Publicacions
Ma emá iques,
Vol
36
(1992),
965-979
.
A
bs ac
ADIC-COMPLETION
AND
SOMEDUAL
HOMOLOGICAL
RESULTS
ANNE-MARIE
SIMON
To
he
memo y
o
Pe e
Menal
Le a be
an
ideal
o
a
commu a i e
ing
A
.
The e
is
a
kind
o du-
ali y
be ween
he
le
de i ed
unc o s
Ui
o
he a-adic
comple ion
unc o ,
called
local
homology
unc o s,
and
he
local
cohomology
unc o s
Há
.
Some
dual
esul s
a e
ob ained
o
hese
Ua, and
also
inequali-
ies
in ol ing
bo h
local
homology
and
local
cohomology
when
he
ing
A
is
noe he ian
o
mo e
gene ally
when
he
U°
and
H
a
-global
dimensions o
A
a e
ini e
.
In
his
pape
A
is
a
commu a i e
ing,
a
an
ideal o
A
and
.
he
A-
modules
a e
gi en
he a-adic
opology
.
The e
is
a
ce ain duali y
be ween
he
le
de i ed
unc o s
Ui
o
he
a-adic
comple ion
unc o
and
he
local
cohomology
unc o s
H,,,
i s
obse ed
by
Ma lis
when
he
ideal
a
is
gene a ed
by
a
( ini e)
egula
sequence,
ue
also
o
any
noe he ian
ing
.
Mo e
ecen ly,
ha
duali y
has
also
been
obse ed
by
C eenlees
and
May
in
a
mo e
gene al
con ex
.
The
pu pose
o his
no e
is
o
pu sue
he
analogy
be ween
he
local co-
homology
unc o s
and
hese
unc o s
Uá,
called
local
homology
unc o s
by
C eenlees
and
May
.
Fi s
we
ha e
dual
esul s
abou
codep h,
a
no ion dual
o
he
no ion
o
homological
dep h
o
g ade
.
To
go
u he ,
we
need
come
noe he ian hypo hesis
in
o de
o
ha e
a
chango
o
ings
heo em
o
he
Ui
;
a lalogous
o
he
co esponding
one
in
local
cohomology
.
This
b ings
us
back
o
he
i s
case
s udied
by
Ma lis,
namely
he
case
o
an
ideal
gene a ed
by a
egula sequence,
and
allows
gene aliza ions
o
some
Ma lis
esul s
.
As a
consequence,
we
ob ain
anishing
esul s
o
he U°,
and
also inequali ies
in ol ing
96
6

A
.-M
.
SIMON
bo h
local
cohomology
and
local
homology
.
So
local
cohomology
and
local
homology
a e
no
only
duals
o
each
o he ,
bu
aleo
in ima ely
connec ed
.
As
gene al
e e ences
o
commu a i e
algeb a
and
homological
ques-
ions,
we
quo e
[1],
[18]
.
The
wo k below
has
been
communica ed
a
he
In e na ional
wo kshop
on
local
cohomology,
geome ic
applica ions
and
ela ed
opics
.
We
ake
his
oppo uni y
o
hank
he
o ganize s
and
he
pa icipan e
o use ul
discussions
.
In his
i s
sec ion
we
ix
no a ions
and
collec
he
ma e ial
we
need
.
Though
pa
o
i
appea ed
in di e en places,
we
hink
i is
mo e
con-
enien
o
ha e
i
a
hand
.
1
.1
.
Comple ion
.
Le
a
be
an
ideal
o
he
commu a i e
ing
A
.
The
A-module
a e
gi en
he
a-adic
opology
.
The
comple ion
o
an
A-module
M
is
deno ed
by
1Vl
:
hus
M
=
lim
M/a'M
.
Le
Tn,1
:
M
-->
M
be
he
na u al
mo phism
.
3J)
.
1
.
P elimina ies
and
no a ions
He e and
in
he
nex
sec ion,
he
ideal
a
is
no
necessa ily
ini ely
gene a ed
;
and
i
migh
happen
ha
he
A-module
M,
comple e
in
i s
na u al
opology,
is
no
comple e
in
i s
a-adic
opology
.
An
example
o
his
can
be
ound
in
([5,
III,
Sec ion
2,
exe cise
12])
o in
([3,
I
;
Sec ion
Recall
howe e
he ollowing
esul
([3,
heo em
1
.3
.1],
o
[13,
heo em
15],
o
[18, 2
.2 .5])
.
Theo em
.
Suppose
he
ideal
a
ini ely
gene a ed
.
Le
M
be
an A-
module and
b
an
open
ideal in he
a-adic
opology
o
A
.
Then
he
mo -
phism- moA/b
:
M/bM
->
M/blVl
is
an isomo phism
.
So
M
is
comple e
in
i s
a-adic
opology
.
1 .2
.
When
is
on o
.
The
a-adic
comple ion
unc o ,
hough
no
igh
exac ,
p ese es
su -
jec ion
.
Howe e ,
we
wan
o
know
p ecisely
when
he
comple ion
o
a
mo phism
is
on o
.
The
ollowing
lemma
was
p o ed
in
([16,
1 .2])
o
a
noe he ian
ing
A,
using
1
.1
.
I
is
ue
in
gene al
.
Lemma
.
Le
:
M
--->
N
be a
mo phism
o
A-modules
.

