A commutative noetherian local ring of embedding dimension 4 having a

arXiv:1702.04262v1 [] 14 Feb 2017

transcendental series of Betti numbers
Jan-Erik Roos
Department of Mathematics
Stockholm University
e-mail: jeroos@math.su.se
February 14, 2017

Abstract
We construct a ring with the properties of the title of the paper. We also construct
some other local rings of embedding dimension 4 with exotic properties. Among the
methods used are the Macaulay2-package DGAlgebras by Frank Moore, combined with
and inspired by results by Anick, Avramov, Backelin, Katthän, Lemaire, Levin, Löfwall
and others.
Mathematics Subject Classification (2010): Primary 13Dxx, 16Dxx, 68W30; Secondary 16S37, 55Txx
Keywords. local ring, Koszul complex, Yoneda Ext-algebra,
Hilbert series, Macaulay2, Betti series, Avramov spectral sequence
0. Introduction
The aim of the present paper is to prove the following (if V is a vector space over a
field k we write |V | instead of dimk V ):
THEOREM 1 – Let k be a field and k[x, y, z, u] the polynomial ring in four variables
of degree one and let
(1)

R=

(x3 , x2 y, (x

k[x, y, z, u]
+ u)(y 2 + z 2 ) + yzu, zu2 , u3 )

be the graded quotient ring. Let
PR (Z) =

X

i
|TorR
i (k, k)|Z

i≥0

be the corresponding series of Betti numbers of R. Then PR (Z) is a transcendental function:
PR (Z) =

(1 + Z)(1 − Z − Z 2 )

Z(1 + Z)2
2n
2
3
4
n=1 (1 − Z ) − 1 + 2Z + 2Z − 4Z − 4Z

Q∞
2

1

If one replaces k[x, y, z, u] with the formal power series ring k[[x, y, z, u]] one obtains a
local ring with the same homological properties. The Theorem might be of interest in
itself (previously known examples of trancendental series needed ≥ 5 variables) but more
interesting are probably the methods used to obtain the example (described in section 1
below) and the methods of proof described in section 2. There is now a possibility to obtain
a new good insight in the theory of the homology of local rings of embedding dimension 4.
In section 3 we will in particular indicate that the following ring which is a simpler variant
of (1)
(2)

R=

(x3 , x2 y, (x

k[x, y, z, u]
+ u)(y 2 + z 2 ), zu2 , u3 )

has a PR (Z) = (1 − Z 2 )(1 + Z)/(1 − 3Z − 3Z 3 − 2Z 5 ) i.e. a rational function but is such
that the Yoneda Ext-algebra Ext∗R (k, k) is not finitely generated as an algebra. This last
phenomenon was also previously only known in the embedding dimension ≥ 5 cases. We
will also deduce some other transcendental results and also show that there are quotients
of k[x, y, z, u] with an ideal with only six cubic generators which is non-Golod but has the
multiplication in Koszul homology equal to zero (the first examples of this last phenomenon
with more relations - monomials of higher degrees and more variables - were obtained by
Lukas Katthän [KAT]).

1. How the example in Theorem 1 was found
In [A] (announced in 1980 [A-CR]), David Anick published the first example (example
7.1 page 29 of [A]) of an R in 5 variables and 7 quadratic relations for which PR (Z) was
transcendental. Here is a variant of his example with only 5 quadratic relations which has
the same property and which will be useful for us:
(3)

R=

k[x1 , x2 , x3 , x4 , x5 ]
2
(x1 , x1 x2 , x1 x3 + x2 x4 + x3 x5 , x4 x5 , x25 )

In this case the Hilbert series of R is H(t) = (1 + 3t + t2 − 2t3 + t4 )/(1 − t)2 and the Hilbert
series of the Koszul dual R! of R is
∞
(1 + t)2 Y 1 + t2n−1
R (t) =
(1 − t − t2 )2 n=1 1 − t2n
!

and
(4)

1/PR (Z) = (1 + 1/Z)/R! (Z) − H(−Z)/Z
2

Remark: For the case studied by Anick [A] we had that H(t) = 1 + 5t + 7t2 and
R! (t) =

