Publicacions
Ma emá iques,
Vol
36
(1992),
293-317
.
A
bs ac
HOPFIAN
AND
CO-HOPFIAN
OBJECTS
K
.
VARADARAJAN
1
The
aim
o
he
p esen
pape
is
o
s udy Hop ian
and
Co-Hop ian
objec s
in
ca ego ies
like
he
ca ego y
o
ings,
he
module
ca e-
go ies
A-mod
and
mod-A
o
any
ing
A
.
Using
S one's
ep esen-
a ion
heo em
any Boolean
ing
can be
ega ded
as
he
ing
A
o
clopen
subse s
o
a
compac
Hausdo
o ally
disconnec ed space
X
.
I
u ns
ou
ha
he Boolean
ing
A
will
be
Hop ian
( esp
.
co-Hop ian)
i
and
only
i
he
space
X
is
co-Hop ian
( esp
.
Hop-
ian)
in
he
ca ego y
Top
.
Fo any
compac
Hausdo
space
X
le
C7,(X)( esp
.
Cc(X))
deno e
he
R( esp
.
C)-algeb a
o
eal
( esp
.
complex)
alued
con inuous
unc ions
on
X
.
Using
Cel and's
ep-
esen a ion
heo em
we
will
p o e
ha
C7¿(X)(Ce(X))
is
Hop ian
( espec i ely
co-Hop ian)
as
an
R(C)-
algeb a
i
and
only
i
X
is
co-Hop ian
( espec i ely
Hop ian)
as
an
objec
o
Top
.
We
also
s udy Hop ian and
co-Hop ian
compac
opological
mani olds
.
In oduc ion
The
no ion
o
a
Hop ian
g oup
[4]
is
by
now
classical
.
Th oughou
he
p esen
pape
he
ings
A
we
conside
a e
associa i e
ings
wi h
an
iden i y
elemen
lA
=,1=
0
.
Any
sub ing
B
o
A
is
equi ed
o
sa is y
he
condi ion
ha
lB
=
lA
.
All
he
modules
conside ed
a e
uni a y
modules
.
A-mod
( esp
.
mod-A)
will
deno e
he
ca e ogy
o
le
( esp
.
igh )
A-modules
.
In
[12]
V
.A
.
Hi ema h
has
in oduced
he
concep
o
Hop ici y
o
a
ing
A
ega ded
as
a
ing
and
also
o
any
MeA-
mod
.
We
will
show
in
he
p esen
pape
ha
A
is
Hop ian
in
A-mod
i
and
only
i i is
Hop ian
in
.
mod-A
(Theo em
1
.3)
.
When
hese
wo
equi alen
condi ions
a e
sa is ied
we
will
simply
say
ha
A
is
Hop ian
as
a
module
.
The e
a e
ob ious
dual
no ions
o
A
being co-Hop ian
espec i ely
as
a
ing,
as
an
objec
in
A-mod
and
as
an
objec
in
mod-A
.
We
ob ain
a
necessa y
and
su icien
condi ion
o
A
o
be
co-Hop ian
in
A-mod
(P oposi ion
1
.4)
.
Unlike
he
Hop ian
case,
by
means
o a
speci ic
1
Resea ch
done
while
he
au ho
was
pa ially
suppo ed by
NSGRC
g an
AS225
.
29
4

K
.
VARADARAJAN
example
we
show
ha có-Hop ici y
is
no
le - igh
symme ic
.
Also,
we
will
gi e
examples
o
show
ha
A
being
Hop ian
( esp
.
co-Hop ian)
as
a
ing
and
A
being
Hop ian
as
a module
( esp
.
co-Hop ian
in
A-mod
o
mod-A)
a e
independen
o
each
o he
.
As
an immedia e
consequence
o
ou
necessa y
and
su icien
condi ion
i
will
ollow
ha
a
nó
necessa ily
commu a i e
in eg al
domain
A
is
co-Hop ian
in
A-mod
as well as
mod-A
i
and
only
i
A
is
a
skew- ield
.
I
is
a
well-known
esul
ha
any
noe he ian
MeA-mod
is
Hop ian
in
A-mod
and
ha
any
a inian
MeA-mod
is
co-Hop ian
([17,
page
42])
.
A gumen s
used
in
p o ing
his
esul
will
show
ha
any
ing
A
wi h
a
.c .c
.
on
wo
sided
ideals
is
Hop ian
as a ing
and any
ing
A
wi h
d
.c .c
on
sub ings
is
co-Hop ian
as a
ing
.
In
pa icula
any
le
noe he ian
(hence
any
le
a inian)
ing
A
is
Hop ian
as a
ing
.
Easy
examples
can be
gi en
o
show
ha
e en
ields
need
no
be
co-Hop ian
as
ings
.
Simila
o
he
esul
ha
any
le
a inian
ing
is
le
noe he ian
we
ha e
he
esul
ha
any
ing
A
which
is
co-Hop ian
in
A-mod
is
au oma ically
Hop ian
in
A-mod,
hence
also
Hop ian
in
nod-A
(P oposi ion
1
.10)
.
Le
n
be
any
in ege
>
1
.
I
is
easy o
p o e
he
ollowing
implica ions
:
a)
M

,
(A)
Hop ian
( esp
.
co-Hop ian)
as a
ing
=>
A
Hop ian
( esp
.
co-Hop ian)
as
a
ing
.
b)
M

(A)
Hop ian
( esp
.
co-Hop ian)
in
M,,
(A)-mod
=>
A
Hop ian
( esp
.
co-Hop ian)
in
A-mod
.
The
analogue
o Hilbe 's basis
heo e a
is
alid
o
Hop ici y,
namely
MEA-mod
is
Hop ian
i
and
only
i
M[X]
is
Hop ian
in
A[X]-mod,
whe e
X
is
an inde e mina e
o e
A
.
This
and
he
analogous
esul
o
M[[X]]
in
A[[X]]-mod
a e
p o ed
in
Sec ion
2 o
he
p esen
pape
(Theo em
2
.1)
.
We
do
no
know
whe he
he
analogous
esul
is
alid
o
M[X,
X
-
1]
in
A[X,
X
-1
]-mod
.
Fo
any
non-ze o
MeA-mod,
i
is
easy
o see
ha
M[X]
( esp
.
M[[X]])
is
no
co-Hop ian
in
A[X]
( esp
.
A[[X]])-mod
.
In
Sec ion
3
we
a e
mainly conce ned
wi h
he
case
when
A
is
commu-
a i e
.
Fo
he
esul s
s a ed
in
he
p esen
pa ag aph
i
will
be
assumed
ha
A
is
a commu a i e
ing
.
Then
i is
well-known
[22],
[24]
ha
e -
e y
.g
.
(abb e ia ion
o
ini ely
gene a ed)
A-module
is
Hop ian
.
I
can
easily
be
shown
ha
M

,(A)
is
Hop ian
in
M

,(A)-mod
o
mod-M,,(A)
o
all
in ege s
n
>_ 1
.
Ou
necessa y
and
su cien
condi ion
o
A
o
be
co-Hop ian
in
A-mod
(Theo em
1 .3)
implies ha
A
is
co-Hop ian
in
A-mod
e-*
A
is
i s
own
o al
quo ien
ing
.
In
his
case
we
will
p o e
ha
M
n
(A)
is
co-Hop ian
in
bo h
M

,(A)-mod
and
mod-M,(A)
.
We
will
also
p o e
ha
An
is
co-Hop ian
in
A-mód
o
all
n
>_ 1
.
The
p oo
o
his
will
depend
on an
auxilia y
esul
asse ing
ha an
A-homomo phism
:
A'
-
A'
is
no
injec i e
i
and
only
i
de
is
a
ze o
di iso in
A,
wha e e
be
he
commu a i e
ing
A
(le ama
3
.1)
.
I is
also
well-known
Hop IAN
AND
Co-Hop ]AN
OBJE
CTS