Then
su7-jec i e
i
and
only
i
N
=
Al
-1-
aN
.
ADIC-COMPLGTION,
DUAL
RESULTS

967
P oo
..
I
N
=
Al
+
aN,
hen
N
=
M
+
anN
o
all
n
>_
1,
and
he
p ojec i e
sys em
o
sequences
0
--->
-1
(
a
' i
N)
la"M
--->
Ml
a'
M
-->
N/a'
N
--->
0
is
exac
.
I
is
easily
seen
o
be
su jec i e
(see [16,
1 .2])
.
So, a e
aking
limi s,
we
ge
an
exac
sequence
and
is
su jec i e
.
Con e sely,
i
is
su jec i e,
we
enso
he
commu a i e
na u al
dia-
g am
M
N
M/aM

>
N/aN,
whe e
he
e ical
composi e
maps
a e he
na u al
p ojec ions,
wi h
Ala
.
The
composi e
e ical
maps
become
he
iden i y,
so
M/aM
->
N/aN
is
su jec i e
and
N
=
M
+aN
.
1 .3
.
The
le
de i ed
unc o s o
he
comple ion
unc o
.
The
le
de i ed
unc o s
o
he
a-adic
comple ion
unc o
a e
deno ed
by
Ui
.
These
we e
i s
s udied
by
Ma lis
when
he
ideal
is
gene a ed
by
a
ini e
egula
sequence
[12],
[13]
.
We
used
hem
in
[16],
whe e
he
ing
is
noe he ian
.
Mo e
ecen ly,
hey
ha e been
compu ed
by
G eenlees
and
May
in a
mo e
gene al
si ua ion
[7]
.
Le
L
1
-
>
Lo
->
A1

+
0
be
a
exac sequence,
wi h
Lo,
L
1
ee
.
By
de ini ion
Uo
(NI)
=
coke
.Í,
so
we
ha e
na u al
mo phisms
A1
Uó
(A1)
->
M
whose
composi e
is
- A
.
The
ollowing
le nma
([16,
5
.1]),
consequence
o
1 .2, is s ill
a ailable
.
Lemma
.
(i)
The
na u al
mo j)hism
UÓ
(M)
->
NI
is
on o
.
(ii)
M
=
0
i
and
only
i
M
=
aA1
i
and
only
i
Uó
(M)
=
0
.
(iii)
I he ideal
a
is
ini ely
gene a ed,
hen
(Uó
(M))^
=
A1
.
1 .4
.
The
class
C
a
.
Le
C,
be
he
class o
modules
M
such
ha
Uó
(M)
=
A7 and
Ui
(M)
_
0
o
i
>
0
.
A
s anda d
homological
a gumen
shows
ha
Uá(A1)
can
be
compu ed
using
a
le
esolu ion
o
A1
wi h
modules
in
C
a
,
and
i is
wo llwile
o
no e
ha
la
nodules
belong
O
C
a
.
968