∞
Y
1
1 + t2n−1
(1 − 2t)2 n=1 1 − t2n

and the formula (4) was also valid in his case.
We now address the problem of constructing a 4-variable version of the ring (3).
We start with the ring S2 = k[x, y, z, u]/(x2, xy, zu, u2 ) where x, y, z, u correspond to
x1 , x2 , x4 , x5 and try to add an extra relation corresponding to x1 x3 + x2 x4 + x3 x5 . How to
do this is not at all evident and instead we try to add a new relation which is a linear combination with coefficients 0 or 1 of the 6 nonzero quadratic monomials yu, xu, z 2 , yz, xz, y 2
in S2 . There are 64 such linear combinations but each of them leads to a rational PR (Z)
(they all occur in the Appendix to [R4]). Instead we turn to the study of cubic relations,
i.e. we start with a new ring
S=

k[x, y, z, u]
(x3 , x2 y, zu2 , u3 )

and try to add a linear combination (coefficients 0 or 1) of the 16 non-zero cubic monomials
yu2 , xu2 , z 2 u, yzu, xzu, y 2u, xyu, x2 u, z 3 , yz 2 , xz 2 , y 2 z, xyz, x2 z, y 3 , xy 2
in S. There are 216 = 65536 cases to study and this can be done automatically using the
following input programme to Macaulay 2 which was written at my request several years
ago by Mike Stillman (it is an elegant version of a programme I wrote for Macaulay 1 a
long time ago (cf. [R3, pp. 294-296])
binaries = (n) -> (
if n === 0 then

{}

else if n === 1 then

{{0}, {1}}

else (
r := binaries(n-1);
join(apply(r, i->prepend(0,i)),
apply(r, i->prepend(1,i)))))
doit = (i) -> (
h := hypers (0,i);
J1 := J + ideal(h);
<< newline << flush;
<< ‘‘--- n = ‘‘ << i << ‘‘ ideal = ‘‘ << hypers (0,i) << ‘‘ ---’’ << flush;
<< newline << flush;

3

E := res J1;
<< newline << flush;
<< ‘‘ ‘‘ << betti E << flush;
<< newline << flush;
A := (ring J1)/J1;
C := res(coker vars A, LengthLimit=>6);
<< newline << flush;
<< ‘‘ ‘‘ << betti C << flush;
<< newline << flush;
<< ‘‘ ‘‘ <<hilbertSeries(A,Order=>12)<< flush;
<< newline << flush;
)
makeHyperplanes = (J) -> (
I := matrix basis(3,coker gens J);
c := numgens source I;
m := transpose matrix binaries c;
I * m)
R = QQ[x,y,z,u]
J = ideal(x^3,x^2*y,z*u^2,u^3)
time hypers = makeHyperplanes J;
time scan(numgens source hypers, i -> doit i);

We obtain 20 possibilities for the PR (z) up to degree 6 presented here in increasing
order (of course there can be variations inside each case Pi (z) due e.g. to different Koszul
homology) :

P1 (z)=1+4z+11z 2 +27z 3 +62z 4 +137z 5 +295z 6 +...

P2 (z)=1+4z+11z 2 +27z 3 +64z 4 +152z 5 +364z 6 +...

P3 (z)=1+4z+11z 2 +28z 3 +69z 4 +168z 5 +407z 6 +...

P4 (z)=1+4z+11z 2 +28z 3 +69z 4 +169z 5 +414z 6 +...

P5 (z)=1+4z+11z 2 +28z 3 +69z 4 +172z 5 +431z 6 +...

P6 (z)=1+4z+11z 2 +28z 3 +70z 4 +173z 5 +427z 6 +...

P7 (z)=1+4z+11z 2 +28z 3 +70z 4 +177z 5 +451z 6 +...

P8 (z)=1+4z+11z 2 +29z 3 +74z 4 +188z 5 +476z 6 +...

P9 (z)=1+4z+11z 2 +29z 3 +75z 4 +193z 5 +496z 6 +...

P10 (z)=1+4z+11z 2 +29z 3 +75z 4 +193z 5 +498z 6 +...

P11 (z)=1+4z+11z 2 +29z 3 +75z 4 +194z 5 +503z 6 +...

P12 (z)=1+4z+11z 2 +30z 3 +80z 4 +213z 5 +567z 6 +...

P13 (z)=1+4z+11z 2 +30z 3 +81z 4 +218z 5 +587z 6 +...

P14 (z)=1+4z+11z 2 +30z 3 +82z 4 +224z 5 +612z 6 +...

4

P15 (z)=1+4z+11z 2 +31z 3 +86z 4 +238z 5 +660z 6 +...

P16 (z)=1+4z+11z 2 +31z 3 +87z 4 +244z 5 +685z 6 +...