295
[2],
[25]
ha
e e y
.g
A-module
is
co-Hop ian
i
and
only
i
e e y
p ime
ideal o
A
is
maximal
.
We
will
explici ly
cons uc
a
commu a i e
ing
A
which
is
i s
own
o al
quo ien
ing
admi ing
a p ime
ideal
which
is
no
maximal
.
In
pa icula
his
ing
will
sa is y
he
condi ion
ha
A'
is
co-Hop ian
in
A-mod
o
cach
in ege
n
>_ 1
and
he e
a e
.g
A-modules
which
a e
no
co-Hop ian
.
The
ing
A
ha
we
cons uc
will
ha e
he
ollowing
addi ional
p ope ies
:
c)
A
is
no
noe he ian
d)
A
does
no
ha e
d
.c .c
o
sub ings
In
[12]
Hi ema h
shows
ha
i
he
Boolean
ing
o
clopen
subse s
o
a
compac
Hausdo
o ally
disconnec ed
space
X
sa is ies
he
condi ion
ha
A
is
Hop ian
as
a
ing
hen
X
is
co-Hop ian
as
a
opological
space
.
He
says
he
does
no
know
whe he
he
con e se
o
his
esul
is
ue
.
Ac ually
we
no
only
show
ha
he
con e se
is
ue
bu
we
also
show
ha
A
is
co-Hop ian
as
a
ing
i
and
only
i
X
is
Hop ian
as
a
opological
space
.
This
is
ca ied
ou
in
Sec ion
4
.
Le
X
deno e
a compac
Hausdo £
space
and
C(X)
deno e
ei he
C7z
(X)
o
Cc
(X)
.
We
ega d Ciz
(X)
as
an
R-algeb a
and
Ce
(X)
as
a
C-algeb a
and
simply
w i e
" he algeb a
C(X
)"
.
Using
Gel and's
ep-
esen a ion
heo em we
show
ha
C(X)
as
an
algeb a
is
Hop ian
( esp
.
co-Hop ian)
i
and
only
i
X
is
co-Hop ian
( esp
.
Hop ian)
in
he
ca -
ego y
Top
o
opological
spaces
.
(Theo em
5
.3)
.
Wc
do
no
ha e
any
cha ac e iza ion
o
compac
Hausdo
spaces
which
a e
Hop ian
( esp
.
co-Hop ian)
.
Howe e
i
is
an
easy
consequence
o
in a iance
o
domain
ha
compac
opological
mani olds wi hou
bounda y
a e
co-Hop ian
.
Among
compac
mani olds wi hou
bounda y
i
can
casily
be
shown
ha
ini e
se s
a e he
only
Hop ian
objec s
.
Among
compac
nani olds
wi h
a
non-emp y
bounda y
he e
a e
no
Hop ian
o
co-Hop ian
objec s
.
I
M
is
a
compac
mani old
wi h
bounda y
(9M
hen
he
pai
(M,
(9M)
is
a
co-Hop ian
objec
in
he
ca ego y
Top
e
o pai s o
opological
spaces
.
We
conclude
ou
in oduc ion
by
poin ing
ou
ha
Hil on
;
Roi be g
e c
.,
ha e
s udied
epimo phisms
and
monomo pliisms
in
he
ho no opy
ca ego y
and
we e
led o
in es iga ing
Hop ian
and
co-Hop ian
objec s
in
he
homo opy
ca ego y
[10],
[18]
.
Finally
we
wish
o
hank
he
e e ee
o
in o ma ion
on
li e a u e
.
In
ac
mos
o
he
ma e ial
in
Sec ion
7
has
been
poin ed
ou
by
he
e e ee
.
Acknowledgemen s
.
Pa o
his
wo k was done
while
he
au ho
was
isi ing
Cen e
de
Rece ca Ma ema ica,
Bella e a
in
Spain
.
The
au ho
would
like
o
hank
P o esso
Cas elle
o
c ea ing
a
e y
con-
duci e
a mosphe e
o
esea ch
.
Also
while
ca ying
ou
his
esea ch
he
au ho
ecei ed
suppo
om
NSERC
g an
A8225
.
29
6

K
.
VARADARAJAN
1
.
Hop ian
and
co-Hop ian
ings
and
modules
Th oughou
we
will
o mula e
ou
esul s
in
he
ca ego y
A-mod
o
le
uni al
A-modules
.
The e
a e
ob ious analogous
esul s
in
he
ca ego y
mod-A
o
uni al
igh
A-modules
.
We
i s
ix
ou
e minology
and
no-
a ion
.
Fo
any
aeA,
£A
(a)
=
{beA
1
ba
=
0}
and
A(a)
=
{beA
1
ab
=
0}
.
By
a
le
( esp
.
igh )
ze o di iso
in
A
we
mean
an
elemen
a
:~
0
in
A
wi h
eA(a)
7~
0
( esp
.
A(a)
~L
0)
.
An
elemen
aeA
will
be
called a
le
( esp
.
igh )
uni
i
he e
exis s
an
elemen
ceA
wi h
ca
=
1
( esp
.
ac
=
1)
.
We
call
aeA
le
( esp
.
igh )
egula
i
Wa)
=
0 ( esp
.
A(a)
=
0)
.
I is
i ial
o see
ha
any
le
( esp
.
igh )
ze o
di iso
is
ne e
a
igh ( esp
.
le )
uni
.
Also
any
le
egula
elemen
a
which
is
a
le
uni
is
au oma ically
a
wo-sided
uni
.
De ini ion
1
.1
.
M A-mod
is
said o
be
Hop ian
( esp
.
co-Hop ian)
i
e e y
su jec i e
( esp
.
injec i e)
homomo phism
:
M
->
M
is
an
isomo phism
.
I
is
wcll-known
ha
any
noe he ian
( esp
.
a inian)
module
is
Hop ian
( esp
.
co-Hop ian)
[17,
Lemma
4,
pago
;
41]
.
P oposi ion
1 .2
.
AeA-mod
is
Hop ian
i
and
only
i
no
le
ze o
di-
iso
in
A
is
a
le
uni in
A
.
This
is
heo em
9
in
[12]
.
Equi alen ly
i
is
welll-known
and
easy
o
see
ha
AeA-mod
is
Hop ian
i
and
only
i
A
is
di ec ly
ini o
(i .e
.
xy
=
1
==>
yx
=
1)
[11]
.
Theo em
1
.3
.
AeA-mod
is
Hop ian
i
and
only
i
Ae
mod-A
is
Hop-
ian
.
P oo
..
Di ec
ini eness
is
clea ly
le
igh
symme ic
.
P oposi ion
1
.4
.
AeA-mod
is
co-Hop ian
i
and
only
i
e e y
le
egula
elemen
aeA
is
a
wo-sided
uni
.
P oo£
Immedia e
consequence
o
he
ac
ha
injec i e
homomo -
phisms
:
A
->
A
in
A-mod
a e
exac ly gi en
by
(a)
_
Aa
wi h
aeA
le
egula
.
Examples
1 .5
.
Conside
he
ing
A=[Z/2Z

Z/2Z
0Z(2)
whe e
Z(2)
is
he
2-localiza ion o
Z, namely
2121
=
{
`cQ
1
n
odd
}
.
71
The
elemen

2

EA
is
easily
checked
o
be
igh
egula
bu
no
(0
in e ible
in
A
.
Hence
A
is
no
co-Hop ian
in
mod-A
.
29
8

