A commutative noetherian local ring of embedding dimension 4 having a transcendental series of Betti numbers

TL;DR

This paper constructs specific examples of commutative noetherian local rings of embedding dimension 4 that exhibit transcendental Betti number series, revealing complex algebraic behaviors.

Contribution

It introduces new examples of local rings with exotic properties, utilizing computational tools and building on prior theoretical results.

Findings

Constructed a local ring with transcendental Betti series.

Developed methods combining Macaulay2 and theoretical insights.

Produced additional rings with unusual algebraic properties.

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 {\tt Macaulay2}-package {\tt DGAlgebras} by Frank Moore, combined with and inspired by results by Anick, Avramov, Backelin, Katth\"an, Lemaire, Levin, L\"ofwall and others.

Equations

$$R=\frac{k[x,y,z,u]}{(x^3,x^{2}y,(x+u)(y^2+z^2)+yzu,z{u}^2,u^3)}$$

$$P_R(Z)=\sum _{i\ge 0}|{\mathrm{Tor}}_i^R(k,k)|Z^i$$

$$P_R(Z)=\frac{Z(1+Z{)}^2}{(1+Z)(1-Z-Z^2{)}^2{\prod }_{n=1}^{\infty }(1-Z^{2n})-1+2Z+2Z^2-4Z^3-4Z^4}$$

$$R=\frac{k[x,y,z,u]}{(x^3,x^{2}y,(x+u)(y^2+z^2),z{u}^2,u^3)}$$

$$R=\frac{k[x_1,x_2,x_3,x_4,x_5]}{(x_1^2,x_1{x}_2,x_1{x}_3+x_2{x}_4+x_3{x}_5,x_4{x}_5,x_5^2)}$$

$$R^{!}(t)=\frac{(1+t{)}^2}{(1-t-t^2{)}^2}\prod _{n=1}^{\infty }\frac{1+t^{2n-1}}{1-t^{2n}}$$

$$1/P_R(Z)=(1+1/Z)/R^{!}(Z)-H(-Z)/Z$$

$$R^{!}(t)=\frac{1}{(1-2t{)}^2}\prod _{n=1}^{\infty }\frac{1+t^{2n-1}}{1-t^{2n}}$$

$$S=\frac{k[x,y,z,u]}{(x^3,x^{2}y,z{u}^2,u^3)}$$

$$y{u}^2,x{u}^2,z^{2}u,yzu,xzu,y^{2}u,xyu,x^{2}u,z^3,y{z}^2,x{z}^2,y^{2}z,xyz,x^{2}z,y^3,x{y}^2$$

$$P_1(z)=1+4z+11z^2+27z^3+62z^4+137z^5+295z^6+\dots P_2(z)=1+4z+11z^2+27z^3+64z^4+152z^5+364z^6+\dots$$

$$P_3(z)=1+4z+11z^2+28z^3+69z^4+168z^5+407z^6+\dots P_4(z)=1+4z+11z^2+28z^3+69z^4+169z^5+414z^6+\dots$$

$$P_5(z)=1+4z+11z^2+28z^3+69z^4+172z^5+431z^6+\dots P_6(z)=1+4z+11z^2+28z^3+70z^4+173z^5+427z^6+\dots$$

$$P_7(z)=1+4z+11z^2+28z^3+70z^4+177z^5+451z^6+\dots P_8(z)=1+4z+11z^2+29z^3+74z^4+188z^5+476z^6+\dots$$

$$P_9(z)=1+4z+11z^2+29z^3+75z^4+193z^5+496z^6+\dots P_{10}(z)=1+4z+11z^2+29z^3+75z^4+193z^5+498z^6+\dots$$

$$P_{11}(z)=1+4z+11z^2+29z^3+75z^4+194z^5+503z^6+\dots P_{12}(z)=1+4z+11z^2+30z^3+80z^4+213z^5+567z^6+\dots$$

$$P_{13}(z)=1+4z+11z^2+30z^3+81z^4+218z^5+587z^6+\dots P_{14}(z)=1+4z+11z^2+30z^3+82z^4+224z^5+612z^6+\dots$$

