# A Short Note on Coproducts of Abelian pro-Lie Groups

Wolfgang Herfort
Institute for Analysis and Scientific Computation
Technische Universität Wien
Wiedner Hauptstraße 8-10/101
Vienna, Austria<EMAIL_ADDRESS>,  Karl H. Hofmann
Fachbereich Mathematik
Technische Universität Darmstadt
Schlossgartenstr. 7
e 64289 Darmstadt, Germany<EMAIL_ADDRESS>and  Francesco
G. Russo
Department of Mathematics and Applied Mathematics
University of Cape Town
Private Bag X1, Rondebosch 7701
Cape Town, South Africa<EMAIL_ADDRESS>
###### Abstract.

The notion of conditional coproduct of a family of abelian pro-Lie groups in
the category of abelian pro-Lie groups is introduced. It is shown that the
Cartesian product of an arbitrary family of abelian pro-Lie groups can be
characterized by the universal property of the conditional coproduct.

###### Key words and phrases:

Pro-Lie groups, coproductsMathematics Subject Classification 2020: 22E20;
22A05

In [HM77] the second author and S. Morris have provided criteria when the
(co)product of a family of locally compact abelian groups exists. For
profinite abelian groups the product of any family $(A_{i})_{i\in I}$ of
profinite groups exists and agrees with the cartesian product $P:=\prod_{i\in
I}A_{i}$. J. Neukirch has shown in [Neu71] that $P$ has a universal property
resembling that of a coproduct (direct sum) in the category of (discrete)
abelian groups. In the present note we present a version of his result, valid
for cartesian products of the much larger family of abelian pro-Lie groups
(see [HM07, Ch. 5]). For formulating our result, we need to adapt the
concepts, originally introduced for families of profinite groups by J.
Neukirch in [Neu71] (see also [RZ10, D.3]), to the category of abelian pro-Lie
groups.

######  Definition 1.

Let $(A_{j})_{\in J}$ be a family of topological groups, $H$ a topological
group, and $\mathbb{F}=(\phi_{j})_{j\in J}$, $\phi_{j}\colon A_{j}\to H$, a
family of continuous homomorphisms. We say that $\mathbb{F}$ is convergent, if
for every identity neighborhood $U$ of $H$ the set $J_{U}:=\\{j\in
J:\phi_{j}(A_{j})\not\subseteq U\\}$ is finite.

######  Example 2.

For any family $(A_{j})_{j\in J}$ of topological groups let $H=\prod_{j\in
J}A_{j}$ be the cartesian product with the Tychonov topology. Let the family
${\mathbb{F}}=(\tau_{j})_{j\in J}$ of natural morphisms $\tau_{j}\colon
A_{j}\to H$ be given by

$\tau_{j}(a)=(a_{k})_{k\in J}\mbox{ with }a_{j}=a\mbox{ and }a_{k}=0\mbox{
otherwise.}$

Then ${\mathbb{F}}$ is convergent.

This follows immediately from the definition of the product topology on $H$.
The morphisms $\tau_{j}$ are called the natural embeddings.

We define the conditional coproduct by means of a universal property,
resembling the one of the coproduct (direct sum) of abelian discrete groups:

######  Definition 3.

In a category ${\mathcal{A}}$ of topological groups we call $G$ a conditional
coproduct of the family $(A_{j})_{j\in J}$ of objects if there is a convergent
family $\tau_{j}\colon A_{j}\to G$, $j\in J$ of morphisms such that for every
convergent family of morphisms $\psi_{j}\colon A_{j}\to H$, $j\in J$ in
$\mathcal{A}$ there is a unique morphism $\omega\colon G\to H$ such that
$\psi_{j}=\omega\circ\tau_{j}$ for all $j\in J$. The morphisms $\tau_{j}$ are
called the coprojections of the conditional coproduct.