A
.-M
.
SIMON
Mo e
gene ally,
le
a¡,,
n
>
0,
be a
dec easing
sequence
o ideals
which o m
a
basis
o
he
a-adic
opology
.
A
module
M
such
ha
To A(A/a

,,
M)=
0 o
all
i
>
0 and
all
n
>
0
belongs
o
C
a ([12,
co olla y
4
.5])
.
When
he
ing
is
noe he ian,
he
comple ion
o
a
ee
module
is
la
([14,
p
.
77],
o
[3,
4,
7],
o
[18,
2
.2
.4])
.
This
can be
used
o
show
ha
comple e
modules
belong
o
C
a
when
A
is
noe he ian
([16, 5
.2])
.
1
.5
.
Local
cohomology
and
Ma lis
duali y
.
Recall he
unc o
H
a'
:
H°(M)
=
{x
E
Mjanx
=
0
o
some
na u al
numbe
n},
whose
igh
de i ed
unc o s
Há
a e
he
local
cohomology
unc o s
.
Recall
also
he
Ma lis
duali y
.
Le
E
be
he
injec i e hull o
he
di ec
sum
o
all
he
A/m
wi h
m,
a
inaximal
ideal
o
A
.
The
Ma lis
duali y
unc o ,
de ined
by
M
=
HOMA(M,
E),
is
ai h ully
exac
[12]
;
[13],
and
we
ha e
he
Ex -To
duali y
:
To A(N,
M)
-
Ex Á(N,
M
)
;
when
N
has
a
p ojec i e esolu ion
composed
o
ini ely
gene a ed
mod-
ules
([6,
VI,
5
.1
;
5
.3])
:
To A(N,
M
)
-
Ex Á(N,
M)
.
When
he
ing
is
noe he ian,
Hó(M)
-
(M
)
^
and
Há(M)
-
U°'(M
)
o
all
i
([16, 4
.2
;
5
.6])
.
This
is
based
on
he
ac
ha ,
o e
a
noe he ian
ing,
la
modules
and
injec i e
modules
a e
in e changed
by
Ma lis
duali y
.
This
was
i s
p o ed
by Ma lis
when
he
ideal a
is
gene a ed
by a
egula
sequence
.
Bu
modules
a e
no
necessa ily
duals,
so in o ma ions
abou
he
local
cohomology
unc o s
Hi,
do
no always
p o ide
in o ma ions
abou
he
local
homology
unc o s
Ui
.
1
.6
.
Fo mal
dep h,
codep h
and
dimension
.
A
sequence
o
co a ian
addi i e
unc o s
G,,,
:
A

+
B,
n
E
Z,
be ween
abelian
ca ego ies
is
a
descending
connec ed
exac
sequence
o
unc o s
i
G
a
=
0
o
n
<
0 and
i
each
exac
sequence
0
-+
M'
~
11N1
->
M"
-
0
in
A
gi es
ise
o
a
long
exac sequence
. . .
->
Gi+i(01)
->
GZ(M')
)
Gi(M)
-,
Gi(M")
->
. . .
in
a
unc o ial
way
.
ADIC-COMYLETION,
DUAL
RESULTS