P17 (z)=1+4z+11z 2 +31z 3 +88z 4 +249z 5 +705z 6 +...

P18 (z)=1+4z+11z 2 +32z 3 +92z 4 +263z 5 +755z 6 +...

P19 (z)=1+4z+11z 2 +32z 3 +93z 4 +269z 5 +780z 6 +...

P20 (z)=1+4z+11z 2 +33z 3 +99z 4 +294z 5 +877z 6 +...

Which one of these cases could give a transcendental PR (Z) ?
One can guess a possibility by studying how the transcendental PR (z) comes up in
five variables when we add linear combinations of quadratic monomials to the ring in 5
variables: Q[x, y, z, u, v]/(x2, xy, uv, v 2) i.e. modifying the previous input programme by
replacing R by QQ[x, y, z, u, v], J by (x2 , x ∗ y, u ∗ v, v 2 ) and the line
I := matrix basis(3,coker gens J);
by the line
I := matrix basis(2,coker gens J);
There are now “only” 211 = 2048 cases to study and now we obtain only 10 possibilities
for the PR (z) up to degree 6 presented here in increasing order:
P1 (z)=1+5z+15z 2 +37z 3 +82z 4 +170z 5 +337z 6 +...

P2 (z)=1+5z+15z 2 +38z 3 +88∗z 4 +192z 5 +406z 6 +...

P3 (z)=1+5z+15z 2 +38z 3 +89z 4 +199z 5 +432z 6 +...

P4 (z)=1+5z+15z 2 +38z 3 +91z 4 +216z 5 +516z 6 +...

P5 (z)=1+5z+15z 2 +39z 3 +95z 4 +222z 5 +506z 6 +...

P6 (z)=1+5z+15z 2 +39z 3 +96z 4 +231z 5 +553z 6 +...

P7 (z)=1+5z+15z 2 +39z 3 +97z 4 +237z 5 +575z 6 +...

P8 (z)=1+5z+15z 2 +39z 3 +99z 4 +254z 5 +659z 6 +...

P9 (z)=1+5z+15z 2 +40z 3 +104z 4 +268z 5 +689z 6 +...

P20 (z)=1+5z+15z 2 +41z 3 +112z 4 +306z 5 +836z 6 +...

There are e.g. 1024 cases of an extra quadratic relation corresponding to P1 (z) and 576
cases corresponding to P4 (z), but only 8 for P2 (z) including the case of the extra relation
xz + yu + zv leading to the transcendental series mentioned in the introduction (the other
7 cases of P2 (z) lead to the same total series). Furthermore, there are also 8 cases for
P6 (z) and they all lead to cases where the Yoneda Ext-algebra Ext∗R (k, k) is not finitely
generated but the series of Betti numbers is rational (this phenomenon was first found by
me in [R1] (inspired by Lemaire [LEM]).
We now try to use this information to try to guess what happens for the 65536
embedding dimension 4 cases above. In this situation there are only 80 cases of P10 (z) and
only two of them correspond to adding a cubic relation with is a sum of 5 cubic monomials
(the other cases need more monomials): case 9257 which corresponds to xy 2 + xz 2 + y 2 u +
yzu + z 2 u and case 13345 which corresponds to xy 2 + xyz + xz 2 + y 2 u + z 2 u. Here one of
5

these cases can be obtained from the other one by permuting x and u. We can therefore
restrict ourselves to the homological study of the ring
(1)

(x3 , x2 y, (x +

k[x, y, z, u]
u)(y 2 + z 2 ) + yzu, zu2 , u3 )

mentioned in the introduction and which will be the subject of the next section.

2. The treatment of the example in Theorem 1
We will now use the programme DGAlgebra of [MOO] on the ring (1).
The Koszul homology of the ring has the Betti numbers given by Macaulay:
total: 1 5
0:
1 .

12 12 4
.
.
.

1:
2:

.
.

.
5

.
2

.
.

3:
4:

.
.

.
.

1
8

.
.
10 3

5:
6:

.
.

.
.

.
1

.
2

.
.