K
.
VARADARAJAN
(b)
Thc
;
o ily
íng
homomo phism
o
i
( esp
.
Q)
ís
he
íden i y
map
.
Hence
Z
and
Q
a e
Hop ian
and
co-Hop ian
as ings
.
Whilc
Z
is
Hop ian
in
Z-mod,
i is
no
co-Hop ian
in
Z-mod
.
Q
is
bo h
Hop ian
and
co-Hop ian
in
Q-mod
(hence
also in
Z-mod)
.
(c)
Fo
any
ing
A
he
unique
ing
homomo phism
cp
:
A[X]
~
A[X]
ca ying
X
o
X
2
and
sa is ying
ep
1
A
=
Id
A
is
an
injec i e
ing
homomo phism
which
is
nó
su jec i e
.
(He e
X
is
an
inde e mi
na e
o e
A)
.
Thus
A[X]
is
no
co-Hop ian
as
a
ing,
wha e e
be
he
ing
A
.
A
simila
a gumen
shows
ha
A[X,
X
-1
]
and
A[[X]]
a e
no
co-Hop ian
as ings
.
(d)
Le
A
=
K[(Xj]a,i
o e
a
coni nu a i e
ing
K
.
Then
A
is
a
co nmu a i e
ing
hence
Hop ian
in
A- nod
.
Le
O
:
J
,
.1
be a
su jec i e
;
nap
which
is
no
1bijec i e
.
Since
J
is
in ini e
such
a
map
exis s
.
The
unique
ing
homomo phism
:
A
-->
A
sa is ying
1
K
=
IdK
and
(X
a
)
=
Xo(a)
is
hen
a
su jec i e
ing
homomo phism
which
is
no
an
isomo phism
.
Thus
A
is
no
Hop ian
as a ing
.
(e)
Le
K
be a
ield
and
L=
K((Xa)a,j)
he
ield
o
a ional
unc-
ions in
an
in ini e-
;
numbe
o inde e mina es
.
Any
ield
is
Hop ian
as a
ing
.
Thus
L
is
Hop ian
as
a
ing
.
I
O
:
J
->
J
is
any
in
jec i e
map,
he e
is
a
unique
homomo phism
cp
:
L
,
L
o
ields
sa is ying
cp(X
.)
=
Xq(a)
and ep/K
=
IdK
.
I
O
is
no
bijec i e,
hen
cp is
an
injec i e
ing
homomo phism
o
L
in
L
which
is
no
su jec i e
.
Hence
L
is
no
co-Hop ian
as a
ing
.
Since
L
i
a
ield
om
ema ks
1
.6(a)
and
(c)
we
see
ha
L
is
bo h
co-Hop ian
and
Hop ian
in
L-mod
.
( )
Fo
any
simple
ing
A
any
ing
homomo phism
:
A-
;
B
is
au oma ically
injec i e
.
Hence
e e y
simple
ing
is
Hop ian
as
a
ing
.
F om
example
(e)
abo e
wc,
see
ha
a
simple
;
ing
(e en
a
iel(1)
need
no
be
c
:o-Hop ian
as a ing
.
(g)
Le
K
be
a
ield
and
V
an
in ini e
dimensional
ec o
space
o e
K
.
Le
A =
EndKV
.
The e
exis
K-linea
su jec ions
:
V
-->
V
which
a e
no
injec i e
.

Choose
such
an
.

Since
V
~
V
~
0
spli s
in
K- nod,
3
a
K-linea
map
h
:
V
-->
V
wi h
o
h
=
IdV
.
This
means
is
a
igh
uni
in
A
.
Since
ke
7~
0
we
can
choose
a,
g
:V ->
V
wi h
g
q¿
0
and
g(V)
C
Ke
.
Then
gEA
sa is ies
o
g
=
0
.
Thus
is
a
igh
ze o
di iso
in
A
which
is
no
a igh
uni
in
A
.
F om
p oposi ion
1
.4
we
see
ha
A
is
no
Hop ian
in
iod-A
and
hence
also
no
in
A-mod
om
heo em
1.3
.
In
case
V
has
coun able
dimension
i
ollows
om
exe cise
14
.13,
page
164
o
[1]
ha
he e
a e
only
wo
non-ze o
ideals in
A=
EndKV
.
Hence
A
is
Hop ian
as a
ing
(see
;
p oposi ion
1
.12)
.

(h)
I
A =
K[(xj~
E
j]
wi h
K
any
commu a i e
ing
and
J
in ini e,
om
1
.8(d)
we
see
ha
A
is
Hop ian
in
A-mod,
bu
no
Hop ian
as a ing
.
When
K
is
a
ield
and
V
a
coun ably
in ini e
dimen
sional
ec o
space
hen
A
=EndK
V
is
Hop ian
as
a
ing
bu
no
Hop ian
as
an
A-module
.
As
al eady
sien
Z
is
co-Hop ian
as
a
ing
bu no
as
a
Z-module
.
When
K
is
a
ield,
A
=
K((X«)«Ei)
he
ield
o
a ional
unc ions
in
an
in ini e
numbe
o
inde e mi-
na es
is
an
example
o a
commu a i e
ing
which
is
co-Hop ian
as
a
module
bu
no
co-Hop ian
as
a
ing
om
1
.8(e)
.
P oposi ion
1
.9
.
HOPFIAN
AND
CO-HOPFIAN
OBJECTS

29
9
(i)
I
A
is
a
ing
sa is ying
a
.
e
.
e
o
wo
sided
ideals
hen
A
is
Hop ian
as a
ing
.
(ii)
I
A
is
a ing sa is ying
d
.c .c
o
sub ings
hen
A
is
o-Hop ian
as a
ing
.
The
p oo
o
his
p oposi ion
is
simila o
ha o
lemma
4,
pago
42 o
[17]
and
hence
omi ed
.
P oposi ion
1
.10
.
Leí
A
be a ing
wi h
he
p ope y
ha
A
is
co-
Hop ian
in
A-mod
.
Then
A
is
au oma ically
Hop ian
in
A-mod
.
P oo -
Le
a be
a
le
ze o
di iso
in
A
.
F om
p oposi ion
1
.2
we
ha e
only
o
show
ha
a
is
no
a
le
uni in
A
.
On
he
con a y
i
a
is
a
le
uni
in
A,
he e
exis s
an
ele ien
ccA
wi h
ca
=
1
.
Then
clea ly
c
is
le
egula
.
Since
A
is
co-Hop ian
in
A-mod,
om
p oposi ion
1
.4
we
see
ha
c is
a wo
sided
uni
in
A
.
Then
ca
=
1
implies ha
a
is
he
in e se
o
c
and
hence
a
is
also a
wo
sided
uni
.
This
con adic s
he
ac
ha
a
is
a
le
ze o
di iso
.
Rema ks
1
.11
.
Hi ema h
[12]
has
al eady
obse ed
ha
a di ec
summand
o
any
Hop ian
module
is
Hop ian
.
The
same
obse a ion
is
alid
o
co-Hop ian
modules
as
well
.
He
ema ks
ha
he
does
no
know
o
any example
o
a
Hop ian
module
wi h
a
submodule
no
Hop ian
.
La e
in
Sec ion
3
we
will
cons uc
such
modules
.
Q
is
Hop ian
and
co-
Hop ian
in
Z-mod,
he quo ien s
Z _
o
Q
a e
no
Hop ian
in
Z-mod
.
La e
esul s
in
Sec ion
3wil also
show
ha
quo ien s
o
co-Hop ian
modules
need
no
be
co-Hop ian'
.
P oposi ion
1
.12
.
Leí
A
be
a
ing
and
n
an
in ege
>
1
.
Then
(i)
M

(A)
Hop ian
( esp
.
co-Hop ian)
as a ing
==>
A
Hop ian
( esp
.
co-Hop ian)
as
a
ing
.
(ii)
M

,(A)
Hop ian
( esp
.
co-Hop ian)
in
M,(A)-mod
=>
A
Hop ian
( esp
.
co-Hop ian)
in
A-mod
.
30
0