We shall prove the following Theorem:

######  Theorem 4.

In the category of abelian pro-Lie groups, the conditional coproduct of a
family $(A_{j})_{j\in J}$ of abelian pro-Lie groups is the cartesian product
$P:=\prod_{j\in J}A_{j}$ for the canonical embeddings $\tau_{j}\colon A_{j}\to
P$.

But first we secure the uniqueness of the conditional coproduct:

######  Proposition 1.

If $G$ and $G^{\prime}$ are conditional coproducts of a family $(A_{j})_{j\in
J}$ of topological groups in a category $\mathcal{A}$ for the convergent
families $\tau_{j}\colon A_{j}\to G$ and $\tau^{\prime}_{j}\colon A_{j}\to
G^{\prime}$, $j\in J$ of morphisms in $\mathcal{A}$, then there is a natural
isomorphism $\lambda\colon G\to G^{\prime}$ such that
$\tau^{\prime}_{j}=\lambda\circ\tau_{j}$ for all $j\in J$.

###### Proof.

By Definition 2, since $G$ is a conditional coproduct of the family
$(A_{j})_{j\in J}$ with the coprojections $\tau_{j}$, $j\in J$, there is a
unique morphism $\lambda\colon G\to G^{\prime}$ such that

(1) $\forall j\in J)\,\lambda\circ\tau_{j}=\tau^{\prime}_{j}\colon A_{j}\to
G^{\prime}.$

Likewise, since $G^{\prime}$ is also a conditional coproduct of the family
$(A_{j})_{j\in J}$ with the coprojections $\tau^{\prime}_{j}$, $j\in J$, there
is a unique morphism $\lambda^{\prime}\colon G^{\prime}\to G$ such that

(2) $(\forall j\in J)\,\lambda^{\prime}\circ\tau^{\prime}_{j}=\tau_{j}\colon
A_{j}\to G.$

Therefore, by (1) and (2), we have

(3) $(\forall j\in
J)\,\tau_{j}=\lambda^{\prime}\circ\tau^{\prime}_{j}=\lambda^{\prime}\circ\lambda\circ\tau_{j}\colon
G\to G.$

However, trivially we also have,

(4) $(\forall j\in J)\,\tau_{j}=\mathop{\rm
id}\nolimits_{G}\circ\tau_{j}\colon G\to G.$

Therefore, by the uniqueness in Definition 3, from (3) and (4) we have

(5) $\lambda^{\prime}\circ\lambda=\mathop{\rm id}\nolimits_{G}.$

Now by exchanging the roles of $G$ and $G^{\prime}$ we also have

(6) $\lambda\circ\lambda^{\prime}=\mathop{\rm id}\nolimits_{G^{\prime}}.$

Hence by (5) and (6), $\lambda$ is an isomorphism, which we had to show. ∎

We note that for profinite groups the conditional coproduct agrees with the
free pro-$\mathcal{C}$ product for $\mathcal{C}$ the variety of abelian
profinite groups, see [Neu71, RZ10].

Let $\mathcal{A}$ be the category of topological abelian pro-Lie groups (i.e.
groups which are projective limits of Lie groups: see [HM07, pp. 160ff. and
Chapter 5]). Each pro-Lie group $G$ has a filterbasis ${\mathcal{N}}(G)$ of
closed normal subgroups such that $G/N$ is a Lie group, and
$G\cong\lim_{N\in{\mathcal{N}}(G)}G/N$. (See e.g. [HM07, p.160, Definition
A.])

Recall that every locally compact abelian group is a pro-Lie group, every
almost connected locally compact group is a pro-Lie group by Yamabe’s Theorem.
Trivially, then, every profinite group is a pro-Lie group. Every cartesian
product $P=\prod_{j\in J}A_{j}$ of pro-Lie groups $A_{j}$ is itself a pro-Lie
group.

######  Lemma 5.

Let $H$ be an abelian pro-Lie group and ${\mathbb{F}}$ be a convergent family
of morphisms $\psi_{j}:A_{j}\to H$. Then, for each $N\in{\mathcal{N}}(H)$, the
set $\\{j\in J:\psi_{j}(A_{j})\not\subseteq N\\}$ is finite.