.
1

From this we can guess that the multiplication in HKR = Tork[x,y,z,u]
(k[x, y, z, u]/J, k),
∗
where J is the ideal in (1) is rather complicated. We determine details about that multiplication as a preparation for using the Avramov spectral sequence ([AV0],[AV1]) where
R̃ = k[x, y, z, u] and R = R̃/J:
(5)

R̃
2
Ep,q
= TorHKR
(k, k) => TorR
p,q
∗ (k, k)//Tor∗ (k, k)

by using the following infile for Macaulay:
loadPackage(‘‘DGAlgebras’’)
R=QQ[x,y,z,u]/ideal(x^3,x^2*y,z*u^2,u^3,x*y^2+x*z^2+y^2*u+y*z*u+z^2*u)
res(ideal R); betti oo
res(coker vars R,LengthLimit => 6); betti oo
HKR=HH koszulComplexDGA(R)
generators HKR
for n from 1 to length(generators HKR) list degree X n
ideal HKR
6

I=ideal(vars HKR)
I^2; trim(oo)
I^3; trim(oo)
res(coker vars HKR,LengthLimit => 3);
betti oo
res(coker vars HKR,LengthLimit => 4);
betti oo
From the Macaulay output of this input file we see that the augmentation ideal I of
HKR satisfies I 3 = 0, and that I 2 is generated by
X4 X5 ,X2 X5 ,X1 X5 ,X3 X4 ,X2 X4 ,X1 X4 ,X2 X3 ,X4 X7 ,X2 X7 ,X1 X7 ,X5 X6 ,X4 X6 ,X3 X6 .X6 X7

Therefore we will be able to calculate the left side of the Avramov spectral sequence (5)
exactly as in [R4], where we used the Theorem 1.3 on page 310 of Clas Löfwall’s paper
[L1], provided we can determine the Hilbert series HKR! (z, z) of the Koszul dual HKR! of
HKR and the Hilbert series HKR(−z, z) of HKR. From the output file we see that HKR
can be presented as follows: It has five generators X1 , X2 , X3 , X4 , X5 of degree (1,3),two
generators X6 , X7 of degree (2,4), three generators X8 , X9 , X10 of degrees (2,5),(2,6),(2,8)
respectively, four generators X11 , X12 , X13 , X14 of degree (3,7), two generators X15 , X16
of degree (3,9), two generators X17 , X18 of degree (4,8) and one generator X19 of degree
(4,10).
Furthermore the first five variables X1 , X2 , X3 , X4 , X5 skew-commute, the variables
X6 , X7 , X8 , X9 , X10 commute among themselves and with X1 , X2 , X3 , X4 , X5 , etc.
The relations are as follows: The product of the last 12 variables X8 , . . . X19 with all
the 19 variables X1 . . . X19 are 0. For the first five generators X1 , X2 , X3 , X4 , X5 we only
have the following three quadratic relations: X3 X5 , X1 X3 + X2 X4 − X4 X5 , X1 X2 , and
if we also take into account the additional variables X6 , X7 we have the extra quadratic
relations X1 X6 , X2 X6 , X3 X7 , X5 X7 , X6 X6 , X7 X7 . It follows that the if
(6)
C=

k[X1 , X2 , X3 , X4 , X5 , X6 , X7 ]
(X3 X5 , X1 X3 + X2 X4 − X4 X5 , X1 X2 , X1 X6 , X2 X6 , X3 X7 , X5 X7 , X6 X6 , X7 X7 )

then the Koszul dual HKR! of HKR can be presented as a coproduct
(7)

C ! ⊔ k < x8 , x9 , x10 , x11 , x12 , x13 , x14 , x15 , x16 , x17 , x18 , x19 >

where the second algebra in the coproduct is the free algebra on the dual generators
x8 , . . . x19 of X8 , . . . X19 .
7

Furthermore, it is well-known (cf. e.g. [LEM]) that for the coproduct A ⊔ B of two
graded connected algebras, we have the following formula for the respective Hilbert series
1
1
1
+1=
+
(A ⊔ B)(z)
A(z) B(z)
(this works also for the bigraded case) and it follows that
1
1
+1= !
+ 1 − 3z 3 − 6z 4 − 3z 5
!
HKR (z, z)
C (z, z)

(8)

and we have therefore reduced our problem to calculate C ! (z, z) where C only involves the
first 7 variables X1 , X2 , X3 , X4 , X5 , X6 , X7
We start with an algebra D that involves only the first 5 variables X1 , X2 , X3 , X4 , X5 :
D=

k[X1 , X2 , X3 , X4 , X5 ]
(X3 X5 , X1 X3 + X2 X4 − X4 X5 , X1 X2 )