K
.
VARADARAJAN
(iii)
M,,(A)
Hop ian
( esp
.
co-Hop ian)
in
A-mod
=>
A
Hop ian
( esp
.
co-Hop ian)
in
A-mod
.
P oo
.
(i)
is
an
immedia e
consequence
o
he
obse a ions
ha
i
:
A
,
A
is
a
ing
homomo phism,
M
(
)
:
M
n
(A)
-->
M,, (A)
de ined
in
he
ob ious
way
is
a
ing
homomo phism
and
ha
M,
,( )
is
su
j
jec i e
( esp
.
injec i e)
~
is
su jec i e
( esp
.
injec i e)
.
(ii)
simila
o
(i)
abo e
excep
o
he
obse a ion
ha
i
:
A
->
A
is
a
map
in
A-mod,
hen
,,( )
:
M
n
(A)
~
M
n
,
(A)
is
a
map
in
M
n
(A)-mod
.
(iii) is
immedia e
om
he
ac
ha
A
is
a di ec
summand
o
M
n
(A)
in
A-mod
.
The
con e ses
o
(ii), (iii)
a e
no
ue
in
gene al
.
Coun e
examples
will
be
gi en
in
Sec ion
7
.
Bu
when
A
is
commu a i e
o
(ii),
(iii)
he
con e ses
a e
ue
and
hey
will
be
p o ed
in
Sec on
3
.
We
do
no
know
whe he
he
con e se
o
(i) is
ue
.
Gi en
any
non-ze o
M6A-mod
i is
known
ha
any
in ini o
di ec
sum
o
copies
o
M
is
nei he
Hop ian
no
co-Hop ian
in
A-mod
.
Any
such
module
will
admi
he
module
N
=
®
>,M,,
as di ec
summand
whe e
M,,
=
M
o
each
n
>_ 1
.
The
shi
map
s
+
which
ca ies
he
n h copy
o
M
o
he
(n
+
1)s
copy
iden ically
is
an
injec i e
map
which
is
no
su jec i e
.
The
shi
nap
s_ which
maps
he
(n
+
1)"
copy
o
he n h
copy
iden ically
o
n
_> 1
and which
maps
he
l`
copy
o
M
o
ze o
is
a
su jec i e
map
which
is
no
injec i e
.
This
ac
will
be
mide
use
o by
us
la e
in
Sec ion
3
o
cons uc ing
a
Hop ian
module
admi ing
a
non-
Hop ian
submodule
.
An
in ini o
di ec
sum
o no -ze o
modules
could
e y
well
be simul aneously
Hop ian
and
co-Hop ian
.
I
P
deno es
he
se
o
all
p imes,
M
=®
PE
p(i/pi)
is
easily
seen
o
be
simul aneously
Hop ian
and
co-Hop ian
in
i-mod,
P oposi ion
1.13
.
Le A[G] deno e
he
g oup
ing
o
a
g oup
G
o e
he
ing
A
.
I
A[G]
is
Hop ian
( esp
.
co-Hop ian)
as
a
ing
hen
A
is
Hop ian
( esp
.
co-Hop ian)
as
a
ing
and
G
is
Hop ian
( esp
.
co-Hop ian)
as a
g oup
.
P oo
.
Le
:
A
-
A
be
a
homomo pl ism
o
ings
and
cp
:
G
-+
G
a
homomo pl ism
o
g oups
.
Then
he
nap
,i
:
A[G]
->
A[G]
de ined
by 0
(I
: g
,c
asg)
=
I
:g,c
(a
y)W(g)
1
is
a
ing
homomo phism
.
Also
i
is
easily
checked
ha
9
is
su jec i e
( esp
.
injec i e)
<=>
and
cW
a e
su jec i e
( esp
.
injec i e)
.
P oposi ion
1
.13
is
an
easy
consequence
o
hese
ac s
.
HOP IAN
AND
CO-HOPFIAN
OBJECTS

30
1
Rema ks
1
.14
.
I
:
A
->
A
is
a
map
in
A-mod
and
W
:G
->
G
is
a g oup
homomo phism
i
is
in
gene al
no
ue ha
/d
:A[G]
-->
A[G]
de ined
~
(I
:,EG
a,g)
=
z9Ec
(as)W(g)
will
be
a
map
in
A[G]-mod
.
In
case
cp
=
MC
i
is
ue
ha
Q
is
a
map
in
A[G]-mod
.
The
analogue
o
1
.13 o
module
ca ego ies
is
no
alid
.
I
A
is
any
commu a i e
ing
and
G
any abelian
g oup,
A[G]
is
Hop ian
in
A[G]-mod
.
G
need
no
be
Hop ian
.
Howe e ,
he
ollowing
can be
p o ed
.
P oposi ion
1
.15
.
I
A[G]
is
Hop ian
( esp
.
co-Hop ian) in
A[G]-
mod,
hen
A
is
Hop ian
( esp
.
co-Hop ian)
in
A-mod
.
P oposi ion
1
.16
.
Le {Aa}a,J
be
any
amily
q
ings
and
A =
IIa,jAa
hei
di ec ,
p oduc
.
(i)
A
is
Hop ian
( esp
.
co-Hop ian)
in
A-mod
<~
:>
each
A
a
is
Hop ian
( esp
.
co-Hop ian)
in
A,-mod
.
(ii)
I
A
is
Hop ian
( esp
.

co-Hop ian)
as
a
ing hen
each
A
u
is
Hop ian
( esp
.
co-Hop ian)
as a
ing
.
P oo
.:
(i)
is
an immedia e
consequence
o
he
ac
ha
any
map
:
A
-
A
in
A-mod
is
uniquely
o
he
o m
II a
:
IIA
a
->
IIA
a
wi h
a
:
A
a
->
A
a
a
map
in
A
a
-mod
and
is
su jec i e
( esp
.
injec i e)
~-
:>
each
a
is
su jec i e ( esp
.
injec i e)
.
(ii)
I
a
:
A<,
-> A,,
is
a
ing
homomo phism
o
each
aE
.I,
hen
=
II a
:
IIA
a
->
IIA,
is
a
ing
homomo phism
.
Mo eo e
is
su jec i e ( esp
.
injec i e)
i
and
only
i
each a
is
su jec i e
( esp
.
injec i e)
.
(ii) is
an
immedia e
consequence
o
hese
ac s
.
Ac ually
p oposi ion
1
.16(1)
can
be
imp o ed
as ollows
:
P oposi ion
1
.17
.
Le {Aa}a,J
be
any
amily
o
ings
and
A
=
II
.EJAa
hei
di ec
p oduc
.
Le
M
a
cA
.-mod
o each
acJ
.
I
M
=
II
á
,jMa
wi h
A-ac ion
de ined
by
a
.m
=
(aama)c
.l
u)hene e
a=
(ac )aEJ
wi h a
a
E
A
a
and
m
=
(ma)aE,1
wi h
mc,
E
M
a
.
Then
M
is
Hop ian
( esp
.
co-Hop ian)
in
A-mod
i
and
only
i
each
M
a
is
Hop ian
( esp
.
co-Hop ian)
in
A,,-mod
.
Again
his is
an
immedia e
éonsequence
o
he
ac ha
any
map
:
M
-
M
in
A-mod
is
uniquely
o
he
o m
II a
:
IIM
a
-
IIM
a
wi h
,,
:
M
a
-
M
aa
map
in
A
a
-mod
.
2
.
Hop ici y
o
he
modules
M[X],M[X]/(Xn)
and
M[[X]]
Gi en
any
MEA-mod
and
an inde e mina e
X
o e
A
we
de ine
30
2

K
.
VARADARAJAN
M[X]eA[X]- nod,
M[X]/(X7')EA[X]/(X7')- nod
a id
M[[X]]c-A[[X]]- nod
as in
Sec ion
1
o
[23],
whe e A[X]
deno es
he
polynomial
ing,
A[X]/(Xn)
o
any
in ege
n
>_
1
deno es
he
unca ed
polynomial
ing
and
A[[X]] he
o mal
powe
se ies
ing
.
The
main
esul
p o ed
in his
sec ion
is
he
ollowing
:
Theo em
2
.1
.
Le ,
McA-mod
.
Then
he
olowing
a e
equi alen
.
(1)
M
is
Hop ian
in
A-mod
.
(2)
M[X]
is
Hop ian
in
A[X]-mod
.
(3)
M[X]/(X
n
)
is
Hop ian
in
A[X]/(X")-mod
.
(4)
M[[X]]
is
Hop ian
in
A[[X]]-mod
.
P oo
..
(2)
=>
(1)
.
Le
:
M
-+
M
be
any
su jec i e
map
in
A-
mod
.