###### Proof.

Let $N\in{\mathcal{N}}(H)$. The Lie group $H/N$ has an identity neighborhood
$V$ in which $\\{0\\}$ is the only subgroup of $H/N$. Now let $p\colon H\to
H/N$ be the quotient morphism and set $U=p^{-1}(V)$.

Therefore $\psi_{j}(A_{j})\not\subseteq N$ implies
$\psi_{j}(A_{j})\not\subseteq U$. However the set of $j$ satisfying this
condition is finite by Definition 1 applying to the conditional coproduct of
the family $(A_{j})_{j\in J}$. This completes the proof of the Lemma. ∎

###### Proof of Theorem 4.

The uniqueness, up to isomorphism, of the conditional coproduct follows from
Proposition 1.

Thus, according to Definition 3, we need to show that given an abelian pro-Lie
group $H$ and a convergent family of morphisms $\psi_{j}:A_{j}\to H$ then
there exists a unique morphism $\omega:P\to H$ with
$\psi_{j}=\omega\circ\tau_{j}$ for all $j\in J$.

We note first that every $x\in P$ has a presentation

(7) $x=(\tau_{j}(a_{j}))_{j\in J}$

for unique elements $a_{j}\in A_{j}$. Denote by $\mathcal{N}(H)$ the set of
all closed subgroups of $H$ such that $H/N$ is a Lie group. It is a
consequence of [HM07, Theorem 3.27] that $\mathcal{N}(H)$ is a filter basis of
closed subgroups of $H$ and that

(8) $H\cong\varprojlim_{N\in\mathcal{N}}H/N$

algebraically and topologically.

Fix $N\in\mathcal{N}(H)$ and let let $J_{N}:=\\{j\in J:\psi_{j}(A_{j})\not\leq
N\\}$. Then, by Lemma 5, the set $J_{N}$ is finite and, taking the
presentation Eq. (7) for $x\in P$ and $\tau_{j}(A_{j})\leq N$ for all $j\notin
J_{N}$ into account, we obtain a well-defined morphism $\omega_{N}:P\to H/N$
by letting

(9) $\omega_{N}(x):=\sum_{j\in J_{N}}\psi_{j}(a_{j})+N.$

For subgroups $M\leq N$ of $H$, both in $\mathcal{N}(H)$, let $\pi_{NM}:H/M\to
H/N$ denote the canonical epimorphism.

For $M\leq N$ one obtains from Eq. (9) the compatibility relation

(10) $\omega_{N}=\pi_{NM}\circ\omega_{M},$

as depicted in the following diagram:

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{N}}$$\scriptstyle{\omega_{M}}$$\scriptstyle{\exists!\,\omega}$$\textstyle{\cdots}$$\textstyle{H/N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{NM}}$$\textstyle{H/M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ldots}$$\textstyle{H=\varprojlim_{N\in\mathcal{U}}H/N}$

Taking the relations in Eq. (10) into account we see that the universal
property of the inverse limit $H=\varprojlim_{N\in\mathcal{U}}H/N$ implies the
existence of a unique continuous homomorphism $\omega:P\to H$, which satisfies
the desired relations

(11) $(\forall j\in J)\ \ \ \psi_{j}=\omega\circ\tau_{j}.$

∎

Notes.

A coproduct of a family of objects in a category $\mathcal{A}$ is a product in
the category obtained by reversing all arrows. Curiously, while products are
usually considered simple concepts, coproducts are often tricky in many
categories $\mathcal{A}$ other than the category of abelian groups. Therefore,
in conclusion of this note, a few general comments may be in order.

One of the early surprises is that in the familiar category of groups, the
coproduct of ${\mathbb{Z}}(2)$ and ${\mathbb{Z}}(3)$ is PSL$(2,{\mathbb{Z}})$.