$$P_{15}(z)=1+4z+11z^2+31z^3+86z^4+238z^5+660z^6+\dots P_{16}(z)=1+4z+11z^2+31z^3+87z^4+244z^5+685z^6+\dots$$

$$P_{17}(z)=1+4z+11z^2+31z^3+88z^4+249z^5+705z^6+\dots P_{18}(z)=1+4z+11z^2+32z^3+92z^4+263z^5+755z^6+\dots$$

$$P_{19}(z)=1+4z+11z^2+32z^3+93z^4+269z^5+780z^6+\dots P_{20}(z)=1+4z+11z^2+33z^3+99z^4+294z^5+877z^6+\dots$$

$$P_1(z)=1+5z+15z^2+37z^3+82z^4+170z^5+337z^6+\dots P_2(z)=1+5z+15z^2+38z^3+88\ast z^4+192z^5+406z^6+\dots$$

$$P_3(z)=1+5z+15z^2+38z^3+89z^4+199z^5+432z^6+\dots P_4(z)=1+5z+15z^2+38z^3+91z^4+216z^5+516z^6+\dots$$

$$P_5(z)=1+5z+15z^2+39z^3+95z^4+222z^5+506z^6+\dots P_6(z)=1+5z+15z^2+39z^3+96z^4+231z^5+553z^6+\dots$$

$$P_7(z)=1+5z+15z^2+39z^3+97z^4+237z^5+575z^6+\dots P_8(z)=1+5z+15z^2+39z^3+99z^4+254z^5+659z^6+\dots$$

$$P_9(z)=1+5z+15z^2+40z^3+104z^4+268z^5+689z^6+\dots P_{20}(z)=1+5z+15z^2+41z^3+112z^4+306z^5+836z^6+\dots$$

$$\frac{k[x,y,z,u]}{(x^3,x^{2}y,(x+u)(y^2+z^2)+yzu,z{u}^2,u^3)}$$

$$E_{p,q}^2={\mathrm{Tor}}_{p,q}^{HKR}(k,k)=>{\mathrm{Tor}}_{\ast }^R(k,k)//{\mathrm{Tor}}_{\ast }^{\tilde{R}}(k,k)$$

$$X_4{X}_5,X_2{X}_5,X_1{X}_5,X_3{X}_4,X_2{X}_4,X_1{X}_4,X_2{X}_3,X_4{X}_7,X_2{X}_7,X_1{X}_7,X_5{X}_6,X_4{X}_6,X_3{X}_6.X_6{X}_7$$

$$C=\frac{k[X_1,X_2,X_3,X_4,X_5,X_6,X_7]}{(X_3{X}_5,X_1{X}_3+X_2{X}_4-X_4{X}_5,X_1{X}_2,X_1{X}_6,X_2{X}_6,X_3{X}_7,X_5{X}_7,X_6{X}_6,X_7{X}_7)}$$

$$C^{!}\bigsqcup k<x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16},x_{17},x_{18},x_{19}>$$

Full text

A commutative noetherian local ring of embedding dimension 4 having a

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 ${\rm dim}_{k}V$):

THEOREM 1 – Let $k$ be a field and $k[x,y,z,u]$ the polynomial ring in four variables of degree one and let

[TABLE]

be the graded quotient ring. Let

[TABLE]

be the corresponding series of Betti numbers of $R$. Then $P_{R}(Z)$ is a transcendental function:

[TABLE]

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 $\geq 5$ 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)

[TABLE]

has a $P_{R}(Z)=(1-Z^{2})(1+Z)/(1-3Z-3Z^{3}-2Z^{5})$ i.e. a rational function but is such that the Yoneda Ext-algebra ${\rm Ext}^{*}_{R}(k,k)$ is not finitely generated as an algebra. This last phenomenon was also previously only known in the embedding dimension $\geq 5$ 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 $P_{R}(Z)$ 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:

[TABLE]