Then
[X]
:
M[X]
-~
M[X]
de ined
by
[X]
(Ek
j=o
ajXj)
_
a

Xj
is
a
su ,
ec i e
ma

in
AX
mod
.
Since
M[X]
is
Ho
lan
in
A[X]-modwe
see
ha
[X]
is
injec i e
.
This
i imedia ely yields
he
injec i i y
o
.
The
p oo s
o
(3)
==>
(1)
and
(4)
=>
(1)
a e simila
and
omi ed
.
(1)

(2)
.
Le
W
:
M[X]
-
M[X]
be
any
su jec i e
A[X]-honio pl ism
.
Le
0
=
coI
M
:M
-
M[X]
.
Then
0
is
an
A-honio no phism
:
Mo eo e
k

k
(4)

cp
ajXj=
X'O(aj)
.
j=0
j=o
Fo any
i
>
0
le
pi
:
M[X]
-~
M
be
de ine(¡
by
i i<k
i
>
k
.
Then
p
i
:
M[X]
-
M
is
a
map
in
A- nod
o
each
i
>
0
.
Since
ep
is
su jec i e,
gi en
any
ceM
he e
exis
an
elemen
E~-o
a
j
Xy
E
M[X]
Wi h
W(L_j=o
a
j
Xj
)
=
c
.
Using
4,
we
see
ha
he
"cons an
e co"
o
O(ao)
is
c
o
equi alen ly
pood(ao)
=
c
.
This
shows
ha
he
mala
po~oO
:M
,
M
is
a
su jec i e
nap
in
A- nod
.
The
Hop ian
na u e
o
M
in
A- nod
iniplies
ha
poOOM
->
M
is
an
iso no phism,
in
pa icula
injec i e
.
Ou
ai i is
o
show
ha
cp
:M[X]
-
M[X]
is
injec i e
.
Le
z
:k
j=o
b
j
XjE
M[X]
sa is y
W(j
:k
=o
bjXj)
=
0
.
Using
4
and
obse ing
ha O(bj)
_
HOP IAN
ANn
Co-HOP IAN
o13JECTS

309
se
I,,
=
{C
e
B(X)
I
x
1C}
hen
x
1--
x is
a
hoeomo phism
o
X
wi h
max
Spec
B(X)
.
De ini ion
4
.3
.
A
opological
space
X
is
said o
be
Hop ian
( esp
.
co-Hop ian)
in
he
ca ego y
Top
i
e e y
su jec i e ( esp
.
injec i e)
con-
inuous
map
:
X
-
X
is
a
homeomo phism
.
The
main
esul
o
his
sec ion
is
he
ollowing
.
Theo em
4
.4
.
A
Boolean
ing
A
is
Hop ian
( esp
.
co-Hop ian)
as a
ing
i
and
onl y
i
XA
is
co-Hop ian
( esp
.
Hop ian)
in he
ca ego y
Top
.
P oo
..
Immedia e
consequence
o
p oposi ion
4
.2
.
Rema ks
4
.5
.
Le
J
be
any
ini e se
.
The
p oduc
space
H"
whe e
H
=
{0,
1}
is
nei he
Hop ian
no
co-Hop ian
.
As
a
se
H
j
is
he
se
o
all
maps
o
.7
in o
H
.
Fo
any
se
heo e ic
map
0
:
J
-
J
we
ha e
an
induced
nap
~-->
o
0 o
H
j
in o
Hj,
which
is
easily
seen
o
be
con inuous
.
I
0
is
an
injec i e ( esp
.
su jec i e)
map
which
is
no
a
bijec ion,
hen
~-->
o
0
s
a
su jec i e
( esp
.
injec i e)
map
which
is
no
bijec i e
.
Since
A
=
B(H
j
)
is
a
commu a i e
ing
in
which
e e y
p ime
ideal
is
maximal,
all
-
gA-modules
a e
simul aneously
Hop ian
and
co-Hop ian
in
he
ca ego y o
A-modules
bu
A
is
nei he
Hop ian
no
co-Hop ian
as
a
ing
.
I
would be
nice o
cha ac e ize
comple ely
he
Hop ian
( esp
.
co-
Hop ian)
compac
Hausdo
o ally
disconnec ed
spaces
.
5
.
Hop ian
and
co-Hop ian
unc ionalgeb as
Le
K
be
a
commu a i e
ing
and
K-alg
deno e
he
ca ego y o
K-
algeb as
.
De ini ion
5
.1
.
A
K-algeb a
A
is
said o
be
Hop ian
( esp
.
co-
Hop ian)
as a
K-algeb a
i
any
su jec i e
( esp
.
injec i e)
K-algeb a
homomo phism
:A
,
A
is
isomo phism
.
Le
R
( esp
.
C)
deno e
he
ield
o
eal
( esp
.
complex)
numbe s
wi h
he
usual
opology
.
Fo
any compac
Hausdo
space
X
le
CR
(X)
( esp
.
Cc(X))
deno e
he
R
( esp
.
C)-algeb a
o
con inuous
unc ions
' om
X
o
R
( esp
.
C)
.
Using
he
Gel and
esp esen a ion
heo em
we
will
de e mine
necessa y
and
su ñcien
condi ions
o
Cn(X)
( esp
.
Cc(X))
o
be
Hop ian
o
co-Hop ian
in
he
ca ego y
R-alg
( esp
.
C-alg)
.
We
will
mainly
concen a e
on
CR(X)
.
Simila
esul s a e alid
o
Cc(X)
.
X
deno es
a
compac
Hausdo
space
and
C(X)
deno es
he
R-algeb a
CR
(X)
.
I is
well-known
ha
he
map
x
-,
(
(x))
EC(x)
is
a
opological

31
0

K
.
VARADARAJAN
imbedding
o
X
in o
II_

EC(x)R
wi h
he
ca esian
p oduc
opology,
whe e
R
=
R
o
each
E
C(X
)
.
Also
x
~--
m~,
=
{
E
C(X)
l
(x)
=
0}
is
a
bijec ion
om
X
o
he
se
o
maximal
deals
in
he
R-algeb a
C(X)
.
I
X~
Y
is
a
con inuous mal)
o
compac
Hausdo
spaces,
he e
is
an
induced
homomo phism
cp *
:
C(Y)
-
C(X)
in
R-alg
gi en
by
p*
(g)
=
goce
o
e e y
g
E
C(Y)
.
Also
gi en
any
R-algeb a
homomo -
phism
a
:
C(Y)
-+
C(X),
he e
is
a
unique
con inuous
mapW
:X
-
Y
sa is ying
a
=
cp*
.
To
see
his,
o
any x
E
X,
a-'
( ¿,
;)
is
a
iaximal
ideal
o
C(Y)
and
hence
a
-
(M
X
)
=
m,,(=)
o
a
unique
elemc
w(x)
E
Y
.
I
jx :X
-
;
II EC(x)R
and
jy
:
Y
->
IIg
E
C(Y)R,,
deno e
hc
:
imbeddings
jx(x)
_
( (x)) EC(x)
and
jy(y)
=
(g(? ))IEC(y)y
espec i ely,
hen
he
se
heo e ic
map
cp
:
X
->
Y
ob ained
abo e
sa is ies
he condi ion
ha
diag am
5.2
Y
II