In any category $\mathcal{A}$ with a well-introduced dual category, such as
the category of locally compact abelian groups, the coproduct
$\coprod_{j}A_{j}$ of a family $A_{j}$, $j\in J$, is naturally isomorphic to
the dual $\widehat{P}$ of $P:=\prod_{j}\widehat{A_{j}}$, the product of its
duals.

Even in special cases, such as the case of compact abelian groups $A_{j}$, the
result is a complicated coproduct, since the character group of an infinite
product of discrete abelian groups may be hard to deal with.

If $\mathcal{A}$ is the category of profinite abelian groups, then its dual is
the category $\mathcal{T}$ of abelian torsion groups. The product in
$\mathcal{T}$ of a family of torsion groups $T_{j}$ is the torsion group
$\mathop{\mathrm{missing}}{tor}(\prod_{j}T_{j})$ of the cartesian product. So
by the time we arrive at the coproduct of, say, an unbounded family of cyclic
groups $A_{j}$ in $\mathcal{A}$, we may have a complicated object
$\coprod_{j\in J}A_{j}$ in our hands.

Therefore, any special situation may be welcome, where a coproduct is
lucid–even when its scope of application may be restricted. An example of such
a situation is our present conditional coproduct in the rather large yet
reasonably well-understood category of abelian pro-Lie groups (see Chapter 5
of [HM07]). The authors encountered such a coproduct in a study of certain
locally compact abelian $p$-groups. Our conditional coproduct covers a
somewhat restricted supply of families of morphisms which we call
“convergent”. Here we encounter the rather extraordinary event that for each
of such families their conditional coproduct agrees with their cartesian
product. Classically, one is familiar with a situation of coproducts agreeing
with products in the category of finite abelian groups which, after all, is
rather special.

## References

  * [HM77] Karl Heinrich Hofmann and Sidney A. Morris.  Locally compact products and coproducts in categories of topological groups.  Bull. Aust. Math. Soc., 17:401–417, 1977.
  * [HM07] Karl H. Hofmann and Sidney A. Morris.  The Lie theory of connected pro-Lie groups, volume 2 of EMS Tracts in Mathematics.  European Mathematical Society (EMS), Zürich, 2007.  A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups.
  * [Neu71] Jürgen Neukirch.  Freie Produkte pro-endlicher Gruppen und ihre Kohomologie.  Arch. Math. (Basel), 22:337–357, 1971.
  * [RZ10] Luis Ribes and Pavel Zalesskii.  Profinite groups, volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer-Verlag, Berlin, second edition, 2010.


# Ideals of generic forms

Ralf Fröberg

###### Abstract

We determine the Hilbert series of some classes of ideals generated by generic
forms of degree two and three, and investigate the difference to the Hilbert
series of ideals generated by powers of linear generic forms of the
corresponding degrees.

Keywords Hilbert series, generic forms, powers of linear forms

##  1 Introduction

Let $R=k[x_{1},\ldots,x_{n}]/(f_{1},\ldots,f_{r})$, where $k$ is a field and
$f_{i}$ is a homogeneous polynomial of degree $d_{i}$, $i=1,\ldots,r$. Then
$R$ is a graded algebra, $R=\oplus_{i\geq 0}R_{i}$, where $R_{i}$ are finite
dimensional vector spaces over $k$. The generating series of these dimensions,
$H_{R}(t)=\sum_{i\geq 0}\dim_{k}R_{i}t^{i}$ is called the Hilbert series of
$R$. In case $R$ is the coordinate ring of a projective variety, the Hilbert
series contains geometric information of the variety, such as the dimension
and the degree. It is shown in [6] that for fixed $n,d_{1},\ldots,d_{r}$ there
are only a finite number of Hilbert series. If we think of $f_{i}$ as a linear
combination of all monomials of degree $d_{i}$, we can view the ideal as a
point in ${\mathbb{A}}^{N}_{k}$, where $N=\sum_{i=1}^{r}{n+d_{i}-1\choose
d_{i}}$. It is further shown in [6] that if $k$ is infinite, there is an
nonempty Zariski open set in ${\mathbb{A}}^{N}_{k}$ where the Hilbert series
is constant, so ”almost all” ideals give the same Hilbert series. We call the
ideal in this set generic. If $k={\mathbb{C}}$, we can think of the case when
all $N$ coefficients are algebraically independent over ${\mathbb{Q}}$.