In this case the Hilbert series of R is $H(t)=(1+3t+t^{2}-2t^{3}+t^{4})/(1-t)^{2}$ and the Hilbert series of the Koszul dual $R^{!}$ of R is

[TABLE]

and

[TABLE]

Remark: For the case studied by Anick [A] we had that $H(t)=1+5t+7t^{2}$ and

[TABLE]

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 $S_{2}=k[x,y,z,u]/(x^{2},xy,zu,u^{2})$ where $x,y,z,u$ correspond to $x_{1},x_{2},x_{4},x_{5}$ and try to add an extra relation corresponding to $x_{1}x_{3}+x_{2}x_{4}+x_{3}x_{5}$. 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 $S_{2}$. There are 64 such linear combinations but each of them leads to a rational $P_{R}(Z)$ (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

[TABLE]

and try to add a linear combination (coefficients 0 or 1) of the 16 non-zero cubic monomials

[TABLE]

in S. There are $2^{16}=65536$ 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;

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^2y,zu^2,u^3)

time hypers = makeHyperplanes J;

time scan(numgens source hypers, i -> doit i);

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

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Which one of these cases could give a transcendental $P_{R}(Z)$ ?

One can guess a possibility by studying how the transcendental $P_{R}(z)$ comes up in five variables when we add linear combinations of quadratic monomials to the ring in 5 variables: ${\bf Q}[x,y,z,u,v]/(x^{2},xy,uv,v^{2})$ i.e. modifying the previous input programme by replacing R by $QQ[x,y,z,u,v]$, J by $(x^{2},x*y,u*v,v^{2})$ and the line

I := matrix basis(3,coker gens J);

by the line

I := matrix basis(2,coker gens J);

There are now “only” $2^{11}=2048$ cases to study and now we obtain only 10 possibilities for the $P_{R}(z)$ up to degree 6 presented here in increasing order:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

There are e.g. 1024 cases of an extra quadratic relation corresponding to $P_{1}(z)$ and 576 cases corresponding to $P_{4}(z)$, but only 8 for $P_{2}(z)$ including the case of the extra relation $xz+yu+zv$ leading to the transcendental series mentioned in the introduction (the other 7 cases of $P_{2}(z)$ lead to the same total series). Furthermore, there are also 8 cases for $P_{6}(z)$ and they all lead to cases where the Yoneda Ext-algebra ${\rm Ext}^{*}_{R}(k,k)$ 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 $P_{10}(z)$ 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 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

[TABLE]

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:

[TABLE]

From this we can guess that the multiplication in $HKR={\rm Tor}_{*}^{k[x,y,z,u]}(k[x,y,z,u]/J,k)$, 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 $\tilde{R}=k[x,y,z,u]$ and $R={\tilde{R}}/J$:

[TABLE]

by using the following infile for Macaulay:

loadPackage(‘‘DGAlgebras’’)

R=QQ[x,y,z,u]/ideal(x^3,x^2y,zu^2,u^3,xy^2+xz^2+y^2u+yzu+z^2u)

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

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

[TABLE]

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 $X_{1},X_{2},X_{3},X_{4},X_{5}$ of degree (1,3),two generators $X_{6},X_{7}$ of degree (2,4), three generators $X_{8},X_{9},X_{10}$ of degrees (2,5),(2,6),(2,8) respectively, four generators $X_{11},X_{12},X_{13},X_{14}$ of degree (3,7), two generators $X_{15},X_{16}$ of degree (3,9), two generators $X_{17},X_{18}$ of degree (4,8) and one generator $X_{19}$ of degree (4,10).

Furthermore the first five variables $X_{1},X_{2},X_{3},X_{4},X_{5}$ skew-commute, the variables $X_{6},X_{7},X_{8},X_{9},X_{10}$ commute among themselves and with $X_{1},X_{2},X_{3},X_{4},X_{5}$, etc.