a
T7
~
1-
~
IE
I
I
n

.
Jc(y)
i~,g
is
comniu a i e,
whe e
a*(( ) eC(x))
=
(S9)9EC(Y)
wi h
sg
=
a(g)
.
Since
a*
composed
wi h
any
p ojec ion
IIR
g
-->
R
g
s
con inuous
we
see
ha
a*
is
con inuous,
hence
cp is
con inuous,
p o ided
we
check
he
comniu a i i y
o
he
diag am
5
.2
.
Bu
i
is
s aigh o wa d
.
Thus
he
se
o
R-algeb a
ho no io phisnis
C(Y)
-
C(X)
is
he
sa c
as
he
se
{W*
:lW
:X
->
Y
con inuous
}
.
The
esul s
quo ed
so
a
a e
well-known
([20,
pagos
327-330]
)
.
P oposi ion
5
.2
.
Le
W:X
->
Y
be
a
con inuous
map
o
compac
Hausdo '
spaces
.
Then
(i)
cp*
:C(Y)
-->
C(X)
is
injec i e
<=>
W:X
-->
Y
is
su jec i e
.
(ii)
cp*
:C(Y)
->
C(X)
is
su jec i e
~-¿
W:X
-->
Y
is
injec i e
.
P oo
(i)
Suppose
cp
:X
-+
Y
is
no
su jec i e
.
ThenW(X)
is
á
p ope
closed
subse
o
Y
.
We
can
pick
an
elemen
b
E
Y
-
W(X)
.
Le
h
:cp(X)
->
R
be any
con inuous
unc ion
.
Then
we
can
ge
con inuous
ex ensions
g
:Y
->
1Z,
g2
:Y
-->
R
o
h
wi h
gi(b)
=
0
and
g2(b)
=
1
(by Tie ze
ex ensio i
heo em)
.
Then
gl :~
92
in
C(Y)
bu
cP*(gi)
=
h
o
co
=
cP*(g2)
Since
g
l
~o(X)
=
g21
;G(X)
=
h
.
Thus
ep
no
su jec i e
=
:>
cp*
no
injec i e
o
equi alen ly
W*
injec i e
=>
co
su jec i e
.
HOP IAN
AND
CO-HOPC'IAN
OT3
.II
CTS

311
I
cp
:X
-
Y
is
su jec i e,
hen
o
any
wo
se
heo e ic
maps
g
:Y ->
R,
92
:
Y
~
R,
we
ha e
hc
;
implica ion
g
o
;o
=
92
o
cp
==>
g
=
.g2
.
In
pa icula
his
implica ion
is
ue
wi h
g ,
g2
in
C(Y)
.
This
p o es
(i)
.
(ii)
Suppose
cp
is
no
i jec i e,
say
x
jA
x2
in
X
sa is y
W(x1)
=
cp(x2)
.
Any
E
C(X)
o
he
o m
g
o
cp
wi h
g
E
C(Y)
has
o
sa is y
(X1)
=
(x2)
.
Howe e ,
we
do
know
ha
3
E
C(X)
wi h
(xi)
=
0
and
(X2)
=
1
.
Thus
ep*
:C(Y)
->
C(X)
is
no
su jec i e
.
Con e sely,
assume
ha
cp is
i jec i e
.
Then
cp
:X
-
W(X)
is
a
ho e-
omo phism
in(¡
W(X)
is
closed
in
Y
.
Gi en
any
E
C(X),
h
:W(X)
->
R
de ined
by
hW(x)
=
(x)
is
con inuous
.
By
Tic ze ex ension
lico c n,
he e
exis s
,g
E
C(Y)
wi h
,gl~o(X)
=
h
.
Then
cp*(g)
=
,
sl owing ha
cp*
:C(Y)
-
C(X)
is
su
j
jec i e
.
Theo em
5
.3
.
Le ,
X
he
a,
compac
Hausdo
; '
space
.
Then
C(X)
is
Hop ian
( esp
.
co-Hop ian)
as
an
R-algeó a
i
and
on1y
i
X
is
co-Hop ian
( esp
.
Hop ian) as a
opologicall
space
.
P oo
:
Immedia e
consequence
o
p oposi ion
5 .2
.
6
.
Hop ianand
co-Hop ian
objec s
in
Top
among
compac
mani o1ds
Fo
each
in ege
n
>_ 1
le
D"
deno e
an
n-disk
.
We
may
ake
D'
=
{x
E
R"1
11
. ,
11<
1}
whe e
11
x
11
deno es
he
usual
no m
in
R"
.
B,y
de ini ion
D
°
consis s
o
a
poin
.
Fo
n >
l,
hc
"
nap
. ,
~--
Zx
is
a
con inuous
i ,jc
;<aion
whic
;h is
no
a
su jec ion
.
The
nap
H
:D"
-
"
D"
gi en
by
2x

'o
jj
x
jj
<

2
0(x)
=
~~

o ~jx
jj_
XI
1

2
is
a
con inuous
su jec ion
which
is
no
i jec i e
.
Thus
D"
is
nei he
Hop ian
no
co-Hop ian
o
n
>_
1
.
Obse e
ha
B
:D"
,
D"
clc ined
abo e
has
he
acldi ional
p ope y
ha
BIS`
=
Id,_,
.
Le
M"
be
;
any
compac
opological
mani old
(wi h
o
wi hou
bounda
.y)
o
di ension
n >
1
.
Then
i bedding a disk
D"
in
M"
wc
can
de ine
a
con inuous
su jec ion
:M"
.
-
"
M"
wi h
i(M"_
In D")
=
Id(m__i ,cl)_)
and
.
¡D"
a
con inuous
su jec ion
o
D"
wi h
i sel
sa is 'ying
.
¡S" -
=
Id,_,
and
ID"
no
i jec i e
.
I
ollows
ha
M"
is
no
Hop ian
.
Thus
wc-
;
ob ain
he
ollowing
.
31
2

K
.
VARADARAJAN
P oposi ion
6
.1
.
Th,e
only
compac ,
mani olds
(wi h
o
wi hou ,)
bounda y
which
a e
Hop ian
a e
ini e
disc e o
spaces
.
As
usual
o any
opological
space
X
we
deno e
he
se ,
o
a cwise
connec ed
componen e
o
X
by IIO(X)
.
Theo em
6
.2
.
Le
Mn
and
Nn
be
a
compac
opological
mani olds
o
he
sane
dimension
n
>_ 1,
boíh o
hem
wi hou
bounda y
.
Suppose
u he
jIIo(Mn)J
=
III°(Nn)j
.
Then
any
con inuous
injec ion
:M
~N
is
a
homeomo phism
.
P oo
.
.-
Since
Mn
and
Nn
a e
compac
wc
:
see
ha
III0(Mn)J=IIIo(Nn)1
<
oo
.
Lc
:
{Mi-}
k
deno e
he
se
o
connec ed
componen e
o
M
n
.
Each
Mi
is
a compac
connec ed
mani old
wi hou
bounda y,
o
dimension
n
.
Hence
(Mi`)
is
a
compac ,
connec ed
subse
o
N
n
.
Since
;
is
injec i e
we
see
ha
¡M`
:Mi`
-+
(Mi`)
is
a
ho eo no phism
.
By
i i a iance
o
do nain
i
ollows
ha
(M`)
is
open
in
Nn
.
Tlms
(M`)
is
open and
closed
in
N"
and
aleo
connec ed
.
He ice
(Mi`)
is
a
connec ed
co iponen
o
Nn
.
F o i
he
injec i i y
o
i
ollows
ha
i
i
:,I
:
j,
(Mi') and
(M
j
`)
a e
dis inc
connec ed
componen e
o
N"`
.
Since
;
IIIo(N"`)1
=
1IIo(M")1
=
k
<
oo,
i
ollows
ha
{
(Mi)}~
a e
all
he
connec ed
componen s
o
Nn,
lienee
:Mn
->
N'
is
in o
.
F om
he
compac
Hausdo
na u e
o
M
and
N
wc
;
see
ha
:
M"
-
N"
is
a
homeo mo phism
.
As
an
in media e
consequence
o
heo em
6
.2
we
ge
Co olla y
6
.3
.
Any
compac
mani old
M"
wi laou
bounda y
is
co-
Hop ian
in
Top
.
P oposi ion
6
.4
.
Any
compac
mani old
M'
wi h
a,
non-emp y
bounda y
8M
is
ne e '
co-Hop ian
in
Top
.
P oo
.
.-
By
Mo on
B own's
colla ing
l eo em,
he e
exis s a
homeo-
mo pl s
0
:BM
x
[0,
1]
->
W
whe e
W
is
a
neighbou hood
o
áM
in
M,
sa is ying
B(x, 0)
=
x
.
o
all
:L
E
&M
.
Le
:M
-->
M
be
de ined
by
( ,)
=
u
o
all
u
E
M
-
0
M
x
[0,
1)),
(0
(x,
))
=
0(x,, ")
o
all
xE
W
and
,
E
[0,
1]
.
Then
:M
,
M
is
a
con inuous
i jec ion
which
is
no
a
su jec ion
.
Le
Tope
deno e
he
ca ego y o
pai s o
opological
spaces
.
De ini ion
6
.5
.
A
pai
(X,
A)
E
Top
e
is
called
Hop ian
( esp
.
co-
Hop ian)
i
any
su jec i e ( esp
.