The variables skewcommute. The Koszul dual D! is the quotient of the free algebra in the
dual variables xi of the Xi :
D! =

k < x1 , x2 , x3 , x4 , x5 >
([x1 , x4 ], [x1 , x5 ], [x2 , x3 ], [x2 , x5 ], [x3 , x4 ], [x1 , x3 ] + [x4 , x5 ], [x2 , x4 ] + [x4 , x5 ]

where the [xi , xj ] are Lie commutators. From this one sees exactly as in Löfwall’s and
mine study of the Anick example [LR1] that the underlying Lie algebra gD of D! sits in
the middle of an extension
0 −→ ⊕∞
i=1 ai −→ gD −→ f1 × f2 −→ 0
i.e. is the extension of the product of two free Lie algebras f1 generated by x1 and x2 and
f2 generated by x3 and x5 with the infinite abelian Lie algebra
a = ⊕∞
i=1 ai
where the ai are onedimensional generated by a1 = x4 , a2 = [x5 , x4 ], a3 = [x5 , [x5 , x4 ]] etc.
which commute and by the 2-cocycle γ : f1 ×f2 → a defined by γ(x1 , x3 ) = −γ(x3 , x1 ) = a2
and γ(x2 , x5 ) = −γ(x5 , x2 ) = a2 . It follows that
(7)

∞
Y
1
1
D (z) =
2
(1 − 2z) n=1 (1 − z n )
!

8

But since we need D! (z, z) we have to replace the z by z 2 in the formula (9) to get
D! (z, z). We next want to incorporate the variables X6 , X7 (which commute with the
X1 , X2 , X3 , X4 , X5 ) into the picture. It turns out that we have to study the underlying
Lie algebra of the Koszul dual C ! of the C in (6), i.e. the quotient of
k < x1 , x2 , x3 , x4 , x5 , x6 , x7 >
with the ideal generated by
[x1 , x4 ], [x1 , x5 ], [x2 , x3 ], [x2 , x5 ], [x3 , x4 ], [x1 , x3 ] + [x4 , x5 ], [x2 , x4 ] + [x4 , x5 ],
as before and the extra ideal generators:
[x1 , x7 ], [x2 , x7 ], [x4 , x7 ], [x6 , x7 ], [x3 , x6 ], [x4 , x6 ], [x6 , x7 ]
where x6 and x7 are dual to X6 and X7 . From the preceding presentation one sees that
the Lie algebra gC of C ! sits in the middle of an extension:
(10)

0 −→ ⊕∞
i=1 ai −→ gC −→ F1 × F2 −→ 0

where F1 is the free Lie algebra generated by x1 , x2 , x6 of degrees 1, 1, 2 and F2 is the free
Lie algebra generated by x3 , x5 , x7 of degrees 1, 1, 2 and the ai :s are as before, and the x6
and x7 operate in the trivial way on the ai . Furthermore we have a new cocycle
Γ : F1 × F2 −→ ⊕∞
i=1 ai
where as before Γ is defined by its value on the generators by −Γ(x3 , x1 ) = Γ(x1 , x3 ) = a2
and −Γ(x5 , x2 ) = Γ(x2 , x5 ) = a2 and being zero for all other pairs of generators. By
calculating H 2 (F1 × F2 , a) for (8) one sees that the relations are those of C ! and therefore
the Hilbert series
(11)

∞
Y
1
1
C (z) =
2
2
(1 − 2z − z ) n=1 (1 − z n )
!

if the variables x1 , x2 , x3 , x4 , x5 , x6 , x7 are given the degrees 1, 1, 1, 1, 1, 2, 2. But for the
calculation of the series of HKR! (z, z) these variables should be given the degrees
2, 2, 2, 2, 2, 3, 3 so that we get the series
(12)

∞
Y
1
1
2
3
2
(1 − 2z − z ) n=1 (1 − z 2n )

9

Using the formulae (8) and (12) we obtain
∞
Y
1
2
3 2
(1 − z 2n ) − 3z 3 − 6z 4 − 3z 5
= (1 − 2z − z )
HKR! (z, z)
n=1

(13)

But I 3 = 0 and
I 2 =(X4 X5 ,X2 X5 ,X1 X5 ,X3 X4 ,X2 X4 ,X1 X4 ,X2 X3 ,X4 X7 ,X2 X7 ,X1 X7 ,X5 X6 ,X4 X6 ,X3 X6 .X6 X7 )

and this gives that HKR(x, y) = 1 + 5xy + 5xy 2 + 6xy 3 + 3xy 4 + 7x2 y 2 + 6x2 y 3 + x2 y 4 ,
so that
HKR(−z, z) = 1 − 5z 2 − 5z 3 + z 4 + 3z 5 + z 6