The relations are as follows: The product of the last 12 variables $X_{8},\ldots X_{19}$ with all the 19 variables $X_{1}\ldots X_{19}$ are 0. For the first five generators $X_{1},X_{2},X_{3},X_{4},X_{5}$ we only have the following three quadratic relations: $X_{3}X_{5},X_{1}X_{3}+X_{2}X_{4}-X_{4}X_{5},X_{1}X_{2}$, and if we also take into account the additional variables $X_{6},X_{7}$ we have the extra quadratic relations $X_{1}X_{6},X_{2}X_{6},X_{3}X_{7},X_{5}X_{7},X_{6}X_{6},X_{7}X_{7}$. It follows that the if

[TABLE]

then the Koszul dual $HKR^{!}$ of $HKR$ can be presented as a coproduct

[TABLE]

where the second algebra in the coproduct is the free algebra on the dual generators $x_{8},\ldots x_{19}$ of $X_{8},\ldots X_{19}$.

Furthermore, it is well-known (cf. e.g. [LEM]) that for the coproduct $A\sqcup B$ of two graded connected algebras, we have the following formula for the respective Hilbert series

[TABLE]

(this works also for the bigraded case) and it follows that

[TABLE]

and we have therefore reduced our problem to calculate $C^{!}(z,z)$ where C only involves the first 7 variables $X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7}$

We start with an algebra $D$ that involves only the first 5 variables $X_{1},X_{2},X_{3},X_{4},X_{5}$:

[TABLE]

The variables skewcommute. The Koszul dual $D^{!}$ is the quotient of the free algebra in the dual variables $x_{i}$ of the $X_{i}$ :

[TABLE]

where the $[x_{i},x_{j}]$ 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 $g_{D}$ of $D^{!}$ sits in the middle of an extension

[TABLE]

i.e. is the extension of the product of two free Lie algebras $f_{1}$ generated by $x_{1}$ and $x_{2}$ and $f_{2}$ generated by $x_{3}$ and $x_{5}$ with the infinite abelian Lie algebra

[TABLE]

where the $a_{i}$ are onedimensional generated by $a_{1}=x_{4}$, $a_{2}=[x_{5},x_{4}]$, $a_{3}=[x_{5},[x_{5},x_{4}]]$ etc. which commute and by the 2-cocycle $\gamma:f_{1}\times f_{2}\rightarrow a$ defined by $\gamma(x_{1},x_{3})=-\gamma(x_{3},x_{1})=a_{2}$ and $\gamma(x_{2},x_{5})=-\gamma(x_{5},x_{2})=a_{2}$. It follows that

[TABLE]

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 $X_{6},X_{7}$ (which commute with the $X_{1},X_{2},X_{3},X_{4},X_{5}$) 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

[TABLE]

with the ideal generated by

[TABLE]

as before and the extra ideal generators:

[TABLE]

where $x_{6}$ and $x_{7}$ are dual to $X_{6}$ and $X_{7}$. From the preceding presentation one sees that the Lie algebra $g_{C}$ of $C^{!}$ sits in the middle of an extension:

[TABLE]

where $F_{1}$ is the free Lie algebra generated by $x_{1},x_{2},x_{6}$ of degrees $1,1,2$ and $F_{2}$ is the free Lie algebra generated by $x_{3},x_{5},x_{7}$ of degrees $1,1,2$ and the $a_{i}$:s are as before, and the $x_{6}$ and $x_{7}$ operate in the trivial way on the $a_{i}$. Furthermore we have a new cocycle

[TABLE]

where as before $\Gamma$ is defined by its value on the generators by $-\Gamma(x_{3},x_{1})=\Gamma(x_{1},x_{3})=a_{2}$ and $-\Gamma(x_{5},x_{2})=\Gamma(x_{2},x_{5})=a_{2}$ and being zero for all other pairs of generators. By calculating $H^{2}(F_{1}\times F_{2},a)$ for (8) one sees that the relations are those of $C^{!}$ and therefore the Hilbert series

[TABLE]

if the variables $x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}$ 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

[TABLE]

Using the formulae (8) and (12) we obtain

[TABLE]

But $I^{3}=0$ and

[TABLE]

and this gives that $HKR(x,y)=1+5xy+5xy^{2}+6xy^{3}+3xy^{4}+7x^{2}y^{2}+6x^{2}y^{3}+x^{2}y^{4}$, so that