injec i e)
map
:(X,
A)
-
(X,
A)
o
pai s
is
a
ho neomo pl ism
.
Fo any
spaee
X
le
H
j
(X)
deno e
hc
:
singula
ho nology
wi h
in ege
coe ñcien s
.
HoPPIAN
AND
Co-Hop IAN
on,I
;e s

313
Theo em
6
.6
.
Le ,
M,
N
be
compac ,
mani olds
wi h,
bounda y
sa is-
ying
he
ollowing
condi ions
.
(i)
dim
M
=
di n
N
(ii)
ank
Ho(OM)
=
ank
Ho(OM),
ank
Ho(M)
=
ank
Ho(N)
and
ank
H
o
(M,
(9M)
=
ank
HO(N,
&N)
.
Then
any
injec i e
con inuous
map
:(M,am)
-
(N,aN)
is
a
homeo-
mo phism
.
P oo
..
Le
V
deno e
he
double
M
+
Uam
M_
o
M
.
Le
=
ank
Ho(M)
and
s
=
ank
Ho(M,
(9M)
.
I
i
:&M
,M
deno es
he
inclusion,
hen om
he
,
exac
sequcnce
Ho(¿)M)
i
Ho(M)
-
Ho(M, 9M)
-
0
we
see
ha
ank
Im
i
=
-
s
.
F om
he
Maye -Vie o is
sequence
Ho(aM)
lx+

1
Ho(M+)
®
Ho(M-)
-
HO(V)
-'
0whe e
i
+:c7M
M
+
,
i_
:áM
,
M_
a e
he
espec i e
inclusions,
we
sce
l a
ank
HO(V)
=
2 -
ank
o i nage
((i
+
)


Howe e
I n((i+)

(i_),)
¡s
-
he
same
as
he
diagonal
subg oup
o
Im
i *
® Im
i

he ce
has he
same
ank
as
Im
i,
.
Thus
ank
Ho(V)
=
2
-
(
-
s)
=
+
s
.
Si nila ly
i
W
deno es
he
doulbe
N
+
Uai
N_
we
ha e
ank
H
o
(W)
=
+
s
.
In
pa icula
we
ge
1 o«)M)1
=
ank
Ho(9M)
=
ank
H
o
(8N)
_
1 a( 7N)1
and
1 o(V)1
=
+
s=
1 o(W)1
.
lOM
:OM
,
iN
is
a
.n
injec i e
con inuous
nap
and
1 o(l9M)1
=
l o(aN)1
.
Henee
heo e
6
.2
iníplies
ha
JOOM
is
a
ho neo no phism
.
The e
is
a
well-de ined
con inuous
map
g
:V
->
W
sa is ying
gIM
+
:M
+
->
N
+
and
glM-
:M-
-
N-
a e
he
same
as
.
Then
g
is
injec i e
and
l7 o(V)j
=
j7 o(W)j
.
F om
l eo em
6
.2
again
we
see
ha
g
:
V
-
W
is
a
homeomo phism
.
I
ollows i media ely
ha
:
(M,
(9M)
->
(N,BN)
is
a
homeomo phism
.
Co olla y
6
.7
.
I
M
is
any
compac ,
mani old
wi h
bounda 'y
OM
hen
(M,
¿9M)
is
a co-Hop a e
ohjec in
Top
e
.
P oo
.
I n nedia e
consequence
o
heo em 6
.6
.
Theo em
6
.8
.
(i)
I
M
is
any
compac
mani old
wi hou
bounda y
hen
C(M)
is
Hop ian
in
R-alg
.
(ii)
I
M
is
an,y
compac ,
opological
mani old
wi h
a,
non-em ) y
bou ad-
a y OOM,
hen
C(M)
is
nei he -
Hop can
no
co-Hop ian
in
R-alg
.
(iii)
I
M
is
a
compac ,
mani old,
hen
C(M)
is
co-Hop ia
in
R-alg
i
and
only
i
M
is
a, ini e
se ,
.
31
4

K
.
VARADARAJAN
P oo
.
:
I nmedia e
consequence
o
heo em
5
.3,
co olla y
6
.3
and
p opo-
si ions
6
.1
ancl
6
.4
.
7
.
Some
ela ed
esul s
and
coun e
examples
Recall
ha
a
ing
A
is
said
o
be
le
( esp
.
igh )
7 - egula
i
gi en
any
aeA
he e
exis s
a,n
elemen
bEA and an
in ege
n
>
1
sa is ying
a'L
=
ba"'
( esp
.
a'
=
a"+
l
b)
.
F
.
Dischinge
[7],[8]
has
sl own
ha
7 -
egula i y
is
le
igh
syn me ic
.
Be o e
Dischinge
ob ained
his
esul ,
G
.
Azumaya
[3]
e e ed o
a
ing
which
is
bo h
le
and
igh
7 - egula
as
a
s ongly
7 - egula
ing
.
By
Dischinge 's
esul
A
is
le
7 - egula
i
and
o ily
i
A
is
igh
n- egula
i
and
o ily
i
A
is
s ongly
7- egula
.
In
[7], [8]
Dischinge
also
ob ained
he
ollowing
esul s
:
1
.
A
ing
A
is
s ongly
i - egula
i
and
only
i
e e y
cyche
le
;
o
igh
A-module
is
co-Hop ian
.
2
.
Fo
a
ing
A
lhe
'ollowing
condi ions
a e
equi alen
.
(i)
E e y
ini ely
gene a ed
le
A-module
is
co-Hop ian
.
(ii)
E e y
ini ely
gene a ed
igh
A-module
is
co-Hop ian
.
(iii)
M,,(A)
is
s ongly
7 - egula
'o
all
in ege s
n
>
1
.
In
[9]
K
.R
.
Goodea l
in oduced
he
concep
o
a
le
epe i i e
ing
.
A
ing
A
is
said o
be
le
epe i i e
i
gi en
any
aeA
and
any
g
le
ideal
I
o A,
he
le
ideal
Ia"
is .
-
g
.
One
o
hc
;
esul s
p o ed
by
Goodea l
in
[9] is
he
ollowing
:
3
.
E e y
-gMcA- nod
is
Hop ian
i
and
only
i
M,, (A)
is
le
epe -
i i e
o
all
in ege s
n >
1
.
A
good
epo
on
hesc
:
ques ions including
new
p oo
;s
and ew
esul s
can
be
'omid in
[13]
.
Examples
7 .1
.
(a)
I
is
clea
ha
MeA-mod
Hop ian
=>
En(¡
(AM)
di ec ly
ini e
.
In
[19]
J .C
.
S ephe dson
gi es
examples
o
di ec ly
ini e
A
wi ii
M,
.,,
(A) no
di ec ly
ini e
o
Bo ne
in ege
n
>_
2
.