96
9
When
we
ha e
such
a
sequence,
as in
([18,
1
.1]
o
[17]),
we
pu
g_(M)
=
in {¡IG
i(
M)
=,,~
o}
o
each
objec
M
in
,A
(so
ha
o
<_
g_
(M)
<
oo)
.
Dually,
we
de ine
-
(M)
o
an
ascending
connec ed
exac
sequence
o
unc o s
F"
in
he
saíne
way
.
These
numbe s
can
be
iewed
as
a
kind
o
codep h
o
dep h
espec-
i ely
.
When
a
is
an
ideal
o
he
ing
A,
we
a e
me ely
conce ned
wi h
he
sequence
Ex
;~(A/a,
.),
To A(A/a,
.),
H,'~(
.),
Ui
(-)
and
wi h
he
co e-
sponding
numbe s
.
He e
a e
some
i s
ema ks
abou hem,
which
will
be
comple ed
la e
(1
.7,
2
.4)
.
P oposi ion
.
Le ,
a
.
C
b
be
ideals
in
he
ing
A,
le
M
an
A-module
(i)
ex
A
(A/a,
AI)
=
ha
(Ah)
(ii)
o '(A/a,
M)
=
ex -(A/a,
M")
(iii)
ex A
(A/a,
M)
<
ex
A
(A/b,
Al1)
(i )
o A
(A/a,
M)
<
o A
(A/b,
M)
( )
he
numbe s
ex
A
(A/a,
M),
o A(A/a,
M)
only
depend
on
he
opology
de ined
by
he
ideal
a
.
Fo
(i)
and
(iii),
see
([18, 5
.3
.1
.5,
5
.3
.11])
;
(ii) is
a
di ec
consequence
o
he
Ex -To
duali y
1
.5
;
(i )
ollows
om
(ii)
.
and
(iii)
.
Since
local
cohomology
only
depends on
he
opology
de ined
by
he
ideal
a,
so
does
ex A(A/a,
.)
in
iew
o
(i),
and
so
does
also
o A(A/a,
.)
in
iew
o
(ii)
.
We
also
de ine
g+
(M)
=
sup{ilC
i
(M)
:y~
0}
(so
ha
g+(M)
=
-oo
o
0
_<
g
+
(A11)
<
oo)
and
+(AI)
in
he
saíne
way
.
These
las
numbe s
a e
ela i e
homological
dimensions
.
1.7
.
The
dep h-codep h
sensi i i y
o he
Koszul
complex
.
The
dep h
sensi i i y
o
he
Koszul
complex
was
p o ed
by
Ba ge
and
Hochs e
o
a
cohe en
ing
[2], [8],
by Ki by
and
Meh an
o
any
commu a i e
ing
[10]
.
An
app oach
in ol ing
bo h
dep h
and
codep h
can
be
ound
in
([18,
6
.1]
o
[17])
.
Le
x
=
x
I
,. . .
,
x,
1
be a
sequence
o
eleinen s
o
he
ing
A
gene a -
ing
an
ideal
a,
and
le
K
.
(x)
be
he associa ed
Koszul
complex
.
Fo
an
A-module
M,
; e
conside
he
descending
and
ascending
Koszul
com-
plexes
K
.
(x,
All)
=
K
.
(x)
®A
A7,
K"
(x,1Vl)
=
Ho nA
(K
.
(x),
M)
;
we
no e
hei
homologies
by
I-Ii
(x,
M)
and
IIi (x,
All)
espec i ely
.
These
unc-
o s
Hj(x,
.)
and
H'(x,
.)
a e
descending
;
ascending
connec ed
exac
se-
quences
o
unc o s
.
In
he
no a ions
o
1
.6,
we
ha e
he
ollowing
esul
([18
.,
6
.1 .6
;
6
.1
.7])
.

97
0

A
.-M
.
SIMON
Theo em
.
Le
x
=
xl,
. . . ,
xn
.
gene a e
an
ideal
a
in he
ing
A
and
¡el
M
be
an
A-module
.
Then
h_
(x,
M)
=
o '
(A/a,
M),
h-
(x,
M)
_
ex í
(A/a,
M)
.
Co olla y
.

