[n] + (x2j − 2σ, xi xj )1≤i<j≤n,

Pn

j=m+1

yj2 .

j≥m+1

where σ ∈ S[m] has order deg(G) and σ ◦ G = 1.
Proof. It suffices to apply Lemma 3.1 taking into account that Ann(H) = (x2j −
x2m+1 , xi xj )m+1≤i<j≤n, j≥m+1 and that x2m+1 ◦ H = 2.

4. Rationality of Poincaré series
We now focus on the Poincaré series PA (z) of the algebra A defined in the introduction: we will generalize some classical results (see [16], [10], [6]). Out of the
decomposition results proved in the previous section, the main tools we use are the
following ones:
• for each local Artinian, Gorenstein ring C with emdim(C) ≥ 2
(4)

PC (z) =

PC/Soc(C) (z)
1 + z 2 PC/Soc(C) (z)

(see [3]);
• for each local Artinian ring C with maximal ideal N, if c1 , . . . , ch ∈ N \ N2
are linearly independent elements of Soc(C), then
(5)

PC (z) =

PC/(c1 ,...,ch ) (z)
1 − hzPC/(c1 ,...,ch) (z)

(see [13]).
Let A be a local, Artinian, Gorenstein, k–algebra
Pn with2 s = sdeg(A) and n =
HA (1). Assume A = S[n]/Ann(F ) where F = G+ j=m+1 yj ∈ P [n] with G ∈ P [m].
Thanks to Corollary 3.2 we have
Ann(F ) + (σ, xm+1 , . . . , xn ) = Ann(G)S[n] + (σ, xm+1 , . . . , xn ),
thus
S[n]
S[m]
∼
.
=
Ann(F ) + (σ, xm+1 , . . . , xn )
Ann(G) + (σ)
Trivially S[m]/Ann(G) is a local, Artinian, Gorenstein, k–algebra.
Since Soc(A) is generated by the class of σ, it follows from formula (4) that
PA (z) =

PS[n]/Ann(F )+(σ) (z)
.
1 + z 2 PS[n]/Ann(F )+(σ) (z)

Notice that xi xj ∈ Ann(F ) + (σ), i = 1, . . . , n, j = m + 1, . . . , n, i ≤ j. In particular
xm+1 , . . . , xn ∈ Soc(S[n]/Ann(F ) + (σ)). It follows from formula (5) that
PS[n]/Ann(F )+(σ) (z) =

PS[n]/Ann(F )+(σ,xm+1 ,...,xn ) (z)
.
1 − (n − m)zPS[n]/Ann(F )+(σ,xm+1 ,...,xn ) (z)

RATIONALITY OF THE POINCARÉ SERIES

7

The inverse formula of (4) finally yields
PS[n]/Ann(F )+(σ,xm+1 ,...,xn ) = PS[m]/Ann(G)+(σ) (z) =

PS[m]/Ann(G) (z)
.
1 − z 2 PS[m]/Ann(G) (z)

Combining the above equalities we finally obtain the following
P
Proposition 4.1. Let G ∈ P [m], F := G+ nj=m+1 yj2 and define A := S[n]/Ann(F )
and B := S[m]/Ann(G). Then
PA (z) =

PB (z)
.
1 − (HA (1) − HA (2))zPB (z)

A first immediate consequence of the above Proposition is the following corollary.
P
Corollary 4.2. Let G ∈ P [m], F := G + nj=m+1 yj2 and define A := S[n]/Ann(F )
and B := S[m]/Ann(G). The series PB (z) is rational if and only if the same is true
for PA (z).
Now assume that m ≤ 4. Since the Poincaré series of each local Artinian, Gorenstein ring with embedding dimension at most four is rational (see [17], [19], [20],
[15]) we also obtain the following corollary.
P
Corollary 4.3. Let G ∈ P [4], F := G + nj=5 yj2 and define A := S[n]/Ann(F ).
Then PA (z) is rational.
Let A be a local, Artinian, Gorenstein k–algebra with n := HA (1).
Corollary 4.4. Let A be a local, Artinian, Gorenstein k–algebra such that f3 ≤ 4.
Then PA (z) is rational.
Proof. If s := sdeg(A), then

A∼
= S[n]/Ann(F )
2
where F :=
i=2 Fi +
j=f3 +1 yj , Fi ∈ P [fi ]i , i ≥ 3 and F2 ∈ P [f3 ]2 (see [7],
Remark 4.2). Thus the statement follows from Corollary 4.3.