Fo
any
such
ing
A, wc
;
lla e
,
A
Hop ian
in
A- nod
.
Also
M,,(A)
is
no
Hop ian
in
M,,(A)- nod
.
Since
A"
is
a
di ec
su imand
o
Mk(A)
in
A-
nod,
whene e
k
2
>
n,
we
also
sc
:e
ha
Mk(A)
is
no
hop ian
in
A-mod
whene e
k
2
> n
.
(b)
In
pa
C
o
[5]
G
.M
.
Be gman
cons uc s
o
each
in ege
n
>
1
a
ing
A
wi h
he
p ope y ha
all
egula
elen en s
in
A
a e
in e -
ible,
bu
M"(A)
is
no
i s
own
classical
ing o
quo ien s
.
In
[13]
P
.
Menal
cons uc s
a
ing
A
which
is
i s
own
classical
quo ien
ing
bu
M,,(A)
is
no
O e,
henee
M,,(A)
does
no
ca en
ha e
a
classical
ing o
quo ien s
.
A
ca e ul
inspec ion
shows
ha
in
bo h

HOP IAN
ANn
Co-Hop IAN
Oi3JECTS

315
hese
examples
he
le
egula
and
he
igh
egula
elemen s
o
A
coincide
.
Hence by
p oposi ion
1 .4
in
ou
p ese i
pape
A
is
co-
Hop ian
in
bo h
A-mod
and
mod-A
.
Howe e ,
M,,(A)
is
nei he
co-Hop ian
in
M,,(A)-mod
no
co-Hop ian
in
mod-M,(A)
.
(c)
In
sec ion
5
o
[16]
i is
ema ked
ha
W
.L
.
May
has
a
me hod
o
ob aining
an
in ini e
abelian
Hop ian
g oup
G
such
ha
he
complex
g oup
algeb á
C(G)
is
no
Hop ian
as
a
C-algeb a,
hence
no
Hop ian
as
a
ing
.
In his
example C
is
Hop ian
as a
ing,
G
is
Hop ian
as
a g oup
bu
C[G]
is
no
Hop ian
as
a ing
.
8
.
Open
p oblems
1
.
I
A
is
Hop ian
as a
ing,
is
A[X]
Hop ian
as
a
ing?
2
.
I
A
is
c
:o-Hop ian
as
a
ing
and
G
a
.
co-Hop ian
g oup
is
A[G]
c
:o-Hop ian
as a
.
ing?
3
.
I
A
is
Hop ian
in
A- nod
and
G
a
Hop ian
g ouli
is
A[G]
Hop ian
in
A[G]- nod?
4
.
I
A
is
co-Hop ian
in
A-mod
and
G
a
co-Hop ian
g oup
is
A[G]
co-Hop ian
in
A[G]-mod?
5
.
I
A
is
Hop ian
( esp
.
co-Hop ian)
as a ing
is
i
ue
ha
M,, (A)
is
Hop ian
( esp
.
co-Hop ian)
as
a
ing?
6
.
Cha ac e i e
he
Hop ian
( esp
.
co-Hop ian)
objec s
in
Top
a iong
compac
Hausdo
o ally
disconnec ed
spaces
.
7
.
I
M
E
A- nod
is
Hop ian
is
M[X,
X
-i
]
Hop ian
in
A[X,
X
-
i]-
nod?
Re e ences
1
.
F.W
.
ANDERSON
ANn
K
.R
.
FULLE ,
"Rings
and
.
Ca egoHes
o
Modules,"
G adua e
Tex s
in
Ma he na ics
13,
Sp inge -Ve lag,
New
Yo k,
1973
.
2
.

E
.P
.
A imENDARIZ,
Jo
;,,
W
.
FisilER
AND
Rom
;i
L
.
SNini
iz,
On
¡Ajec i e
and
su jec i e
cndomo plüs is
o
ini ely
gene a ed
mod-
ules,
Communica ions
in
A
.lgeb a
6
(1978),
659-672
.
3
.

G
.
AZUMAYA,
S ongly
n- egula
ings,
J
.Fac
.
Se¡
.,
Hokkaido
Uni
.
13
(1954),
34-39
.
4
.

G
.
BAUMSLAG,
"Topics
in
Abelian
G oups,"
Edi ed
by
J
.
I win
and
E
.A
.
Walke ,
Sco
Fo esmann
and
Company,
1963,
pp
.
331-335
.
5
.
G
.M
.
BE1 CMAN,
So ne
examples
in
PI
Ring
Theo y,
Is ael
J
Ma h
.
18
(1974),
257--277
.
31
6

K
.
VARADARAJAN
6
.

A
.W
.
CI ATTCRS
AND
C
.R
.
HAJARNAVIS,
"Rings
wi h
chain
con-
di ions,"
Resca ch
No es
in
Ma hc na ics
44,
Pi nian
Publishing
Limi ed,
1980
.
7
.

F
.
DIscilINGCR,
Su
les
anneaux
o e emen
n- egulie s,
C
.R
.
Acad
.
Sci
.
Pa is,
Se
.
A
283
(1976),
571-573
.
8
.

F
.
DISDIIINGCR,
S a k
7 - egula
Ringe,
Disse a ion,
Ludwig-Ma-
ximilians-Uni e si a ,
Munchen,
1977
.
9
.

K
.R
.
GOODEARL,
Su jec i e
endomo phisms
o
ini ely
gene a ed
modules,
Comm
.
i
n
Algeb a
15
(1987),
589-609
.
10
.
P
.J
.
MLTON
AND
J
.
ROITBGRG,
Rela i e
epimo phisms
and
mono-
mo phisms
in
homo opy
heo ey, Composi ion
Ma hema ica 61
(1987),353-367
.
11
.
Y
.
HIRANO,
On
Fil ing's
Lem na,
Hi oshima
Ma h
.
J
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9
(1979),
623-626
.
12
.
V
.A
.
HIREMATIi,
Hop ian
Rings
and
Hop ian
Modules,
Indian
J
.
Pu e and Appl
.
Ma h
.
17
(1986),
895-900
.
13
.

P
.
MENAL,
Cancella ion
modules
o e
egula
ings,
P occedings
in
ing
Tl eo y,
G anada,, 1986
SLNM
1328
.
14
.

P
.
M NAL,
Mo i a
equi alen e
and
quo ien
ings,
Resul s
in
Ma h
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13
(1988),
137-139
.
15
.

M
.
ORZGGII,
On o
endo no pliisms
a e
iso no pliisms,
Ame
.
Ma h
.
Mon hly 78
(1971),
357-362
.
16
.

M
.
OI z cII
AND
L
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RIBCS,
Residual
ini eness
and
he
Hop
p op-
e y
in
Rings,
J
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Alg
.
15
(1970),
81-88
.
17
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P
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RIBENBOIM,
"Rings
and
Modules,"
T ac s
in
Ma hema ics
24,
In e science
Publishe s,
1969
.
18
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J
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Residually
ini e
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and
co-Hop ian
spaces,
Con-
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37
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131-144
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19
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J
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.
ST PII RDSON,
In e ses
and
ze o
di iso s in
ma ix
ings,
P oc
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London
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(1951),
71-85
.
20
.
G
.F
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SIMMONS,
"In oduc ion
o
Topology
a,nd
Mode n
Analysis,"
McG aw-Hill
Book
Company,
1963
.
21
.

M
.H
.
STONE,
Applica ions
o he heo y o
Boolean
.
Rings
o
Gen-
e al
Topology,
T ans
.
A
.M
.S
.
4
1
(1937),
375-481
.
22
.
J
.R
.
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p ojec i es,
Nagoya
Ma h
.
J
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27
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747-751
.
23
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K
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VARADARAJAN,
A
gene aliza ion
o
Hilbe 's
Basis
Theo em,
Communica ions
in
Algeb a
10
(1982),
2191-2204
.
HOi'FlAN
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CO-HOPFIAN
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
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24
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VASCONCGLOS,
On
ini ely
gene a ed
la
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D-ans
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A
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138
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505-512
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25
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Injec i e
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o
ini ely
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The
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Albe a
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1N4
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6
de
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1992