(14)

and finally the formula
1
= (1 + 1/z)/HKR! (z, z) − HKR(−z, z)/z
PHKR (z, z)
gives that
∞
Y
1
2
3 2
= (1 + 1/z)[(1 − 2z − z )
(1 − z 2n ) − 3z 3 − 6z 4 − 3z 5 )]
PHKR (z, z)
n=1

−(1 − 5z 2 − 5z 3 + z 4 + +3z 5 + z 6 )/z
and if the Avramov spectral sequence degenerated this series should be the same as
(1 + z)4 /PR (z) so the final result should be
PR (z) =

(1 + z)(1 − z − z 2 )

z(1 + z)2
2n
2
3
4
n=1 (1 − z ) − 1 + 2z + 2z − 4z − 4z

Q∞
2

= 1 + 4z + 11z 2 + 29z 3 + 75z 4 + 193z 5 + 498z 6 + 1289z 7 + 3341z 8 + 8663z 9 + 22466z 10 + . . .
as asserted in our Theorem 1. The following calculation in Macaulay2
res(coker vars R, LengthLimit => 11); betti oo
gives the diagram of Betti numbers (16) below whose first row (the total Betti numbers)
gives support to this assertion. But the Theorem 5.9 of [AV0] gives first that the differentials dr of the Avramov spectral sequence are 0 for r > 2 but furthermore since the
dimensions of the two sides of the Avramov spectral sequence are the same in degrees ≤ 10
10

it follows from the proof of Theorem 5.9 in [AV0] and the structure of the matrix Massey
products that there are no non-zero differentials, and Theorem 1 is proved. We can also
get a bigraded more precise version of all this: Indeed, we also have a series
X
p q
PR (x, y) =
|TorR
p,q (k, k)|x y
p,q

which takes into account the extra grading of R (we have that PR (z, 1) is the old PR (z))
given as
(15)

PR (x, y) =

(1 + xy)4
(1 + 1/x)/A − H/x

where
2 3

3 4 2

1/A = (1−2x y −x y )

∞
Y

(1−x2n y 3n )−x3 y 5 −x3 y 6 −x3 y 8 −4x4 y 7 −2x4 y 9 −2x5 y 8 −x5 y 10

n=1

and H is equal to
1−5x2 y 3 −2x3 y 4 −x3 y 5 −x3 y 6 −x3 y 8 −4x4 y 7 −2x4 y 9 −2x5 y 8 −x5 y 10 +7x4 y 6 +6x5 y 7 +x6 y 8
and the expansion of PR (x, y) of (15) up to degree 11 gives the same diagram of graded
Betti numbers that was earlier found by the Macaulay2 calculation:

(16)

total: 1
0:
1
1:
.

4 11 29
4 6 4
. 5 22

75 193 498 1289 3341 8663 22466 58264
1 .
.
.
.
.
.
.
38 32 13 2
.
.
.
.

2:
3:

.
.

.
.

.
.

1
1

22 92
8 34

171 169
135 389

92
689

26
744

4:
5:

.
.

.
.

.
.

.
1

.
6

10
15

87
22

346
96

923
645

1870 2835
2426 5739

3066
9705

6:
7:
8:

.
.
.

.
.
.

.
.
.

.
.
.

.
.
.

10
.
.

65
2
2

186
73
20

348
459
112

964 4302
1446 3296
669 3025

14387
8595
9538

9:
10:

.
.

.
.

.
.

.
.

.
.

.
.

.
1

.
8

30
28

320
62

1668
396

6469
3282

11:
12:

.
.

.
.

.
.

.
.

.
.

.
.

.
.

.
.

15
.

129
3

510
153

1377
1278

13:
14:

.
.

.
.

.
.

.
.

.
.

.
.

.
.

.
.

.
.

3
.

36
.

246
60

15:
16:

.
.

.
.

.
.

.
.

.
.

.
.

.
.

.
.

.
.

1
.

10
.

45
20

11

3
493

.
196

3. Other embedding dimension 4 cases
The example we have just studied is certainly not unique. Let us illustrate this with
two examples: If one adds the relation z 3 to the example in Theorem 1 we obtain the
following
k[x, y, z, u]
Ra = 3 2
(x , x y, (x + u)(y 2 + z 2 ) + yzu, z 3 , zu2 , u3 )
which has Hilbert series
1 + 3T + 6T 2 + 4T 3 − T 4 − 7T 5
1−T
Furthermore the homology of the Koszul complex of Ra is
total: 1 6

18 20 7

0:
1:

1 .
. .