Ps

Pn

5. Examples of algebras with rational Poincaré series
In this section we give some examples of local, Artinian, Gorenstein k–algebras A
with rational PA using the results proved in the previous section.
We start with the following Lemma generalizing a result in [18].
Lemma 5.1. LetPA be a local, Artinian, Gorenstein, 3–stretched k–algebra. If
s−4
HQA (a) (2) ≥ HA (3).
HA (3) ≤ 5, then a=0
Ps−4
Proof. We set m := HA (3) and p := a=0
HQA(a) (2). We have to show that p ≥ m:
assume p ≤ m − 1.
Ps−4
If s = 4, then HQA (0) = a=0
HQA(a) = (1, m, p, m, 1). If s ≥ 5, then we have

if a = 0,
 (1, 1, 1, 1, 1, . . . , 1)
(0, 0, 0, 0, 0, . . . , 0)
if a = 1, . . . , s − 5,
HQA (a) =

(0, m − 1, p − 1, m − 1, 0, . . . , 0)
if a = s − 4.

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G. CASNATI, J. JELISIEJEW, R. NOTARI

Ps−4
In particular a=0
HQA (a) = HQA(0) + HQA(s−4) . Notice that f4 = m.
Macaulay’s growth theorem (see [5], Theorem 4.2.10) and the restriction m ≤ 5
imply that 3 ≤ p = m − 1 necessarily. Thus we can restrict our attention to the two
cases p = 3, 4. We examine the second case, the first one being analogous.
Let n := HA (1), take a polynomial F := y1s + F4 + F3 + F2 , Fi ∈ P [fi ]i , x31 ◦ F4 = 0
Remark 4.2 of [7]) and set B := S[n]/Ann(F≥4 ).
such that A ∼
= S[n]/Ann(F ) (see
Ps−4
We first check that HB = a=0 HQA (a) = (1, 5, 4, 5, 1, . . . , 1). On the one hand,
bA (a) = C
bB (a), a ≤ s − 3, whence
Lemma 1.10 of [14] implies that C
HB (1) ≥

s−4
X
a=0

HQB (a) (1) =

s−4
X

HQA (a) (1) = 5.

a=0

On the other hand, F≥4 ∈ P [f4 ] = P [5], whence 5 = HB (1) ≤ 5. It follows that
equality holds, thus HQB (s−2) (1) = HQB (s−3) (1) = 0. By symmetry we finally obtain
HQB (s−2) = HQB (s−3) = 0. This last vanishing completes the proof of the equality
Ps−4
HB = a=0
HQA (a) = (1, 5, 4, 5, 1, . . . , 1).
Let I ⊆ k[x1 , . . . , xn ] ⊆ S[n] be the ideal generated by the forms of degree at
most 2 inside Ann(tdf(F≥4 )) = ldf(Ann(F≥4 )). We obviously have x6 , . . . , xn ∈ I,
because F≥4 ∈ P [5]. Denote by I sat the saturation of I and set R := k[x1 , . . . , xn ]/I,
Rsat := k[x1 , . . . , xn ]/I sat . Due to the definition of I we know that HR (t) ≥ HB (t)
for each t ≥ 0, and equality holds true for t ≤ 2. Moreover, we know that
HB (2)h2i = HB (3) ≤ HR (3) ≤ HR (2)h2i = HB (2)h2i ,
hence

   
2
4
= HR (2)h2i .
+
HR (3) =
2
3
Gotzmann Persistence Theorem (see [5], Theorem 4.3.3) implies that

 

t−1
t+1
= t + 2,
t ≥ 2.
+
HR (t) =
t−1
t

We infer HRsat (t) = t + 2, t ≫ 0.
When saturating, the ideal can only increase its size in each degree, hence HRsat (t) ≤
HR (t) for each t ≥ 0. Again Macaulay’s bound thus forces HRsat (t) = HR (t) = t + 2
for t ≥ 2. In particular the components It and Itsat of degree t ≥ 2 of I and I sat
coincide.
Since HRsat is non–decreasing, it follows that
HRsat (1) ≤ HRsat (2) = 4 < 5 = HB (1) = HR (1).
In particular there exists a linear form ℓ ∈ I sat \ I. The equality I2 = I2sat forces
ℓxj ∈ I2 ⊆ Ann(tdf(F≥4 )), j = 1, . . . , n. Since x6 , . . . , xn ∈ I, it follows that we can
assume ℓ ∈ S[5] ⊆ S[n]. Moreover we also know that y1s ∈ tdf(F≥4 ), hence ℓ cannot
be a multiple of x1 . In particular we can change linearly coordinates in such a way
that ℓ = x5 .
If j ≥ 2, then xj ◦ F≥4 = xj ◦ F4 , thus the condition xj x5 ∈ I2 ⊆ Ann(tdf(F≥4 )),
j = 2, . . . , 5, and x31 ◦ F4 = 0 imply that x5 ◦ F4 = 0. Such a vanishing contradicts

RATIONALITY OF THE POINCARÉ SERIES

9

the linear independence of the derivatives
x2 ◦ F≥4 ,

x3 ◦ F≥4 ,

x4 ◦ F≥4 ,

x5 ◦ F≥4 .