There is a longstanding conjecture on what this series is.

###### Conjecture 1

[[3]] The Hilbert series of $k[x_{i},\ldots,x_{n}]/(f_{1},\ldots,f_{r})$,
where $k$ is infinite anf $f_{i}$ is a generic form of degree $d_{i}$[ is
$[\frac{\prod_{i=1}^{r}(1-t^{d_{i}})}{(1-t)^{n}}]$. Here
$[\sum_{i=0}^{\infty}a_{i}t^{i}]=\sum_{i=0}^{\infty}b_{i}t^{i}$, where
$b_{i}=a_{i}$ if $a_{j}>0$ for all $j\leq i$ and $b_{i}=0$ otherwise.

The conjecture is trivially true for $r\leq n$. It is proved for $n\leq 3$,
[1], for $r=n+1$, [9], and for some other cases, see [5]. From now on w
suppose that $d_{i}=d$ for all $i$, so the conjectured series is
$[\frac{(1-t^{d})^{r}}{(1-t)^{n}}]$. This note is an amplification of [4],
where the conjecture was proved for $d=2,n\leq 11$ and $d=3,n\leq 8$. In [4]
was also investigated when the series for $r$ generic forms of degree $d$
equals the series for $r$ $d$th powers of generic linear forms.

##  2 Results

Now we extend the results from [4].

###### Theorem 1

Conjecture 1 is true if if $d=2$ and $n=12$ or $n=13$ and if $d=3$ and $n=9$.

It is known that the true Hilbert series is greater or equal to the
conjectured lexicographically, see [3]. Thus it is sufficient to find one
example for the parameters $(n,r,d)$ which gives equality, to prove the
theorem for those parameters. This has been done with computer for the ideals
$(x_{1}^{d},\ldots,x_{n}^{d})+J$, where $J$ is generated by $r-n$ random forms
of degree $d$. The calculations where made in Macaulay2, [7]. More interesting
is perhaps next theorem. Let $Q_{n,r,d}(t)$ be the Hilbert series of
$k[x_{1},\ldots,x_{n}]/(l_{1}^{d},\ldots,l_{r}^{d})$, where $l_{i}$ are
generic linear forms, and let $F_{n,r,d}(t)=[\frac{1-t^{d})^{r}}{(1-t)^{n}}]$
, which in the cases we study is the correct Hilbert series for
$k[x_{i},\ldots,x_{n}]/(f_{1},\ldots,f_{r})$, where the $f_{i}$’s are generic
forms of degree $d$.

###### Theorem 2

For $n=12$, $Q_{n,r,2}(t)=F_{n,r,2}(t)$ unless $r=n+2$. For $n=13$,
$Q_{n,r,2}(t)=F_{n,r,2}(t)$ unless $r=n+2$ or $r=n+3$. For $n=9$,
$Q_{n,r,3}(t)=F_{n,r,3}(t)$ unless $r=n+2$ or $r=n+3$.

The following claim is checked several times on computer.

###### Conjecture 2

$Q_{12,14,2}(t)-F_{12,14,2}(t)=64t^{6}$,
$Q_{13,15,2}(t)-F_{13,15,2}(t)=13t^{6}+t^{7}$,
$Q_{13,16,2}(t)-F_{13,16,2}(t)=t^{6}$,
$Q_{9,11,3}(t)-F_{9,11,3}(t)=t^{8}+154t^{9}+t^{10}$,
$Q_{9,12,3}(t)-F_{9,12,3}(t)=12t^{8}$.

For the geometric implications of $Q_{n,r,d}(t)=F_{n,r,d}(t)$, and a
conjecture when it is true, see [8], [2].

## References

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