Le
a
=
(XI,
. . . ,
xn)
be
a
ini ely
gene a ed
ideal o he
ing
A,
and
le
M
be
an
A-module
.
(i)
o A
(A/a,
M
)
=
ex ~
(A/a,
M)
(ii)
The numbe s h~
(M), ex _
(A/a,
M),
o A
(A/a,
M)
a e
ini e
si-
mul aneously
.
In
ha
case,
ex A
(A/a,
M)
+
o A
(A/a,
M)
<_
n
.
(iii)
I he
numbe s
h~l
(M),
ex ~(A/a,
M),
o A(A/a,
M)
a e
in ini e,
hen,
o
any
ideal
a',
open
in he
a-adic
opology
o
A,
he
num-
be s
ex
-
(A/a',
M),
o A
(A/a',111),
uá'
(M)
a e
also
in ini e
.
(i )
I
:
A
--->
B
is
a
mo phism
o
ings,
i
b
=
(a)B,
hen,
o
each
B-module
N,
we
ha e
hb (N)
=
ex ,
(B/b,
N)
=
ex
A
(A/a,
N)
_
h
a
-
(N),
o B(B/b,
N)
=
o A(A/a,
N)
.
P oo
.
(i)
We
ha e
an
isomo phism
K'(x,M) -K(x,M'),
so
o A(A/a,M
)
=
h_
(x,117")
=
h -
(x,
M)=
ex
A
(Ala,
Al)
.
(ii)
This
is
a
consequence
o
he
sel -duali y
o
he
Kosmil
complex
:
H'(x,
M)
-
H
n_
2
(x,
M)
(see
[18
;
6
.1
.8])
.
(iii)
When
an
ideal
a'
is
open
in
he
a-adic
opology,
we ha e
a'
D
a
o
a
ce ain
na u al
numbe
.
Using
1
.6,
we
ob ain oo
=
o A(A/a,
M)
=
o A(A/a ,
M)
_<
o A
(A/a',
M)
=
oo
.

The
open
ideal
a'
being
ixed
now,
we
ha e
also
o A(A/a",
M)
=
oc
o
all
ideals
a",
open
in
he
a'-adic
opology
.
So
he
module
M
belongs
o
he
class
C
a
,
(1
.4)
and, as
M
=
a'M,
we
ha e
also
Uó'
(M)
=
0
(1
.3)
and
uá'
(M)
=
oo
.
(i )
Take
he
imago y
in
B
o
he
sequence
x
gene a ing
a
:
yi
.
=
(xi)
.
The e
a e
ob ious
isomo phisms
K
.
(y)
-
K
.
(x)
®A
B,
K
.
(y)
®B
N
,_
.,
K
.
(x)
®A
N,
HOMB
(K
.
(y),
N)
-
HomA(K
.
(x),
N)
.
So
his
is
ano he
consequence
o
he
heo em abo e
.
No e
ha
we
ha e
ob ained
he e
a
change
o
ings
esul
o
he
dep h h
a
(-)
(a
ini ely
gene a ed)
in a
si ua ion
whe e
we
don'
ha e a change
o
ings
heo em
o
he
local
cohomology
unc o s
2
.
U-codep h
In
local
cohomology,
we
ha e
he
equali y
h
a
(M)
=
ex
A
(A/a,
M)
al-
eady
men ionned
(1
.6)
.
We
wan
an
analogous
esul
o
he
U-codep h
ADIC-COMPLETION,
DUAL
RESULTS

97
1
u"_
using
To
ins ead
o
Ex
.
To
achie e
his,
we
need
some
p epa a ion
.
2
.1
.
The
ollowing
li ing
p oposi ion
was
p o ed
in ([4,
3
.5])
in
he
local
case
.
P oposi ion
.
Le
a
be
an
ideal
con ained
in
he
Jacobson
adical o
he ing
A,
and
le
F
be a
ía
A-'module
such
ha
F/aF
is
ee as
an
A/a-module
.
I
{ei¡i
E
I}
is
a
se ,
o
elemen s
o
F
such
ha
i s
image
{~ili
E
I}
in
F/aF
is
a basis
o
F/aF,
hen
he
se
{ei¡i
E
I}
gene a es
a
pu e
ee
submodule
L
o
F,
,
and
F
=L
+
aF
.
P oo
...
We
i s
p o e
he
eeness
o
he
el in
F
.

I
n
I
biei
=
0, bi
E
A,
we
pu
b
=
(b
l
, . . . .
b,
y
),
e
=
(e],
. .
.
.
en)
;
in
ma lclal
language,
we
ha e
b
.e'
=
0
.
By
a
la ness
c i e ium
([5,
I
;
Sec ion
2,
p oposi ion
13,
co olla y
1]),
he e
is
a
ma ix
X
E
A'n"
and
a
ec o
E
FI
"1
such
ha
e
=
X
.
c,
b
.X
=
0
.