Indeed 5 = HB (1) = dimk (tdf(F≥4 )1 ) and xj ◦ F≥4 = 0, j ≥ 6.



Using the results proved in the previous section and the Lemma above we are able
to handle the first example of this section, proving the following theorem generalizing
Corollary 2.2 of [8].
Theorem 5.2. Let A be a local, Artinian, Gorenstein k–algebra with HA (2) ≤ 4
and cdeg(A) ≤ 3. Then PA is rational.
Proof. Let us examine the case cdeg(A) = 3, the other ones being similar. Lemma
5.1 yields
(6)

HA (2) ≥

s−4
X

HQ(a) (2) ≥ HA (3).

a=0

If sdeg(A) ≥ 5, then Decomposition (3) is
(1, 1, . . . , 1) + (0, a1 , a2 , a1 , 0) + (0, b1 , b1 , 0) + (0, c1 , 0)
for some integers a1 , a2 , b1 , c1 . Inequality (6) is equivalent to a1 ≤ a2 . We know
that HA (2) = a2 + b1 + 1 ≤ 4, so f4 = a1 + b1 + 1 ≤ 4 and the argument follows
from Corollary 4.4. In the case sdeg(A) = 4, the decomposition (3) changes, but
the argument stays the same.

Now we skip the condition cdeg(A) = 3 but we impose a restriction on the shape
of HA . The following theorem generalizes a well–known result proved when either
m = 1, 2 (see [16] and [10] respectively) or m ≤ 4 and s = 3 (see again [8]).
Theorem 5.3. Let A be a local, Artinian, Gorenstein k–algebra such that HA (i) =
m, 2 ≤ i ≤ cdeg(A). If m ≤ 4, then PA is rational.
Proof. Let c := cdeg(A), n := HA (1), take a polynomial F := y1s + Fc+1 + . . . ,
Fc+1 ∈ P [fc+1 ]c+1 = P [m]c+1 such that A ∼
= S[n]/Ann(F ) (see Remark 4.2 of [7])
and set B := S[n]/Ann(F≥c+1 ) so that QA (a) = QB (a) for a ≤ s − c − 1 (again
by Lemma 1.10 of [14]). In particular HB (c) = m, thus Decomposition (3) implies
HB (1) ≥ m. Since we know that F≥c+1 ∈ P [m], it follows that HB (1) ≤ m, hence
equality must hold.
As in the proof of the previous lemma one immediately checks that either s = c+1,
and HQA(0) = (1, m, . . . , m, 1), or s ≥ c + 2, and

if a = 0,
 (1, 1, . . . , 1, 1, . . . , 1)
(0,
0,
.
.
.
,
0,
0,
.
.
.
,
0)
if
a = 1, . . . , s − c − 2,
HQA(a)

(0, m − 1, . . . , m − 1, 0, . . . , 0)
if a = s − c − 1.
Assume that HB (i) ≤ m − 1 ≤ 3 for some i = 2, . . . , c − 1. Let i0 be the maximal
of such i’s. We know that there are k(i0 ) > k(i0 − 1) > k(i0 − 2) > . . . such that

 
 

k(i0 − 2)
k(i0 − 1)
k(i0 )
+ · · · ≤ m − 1 ≤ 3.
+
+
HB (i0 ) =
i0 − 2
i0 − 1
i0

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G. CASNATI, J. JELISIEJEW, R. NOTARI

If i0 ≥ 3, it would follow k(i0 ) ≤ i0 , thus Macaulay’s bound implies
HB (i0 + 1) ≤ HB (i0 )hi0 i =

 
 

k(i0 − 2) + 1
k(i0 − 1) + 1
k(i0 ) + 1
+··· =
+
+
=
i0 − 1
i0
i0 + 1
= HB (i0 ) ≤ m − 1,
a contradiction. We conclude that i0 = 2.
Due to the symmetry of HQB (s−c−1) we deduce that c = 3. If HQB (s−3) (2) = q, the
symmetry of HQB (s−3) implies HQB (s−3) (1) = q, hence Decomposition (3) implies
m = HB (1) =

s−2
X

HQB (a) (1) = m + q + HQB (s−2) (1).

a=0

It follows that q = HQB (s−2) (1) = 0, whence HB = (1, m, p, m, 1, . . . , 1) where
p ≤ m − 1 which cannot occur by Lemma 5.1.
We conclude that HQA(s−c−1) (i) = HQB (s−c−1) (i) = m − 1 for each i = 2, . . . , c,
then the hypothesis on HA (i) and Decomposition (3) yield

(0, 0, 0, . . . , 0, 0, . . . , 0)
if a = s − c, . . . , s − 3,
HQA (a) =
(0, n − m, 0, . . . , 0, 0, . . . , 0)
if a = s − 2,
Ps−3

whence f3 = a=1 HQ(a) (1) = m ≤ 4.
As third example we skip the condition on the shape of HA but we put a limit on
dimk (A), slightly extending the result proved in [7].