.
.

2:
3:
4:

.
.
.

2 .
.
2 .
.
14 20 7

6
.
.

.
.

.
.

We get same Hilbert series and the same diagram of Koszul homology if we replace z 3 with
y 3 + z 3 to get the ring
Rb =

(x3 , x2 y, (x +

k[x, y, z, u]
+ z 2 ) + yzu, y 3 + z 3 , zu2 , u3 )

u)(y 2

Both these ring have transcendental series of Betti numbers. These two series can be
determined and they are different. For the case of Ra we have using DGAlgebras that
HKRa is generated by 6 skew-commuting variables X1 , X2 , X3 , X4 , X5 , X6 of degree (1,3)
(and 27 variables of higher degrees). The first 6 variables satisfy the following relations
X5 X6 , X3 X6 , X3 X5 , X1 X3 +X2 X4 −X4 X6 , X1 X2 and the Hilbert series of the corresponding Koszul dual is
1
Q∞
(1 − 2z)(1 − 3z) n=1 (1 − z n )
On the other hand, using DGAlgebras for Rb we find that HKRb is still generated by 6
variables X1 , X2 , X3 , X4 , X5 , X6 of degree (1,3) (but only 24 variables of higher degrees).
In this case the first 6 variables satisfy the following relations X3 X6 , X1 X3 + X2 X4 −
X4 X6 , X1 X2 and in this case the Hilbert series of the corresponding Koszul dual is
(1 −

2z)2 (1

1
Q∞
− z) n=1 (1 − z n )
12

Now we can continue our analysis as in section 2 and we obtain the following formulae:
PRa (Z) =

Z(1 + Z)2
Q
∞
(1 − Z − Z 2 )(1 − 3Z 2 − Z 3 ) n=1 (1 − Z 2n ) − 1 + 2Z + 3Z 2 − 6Z 3 − 7Z 4

and
PRb (Z) =

(1 − Z)(1 − 2Z 2 − Z 3 )

Z(1 + Z)2
2n
2
3
4
n=1 (1 − Z ) − 1 + 2Z + 3Z − 6Z − 7Z

Q∞
2

Note also that we could have started with many more examples in embedding dimension
4 than just our variation of the Anick case that we took in section 1.
Now let us also briefly analyze the case when
S=

(x3 , x2 y, (x

k[x, y, z, u]
+ u)(y 2 + z 2 ), zu2 , u3 )

Using DGAlgebras we obtain as before that
PS (Z) =

(1 − Z)(1 + z)2
1 − 3Z + 3Z 3 − 2Z 5

but we can also prove that neither Ext∗S (k, k) nor Ext∗HKS (k, k) are finitely generated as
algebras equipped with the Yoneda product. The case S is the 4-variable of the 5-variable
k[x, y, z, u, v]/(x2, xy, xz + zv, uv, v 2 ), a variant of which we treated in [R1], inspired by
Lemaire [LEM]. An alternative way to treat S is to observe that the quotient map
k[x, y, z, u]
(x3 , x2 y, zu2 , u3 )

−→ S

is a Golod map [LEV] so that k[x, y, z, u] → S is a composite of three Golod maps, and
use [R2].
All this leads support to the surmise that the embedding dimension 4 case could be
equally complicated as the general case, and one could even pose the
P
R
n
QUESTION 1: Let E4 be the set of series
n≥0 |Torn (k, k)|Z for (R, m) a local
commutative noetherian ring of embedding dimension ≤ 4 where k = R/m. Can E4 be
added to the list of 17 series of [A-Gu] that are all rationally related ?
One would certainly get a smaller set than E4 if one restricted oneself to rings of the
form k[x, y, z, u]/(f1, . . . fs ) where the relations fi were of degree ≤ 3 or ≤ 4 .
13

So far no embedding dimension 4 variant of the rings in [LR2] has been found. But
even in this restricted case of degree 3 relations one can get surprizes: The ring
SI = Q[x, y, z, u]/(u3, xy 2 , (x + y)z 2 , x2 u + zu2 , y ∗ u + xzu, y 2 z + yz 2 )
constructed in [R5] (inspired by Lukas Katthän [KAT]) which has only six relations of
degree 3, has as homology of the Koszul complex:
Total:

1

6 12 9

2

0:
1:

1
.

.
.