[TABLE]

and finally the formula

[TABLE]

gives that

[TABLE]

[TABLE]

and if the Avramov spectral sequence degenerated this series should be the same as

$(1+z)^{4}/P_{R}(z)$ so the final result should be

[TABLE]

[TABLE]

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 $d^{r}$ 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 $\leq 10$ 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

[TABLE]

which takes into account the extra grading of $R$ (we have that $P_{R}(z,1)$ is the old $P_{R}(z)$) given as

[TABLE]

where

[TABLE]

and $H$ is equal to

[TABLE]

and the expansion of $P_{R}(x,y)$ of (15) up to degree 11 gives the same diagram of graded Betti numbers that was earlier found by the Macaulay2 calculation:

[TABLE]

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

[TABLE]

which has Hilbert series

[TABLE]

Furthermore the homology of the Koszul complex of $R_{a}$ is

[TABLE]

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

[TABLE]

Both these ring have transcendental series of Betti numbers. These two series can be determined and they are different. For the case of $R_{a}$ we have using DGAlgebras that $HKR_{a}$ is generated by 6 skew-commuting variables $X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}$ of degree (1,3) (and 27 variables of higher degrees). The first 6 variables satisfy the following relations $X_{5}X_{6},X_{3}X_{6},X_{3}X_{5},X_{1}X_{3}+X_{2}X_{4}-X_{4}X_{6},X_{1}X_{2}$ and the Hilbert series of the corresponding Koszul dual is

[TABLE]

On the other hand, using DGAlgebras for $R_{b}$ we find that $HKR_{b}$ is still generated by 6 variables $X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}$ of degree (1,3) (but only 24 variables of higher degrees). In this case the first 6 variables satisfy the following relations $X_{3}X_{6},X_{1}X_{3}+X_{2}X_{4}-X_{4}X_{6},X_{1}X_{2}$ and in this case the Hilbert series of the corresponding Koszul dual is

[TABLE]

Now we can continue our analysis as in section 2 and we obtain the following formulae:

[TABLE]

and

[TABLE]

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

[TABLE]

Using DGAlgebras we obtain as before that

[TABLE]

but we can also prove that neither ${\rm Ext}^{*}_{S}(k,k)$ nor ${\rm Ext}^{*}_{HKS}(k,k)$ are finitely generated as algebras equipped with the Yoneda product. The case $S$ is the 4-variable of the 5-variable $k[x,y,z,u,v]/(x^{2},xy,xz+zv,uv,v^{2})$, 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

[TABLE]

is a Golod map [LEV] so that $k[x,y,z,u]\rightarrow S$ 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

QUESTION 1: Let ${\cal E}4$ be the set of series $\sum_{n\geq 0}|{\rm Tor}^{R}_{n}(k,k)|Z^{n}$ for $(R,m)$ a local commutative noetherian ring of embedding dimension $\leq 4$ where $k=R/m$. Can ${\cal E}4$ be added to the list of 17 series of [A-Gu] that are all rationally related ?

One would certainly get a smaller set than ${\cal E}4$ if one restricted oneself to rings of the form $k[x,y,z,u]/(f_{1},\ldots f_{s})$ where the relations $f_{i}$ were of degree $\leq 3$ or $\leq 4$ .

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

[TABLE]

constructed in [R5] (inspired by Lukas Katthän [KAT]) which has only six relations of degree 3, has as homology of the Koszul complex:

[TABLE]

Furthermore the multiplication of elements of positive degree in this homology is zero (use DGAlgebra), and

[TABLE]

whereas the series of the total Betti numbers of $S_{I}$ is given by Macaulay2 as:

[TABLE]

so that $S_{I}$ 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 $R_{a}$ and $R_{b}$ essentially the only examples where we have a transcendental $P_{R}(Z)$ ?

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 Poincaré-Betti non rationnelles, C.R. Acad. Sc. Paris 290, (1980), pp. A733-A736.

[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 266–289.

[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. 291-338.

[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. 204–230.

[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.