Theorem 5.4. Let A be a local, Artinian, Gorenstein k–algebra with dimk (A) ≤ 16
and HA (2) ≤ 4. Then PA is rational.
Proof. Thanks to [15] we can restrict our attention to algebras A with HA (1) ≥ 5.
The rationality of the Poincaré series of stretched algebras is proved in [16]. For
almost stretched algebras see [10]. For the case of algebras A with sdeg(A) = 3 and
HA (2) ≤ 4 see [8]. Finally the case HA (i) = m, 2 ≤ i ≤ cdeg(A) with m ≤ 4 is
covered by Theorem 5.3 above.
There are several cases which are not covered by the aforementioned results. In
each of these cases one can check that the condition f3 ≤ 4 of Corollary 4.4 is fulfilled.
We know that necessarily HA (2) ≥ 3, otherwise A is almost stretched by Macaulay’s
bound. The restriction HA (2) ≤ 4 implies HA (3) ≤ 5 again by Macaulay’s bound.
Theorem 5.2 deals with the case sdeg(A) = 4. Let us analyze the case sdeg(A) = 5
and dimk A ≤ 16. The decomposition is
(1, a1 , a2 , a2 , a1 , 1) + (0, b1 , b2 , b1 , 0) + (0, c1 , c1 , 0) + (0, d1, 0)
for some integers a1 , a2 , b1 , b2 , c1 , d1. If a1 = 1 then the algebra is 3-stretched, so we
may suppose a1 ≥ 2. We know that HA (2) = a2 + b2 + c1 ≤ 4 and we would like to
prove a1 +b1 +c1 ≤ 4. Suppose a1 +b1 +c1 ≥ 5, then the inequality on the dimension
of A shows that 2 · a2 + b2 ≤ 4, in particular a2 ≤ 2 and from Macaulay’s bound
it follows that a1 = a2 = 2. It follows that b2 = 0 and once again from Macaulay’s
bound b1 = 0. This forces a1 + b1 + c1 = 2 + c1 = a2 + b2 + c1 ≤ 4, a contradiction.

RATIONALITY OF THE POINCARÉ SERIES

11

Let us now suppose that sdeg(A) = 6. Look at the first row of the symmetric
decomposition (3): (1, a1 , a2 , a3 , a2 , a1 , 1).
• If a1 ≥ 3, then a2 , a3 ≥ 3 and the sum of the row is at least 17.
• If a1 = 2 then a2 = a3 = 2 and the sum of the row is 12. If we suppose
that f3 ≥ 5, then the sum of the first column of the remaining part of the
decomposition will be at least three, so the sum of whole remaining part will
be at least 2 · 3 = 6 and the dimension will be at least 12 + 6 > 16.
• Suppose a1 = 1 and look at the second row (0, b1 , b2 , b2 , b1 , 0). If b1 = 0
then the algebra is 3-stretched so the result follows from Theorem 5.2. From
HA (2) ≤ 4 it follows that b2 ≤ 3. If b2 = 3, then b1 ≥ 2 so the dimension is
at least 7 + 10 > 16. If b2 ≤ 2 then b1 ≤ b2 from Macaulay’s bound. Hence,
the same argument as before applies.
Let us finally suppose that sdeg(A) ≥ 7. Take the first row, beginning with
(1, a1 , a2 , . . . ). If a1 ≥ 3 then its sum is at least 3 · sdeg(A) − 1 > 16. If a1 =
2, the sum of this row is 2 · sdeg(A) ≥ 14. Then one can argue as in the case
sdeg(A) = 6, a1 = 2. A similar reasoning shows that when a1 = 1 the algebra has
decomposition (1, 1, . . . , 1) + (0, 4, 4, 0) and so HA (2) ≥ 5.

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[17] J.P. Serre: Sur la dimension homologique des anneaux et des modules noethériens, in Proceedings of the International Symposium on Algebraic Number Theory, Tokyo, (1955), 175–189.
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Gianfranco Casnati,
Dipartimento di Scienze Matematiche, Politecnico di Torino,
corso Duca degli Abruzzi 24, 10129 Torino, Italy
e-mail: gianfranco.casnati@polito.it
Joachim Jelisiejew,
Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw,
Banacha 2, 02-097 Warsaw, Poland
jj277546@students.mimuw.edu.pl
Roberto Notari,
Dipartimento di Matematica, Politecnico di Milano,
via Bonardi 9, 20133 Milano, Italy
e-mail: roberto.notari@polimi.it