Deno ing
he
images
modulo
he
ideal
a by
-),
we
ha e
é=
Ñ
j'
.
Bu
he
éi
o n
a
basis
o
F/aF,
he
ma ix
X
is
hus
igh -in e ible,
and
so
is
he
ma ix
X
since
a
is
con ained
in
he
Jacobson
adical o
A
.
F om
b
.X
=
0
we
deduce
b
=
0
and
he
eeness
o
he
el
in
F
.
We
now
p o e
he pu i y
o
L
in
F
.
As
F
is
la ,
i
is
enough
o
check
he
injec i i y
o
he
mapsL/eL
-->
F/cL
o
each
ideal c
o
A
.
As
he
image
ei
o
he
elemen s
e
l
o
L
o m
a
basis
o
F/aF,
he
na u al
mo phism
L/aL
->
F/aF
is
an
isomo phism,
and
so
is
L/(a
+
c)L
->
F/(a
+
c)F
.
Bu
he
ideal (a
+
c)/c
o
A/c
is
con ained
in
he
Jacobson
adical
o
A/c
.
We
apply
he
i s
pa
o
he
p oo
o
he
ía
A/c-module
F/cF
and
o
he
images
o
he
el
in
F/cF
:
he e
images
gene a e
a
ee
submodule
o
F/cF,
so
he
mo phism
L/cL
->
F/cL
is
injec i e
and
L
is
pu e
in
F
.
Now
F
=
L+
aF
is
clea
.
2
.2
.
P oposi ion
.
Le
a
be
an
ideal
con ained
in
he
Jacobson
adical
o
he
ing
A,
and
le
M
be
an
A-module
wi h
M
=
aM
.
Then
he e
exis s
an epimo phism
P
->
M
whe e
P
is
a
ía
A-module
wi h
P=
aP
.
P oo
.
Le
0
-->
K
->
F
->
M
->
0
be
an
exac
sequence,
whe e
F
is
ee
.
As
M
=
aM,
he
sequence
K/aK
->
F/aF
-->
0
is
exac
.
Choose
in
K
elemen s
yi
whose
images
in
he
ee
A/a-module
F/aF
o n a
basis
97
2

A
.-M
.
SIMON
o
F/aF
.
By
2
.1,
hese
yi
gene a e
a
ee
pu e
submodule
L
o
F,
and
L C
K
.
So
P
= F/L
is
ia ,
and
P=
aP
since
F=
L+aF
.
Now
he
epimo phism
F
->
M
induces
an
epimo phism
P=
F/L
->
M
.
2
.3
.
In
he
p eceding
p oposi ion,
he
condi ion
M
=
aM
means
ha
he
To -codep h
and
he
Ua-codep h
o
M
a e
posi i e
:
o
A
(A/a,
M)
>
0,
ua
(M)
>
0
(1
.3)
.
On
he
o he
hand,
o
he
la
module
P,
we
ha e
o A(A/a,
P)
=
oo
=
uQ
(P)
(P
belongs
o
C
a
,
see 1
.4)
.
So
his
shows
ha
he unc ions
o A(A/a,
.)
and
ua
(
.)
sa is y
he
duals
o
he
axioms
o I oh
cha ac e izing
a
homological
g ade
[9]
.
I
will
be
used
o
p o e
he
equali y
be ween
he
U'-codep h
and
he
To -codep h
.
When
he
ing
is
noe he ian,
we
ge
id o
he
assump ion
on
he
ideal
a
by
enso izing
wi h
Á
.
Indeed,
in
ha
case,
Á
is
A- la ,
h
n
=
OÁ,
á
is
con ained
in
he
Jacobson
adical
o
Á
[1],
and
we
ha e
he
ollowing
easy
obse a ion,
ex ending
([18, 2
.2
.2])
.
Lemma
.
Le
a
be
an
ideal o he
noe he ian
ing
A
.
Then,
o
each
A-module
M,
he
module
11!1
is
isomo phic
o he
á-adic
comple ion
o
he
Á-module
Á
®A
M,
and
Ua(M)
-
UA(Á
®A
M),
To A(A/a
n
,
M)
-
To A(Á/á',
Á
(DA
M)
o
all
n>
0
.
(I
L
.
-->
M
->
0
is
a
ee
esolu ion
o
M,
hen
A
OA
L
is
a
ee
esolu ion
o
he
Á-module
Á
®A
M,
and
Á/án
®á
(Á
&A
L
.)
-
Á/án
®A
L
.
--
A/an
(DA
L
.
.
This
gi es
he
esul ,
a e
aking
limi s
o
he
U-pa
o
i )
.
2
.4
.
Theo em
.
Le
a
be
an
ideal
o
he
ing
A
.
a
is
con ained
in
he
Jacobson
adical
o
A
o
i
A
is
noe he ian,
hen,
o
each
A-module
M
;
uá
(M)
=
o A
(A/a,
M)
.
P oo
.
.
We
al eady
know
ha
uá
(M) and
o A
(A/a,
M)
anish
simul ane-
ously,
exac ly
when
M
=A a
.1V1
(1
.3)
.
Wi h
2
.3,
we
a e
educed
o
he
i s
case,
whe e a
is
con ained
in
he
Jacobson
adical
o
A
.
In
ha
case,
i
one
o
he
numbe s
abo e
is
posi i e
ini e,
we
ha e an
exac sequence
0
-->
M
l
-->
P
-->
M
->
0,
whe e
P
is
la
and
P
= aP
(2
.2)
.
The
long
exac
sequences
associa ed
wi h
i
shows
o '
(A/a,
N1
1
)
=
o '
(A/a,
M)
-
l,
.ua
(Ml)
=
ua
(M)
-
.l
.
So
2
.5
.
ADIC-COMPLETCON,
DUAL
RESULTS