.
.

.
.

2:
3:

.
.

6 .
.
. 12 5

.
1

4:
5:

.
.

.
.

.
1

.
.

.
.

4
.

Furthermore the multiplication of elements of positive degree in this homology is zero (use
DGAlgebra), and
(1 + z)4
= 1+4z +12z 2 +40z 3 +130z 4 +422z 5 +1376z 6 +4476z 7 +. . .
2
3
4
5
(1 − 6z − 12z − 9z − 2z )
whereas the series of the total Betti numbers of SI is given by Macaulay2 as:
1 + 4z + 12z 2 + 40z 3 + 130z 4 + 421z 5 + 1371z 6 + 4454z 7 + . . .
so that SI is not a Golod ring. In [R5] we show that if one accepts relations of degree 3
and 4, there are hundreds of such exotic non-Golod rings.
QUESTION 2: For local rings R of embedding dimension 4 with only cubic relations,
are my R in Theorem 1 and the Ra and Rb essentially the only examples where we have
a transcendental PR (Z) ?
4. References
[A] D.J. Anick, A counterexample to a conjecture of Serre, Ann. of Math. 115, 1982, pp.
1-33; Correction, Ann. of Math. 116 , p. 661.
[A-CR] D.J. Anick, Construction d’espaces de lacets et d’anneaux locaux à séries de PoincarBetti non rationnelles, C.R. Acad. Sc. Paris 290, (1980), pp. A733-A736.
14

[A-Gu] D.J. Anick and Tor H. Gulliksen, Rational dependence among Hilbert and Poincare
series, Journal of Pure and Applied Algebra 38, 1985, pp. 135-157.
[AV0] Avramov, L. L.,The Hopf algebra of a local ring. (Russian), Izv. Akad. Nauk SSSR Ser.
Mat. 38, (1974), pp. 253-277. English translation [Math. USSR-Izv.8 (1974), 259-284].
[AV1] Avramov, L. L.,Obstructions to the existence of multiplicative structures on minimal
free resolutions. Amer. J. Math., 103 (1981), pp. 1-31.
[KAT] L. Katthän, A non-Golod ring with a trivial product on its Koszul homology,
http://arxiv.org/pdf/1511.04883.pdf
[LEM] Lemaire, Jean-Michel,Algèbres connexes et homologie des espaces de lacets, Lecture
Notes in Mathematics, 422, Springer-Verlag, Berlin-New York, 1974.
[LEV] Levin, Gerson, Local rings and Golod homomorphisms, Journal of Algebra 37, 1975,
pp 266289.
[L1] Löfwall, C., On the subalgebra generated by one-dimensional elements in the Yoneda
Ext–algebra, in Algebra, algebraic topology, and their interactions, ( J.–E. Roos, ed),
Lecture Notes in Math., vol.1183, Springer-Verlag, Berlin–New York, 1986, pp. 291338.
[LR1] C. Löfwall and J.-E. Roos, Cohomologie des algèbres de Lie graduées et séries de
Poincaré-Betti non rationnelles, C.R. Acad. Sc. Paris 290, 1980, pp. A733-A736.
[LR2] C. Löfwall and J.-E. Roos, A nonnilpotent 1-2-presented graded Hopf algebra whose
Hilbert series converges in the unit circle, Adv. Math.130, 1997, pp. 161-200.
[MOO] Moore, Frank, DGAlgebras. A package for Macaulay2
http://www.math.uiuc.edu/Macaulay2/Packages/
[R1] J.-E. Roos, Relations between the Poincaré-Betti series of Loop Spaces and of Local
rings, Springer Lecture Notes in Math.740,1979, 285-322.
[R2] J.-E. Roos, On the use of graded Lie algebras in the theory of local rings, Commutative
algebra: Durham 1981 (R. Y. S harp, ed.) London Math. Soc. Lecture Notes Ser. vol.
72, Cambridge Univ. Press, Cambridge, 1982, pp. 204230.
[R3] J.-E. Roos, A computer-aided study of the graded Lie-algebra of a local commutative
noetherian ring (with an Appendix by Clas Löfwall), Journal of Pure and Applied
Algebra 91 , 1994, pp. 255-315.
[R4] J.-E. Roos, Homological properties of the homology algebra of the Koszul complex of a
local ring: Examples and questions, Journal of Algebra 465, 2016, pp. 399-436.
[R5] J.-E. Roos, On some unexpected rings that are close to Golod rings, in preparation.
15