97
3
by an
induc ion
a gumen
we
ha e
u'
(AI)
=
o A(A/a,
Al1)
.
This
shows
also
lia
hese
wo
numbe s
a e
in ini o
simul aneously
.
P oposi ion
.

Unde
he hypo hesis
o
2
.4,
i
=
u°_
(M)
<
oo and
i
he ideal
a
is
ini ely
gene a ed,
hen
U,"(M)
^
=
lim
To
(A/a",
M)
.
P oo
..
This
is
done by
induc ion
on
,
using
an
exac
sequence
as
in
2
.4
(a e
ha ing
enso ed
by
A
in
he
noe he ian
case),
he
case
=
0
is
1
.3
.
3
.
U-dimension
and
H-dimension
o e
a
noe he ian
ing
We
now
s udy
he
dimensions
u+
(M)
and
há
(M)
as
de ined
in
1
.6
.
Ou
ings
a e
now
noe he ian
.
In
local
coho nology,
i is
known
ha
h
a
(M)
<_
di n
A4I
o
each
A-
module
M
[15]
(mo eo e ,
i
All
is
ini ely
gene a ed
and
i
a
=
m
is
he
maximal
ideal o
a
local
ing
;
hen
h
,-
4
,
-
,(M)
=
dim AJ
[11])
.
We
s ablish
an
analogous
inequali y
o
he
U-dimension
.
This
in
u n
allows
i_is
o
e ine
he
inequali y
abo e
.
To
achie e
his,
we
need
a
chango
o
ings
heo em
o
he
U
¡
',
analogous
o
he
co esponding
one
in
local
cohomology
.
3
.1
.
Le
Ali,
i
E
I,
be
a
amily
o
modules
o e
he
noe he ian
ing
A
.
Then
(®ilhli)
^
=
{w E
lliÑ
i
l
o
all
n,
all
bu
ini ely
many
compo-
nen s
i i
o
w
belong
o a n
ú¿}
([16,
9
.4]),
so
(®iA4i)
^
=
When
1Vl
i
-
A1
o
all
i,
we
w i e
as
usual
AI(')
=
®
i
M
i
,
MI=
IIiMi
.
.
The
ollowing
lemina
was
obse ed
in ([14,
p
.
77
.
2
.4
.2])
.
Lemma
.
Le
I be
a
se
.
Each
sho exac
sequence
0
->
M'
-
Al1
->
AI"
->
0
o
ini ely
gene a ed
modules
o en
he
noe he ian ing
A
gi es
ise
o
an
exac
sequence
0
-)

(M1(1))^
--
*U
,
(Aj(I))^
--
V
-->
(A4,i (1))^

>
0