Publicacions
Ma emá iques,
Vol
36
(1992),
189-195
.
GLOBAL
DIMENSION
IN
NOETHERIAN
RINGS
AND
RINGS
WITH
GABRIEL
AND
KRULL
DIMENSION
A
bs ac
JUAN
JACOBO
SIMÓN
PINCRO
In
his
pape we
compu e
he
global
dimension
o
Noe he ian
ings
and
ings
wi h
Gab iel
and
K ull
dimension
by
aking
a
subclass
o
cyclic
modules
de e mined
by
he
Gab iel
il a ion
in
he
la ice
o
he edi a y
o sion
heo ies
.
In oduc ion
In
his
pape
we
exhibi
a nice
subclass
o cyclic
modules
o
compu e
he
global
dimension
o
a ing
(see
[9],
[12], [13],
[15])
whose
o igins
a e
in
[3]
and
[11]
.
In
he
i s
pa ,
he
le
global
dimension
o a
noe he ian
ing,
R,
is
compu ed
in
e ms
o
he
injec i e
dimensions
o
he
ollowing
subclass
o
R-mod
.
I
T-,
<
-
o
<
...
<
T,
is
he
Gab iel
il a ion
in
he
la ice
o
he edi a y
o sion
heo ies
o R,
Le
.
R- o s
[2],
hen
he
subclass
consis
o
all
he
cyclic
,,-coc i ical
le
R-modules
whose
injec i e
dimension
equals he
injec i e
dimension
o
e e y
one
o
i s
submodules,
wi h
p
anging
o e
all
he
o dinals
less
han
,Q
.
Also,
we
ob ain
some
o
he
classical
esul s
o
noe he ian
ings as
consequences
o
ou
esul s
.
In
he
second
pa
we
no e ha
all
ou
esul s
can
be
dualized
.
Th oughou
his
pape ,
R
will
deno e an
associa i e
ing
wi h
1
and
R-mod
he
ca ego y o
all
uni a y
le
R-modules
.
To sion
classes
and
o sion
heo ies
will
always
be
he edi a y
;
all
e minology conce ning
o sion
heo ies
is
quo ed
om
[2]
.
Gi en a
nonze o
M
c
R-mod,
Id(M)
and
Pd(M)
deno e
espec i ely,
he
injec i e
and
p ojec i e
dimensions
o
M,
se ing
Id(0)
=
Pd(0)
=
-oo
.
The
le
global
dimension
o
R
will
be
deno ed
by
lgl
dim(R),
and
he
Gab iel
dimension
G
dim(R)
.
Fo
u he
de ails
on
each
o
hese
dimensions
we
e e
espec i ely
o
[13]
and
[2]
.
19
0

J
.J
.
SIMÓN
PINERO
1
.
Injec i e
dimension
The
S ong
Injec i e
Dimension
o a
le
R-module,
M
is
de ined
as
Sid(M)
=
sup{Id(M%0
-+
M'
-
M
is
exac }
.
Following
[11],
gi en
n E
PNl,
we
will
deno e
by
G,,
he
class
o
le
R-modules
M,
wi h
Sid(M)
<_
n
.
We
de ine
Sid(M)
=
oo
when
o
all
n E
IN,
he e
ex-
is s
a
submodule
M'
C_
M
such ha
Id(M)
>_
n
(wi h
he
con en ion
n<
oo)
.
Obse e
ha
i
he e
exis
M'
C_
M
wi h
Id(M)=
oo,
hen
Sid(M)
=
oo
.
We
ema k
[11]
ha
i
R
is
a
le
noe he ian
ing
hen
he
classes
L
n
(n
=
0,
1
. . . .
)
a e
o sion
classes
and
[11]
i
RR
is
he
ing
R
aken
as
le
R-module
he
Sid(RR)
=
lgl
dim(R)
.
No e
ha
he e
is
a
chain
o
o sion
classes
Co
C_
L
1
C_
. . .
C_
L
n
C
.
.
.
We
add
G,
=
{O}
and
,C,,,
=
R-mod
.
Le
un
be
he
o sion
heo y
co esponding
o
En
.
No e
ha
M
E
R-mod
is
a,,- o sion ee
i
and
only
i
o
all
submodules 0
=~
M'
C_
M
he e
exis
a
submodule
N
C
M'
such
ha
Id(N)
>n
.
Wi hin
he
abo e
chain he e
exis s
a
s ic ly
inc easing
subchain
;
ha
is, i
no
=
-oo,
hen
Lno
C
...
C
Cni
C
...
whe e
he
leng h o
he
subchain
is
a
mos
w
.
1
.1
Examples
.
(i)
In
7L
we
ha e
,C_,,,
=
L
o
C
:,Cl
=
7L-mod
.
u

Le
K
be
a
ield
.
Fo
R
=

K

K[X,
Y]

we
ha e
G_
.
C
L
o
C
(
.
.)

0

K[X,
Y]
,C
l
C
£2
=
R-mod
.
(iii)
Le
R
be
a commu a i e
noe he ian
egula
local
ing,
wi h
J
maximal
ideal
.
Suppose
ha
Id(R/J)
=
n,
hen
we
ha e
in
his
case
,C_
.=
Co
=
. . .
=,c,1
C
Cn
=
R-mod
.
(i )
Le
R
be
a
le
a inian
le
local
ing
[2]
.
Then
R-mod
has
a
chain
as in
(iii)
.
( )
In
[3],
he e
a e
examples
whe e
all
he
inclusions
in
he
chain
a e
p ope
.
( i)
In
any
le
a inian
ing
which
has
a
leas
wo
simple
le
R-
modules
wi h
di e en
injec i e
dimensions
(as
Z12)
he
Gab iel
il a ion
has
less
e ms
han
he
subchain
.
1
.2
Lemma
.
Le
R
be
a
le
noe he ian
ing
and
L
n
,
(j
=
-oo,
0,
1,
. .
.)
a
e m
in
he
subehain such
ha
Lni
=,A
R-mod
.
Then
he e
exis s
a
Qnj
-coc i ical
le
R-module
M,
such
ha
n,
<
Id(M)
=
Id(M1)
o
all
submodules
0
=~
M'
C
M
.
GLOBAL
DIMCNSION

19
1
P oo
..
Since
R
is
noe he ian
and
G,,,
zA
R-mod
hen
he e
exis
a
Q
ni
-coc i ical
le
R-module
M
.
(a)
I
Id(M)
>
nj
(including
oo)
hen
in
each
exac
sequence
0 ->
M'
-+
M
-
M"
-
0
we
ha e
Id(M")
<
nj
and
Id(M)
> n
;,
and
so
Id(M')
=
Id(M)
;
hence
M
is
he
equi ed
objec
.
(b)
I
Id(M)
<
n
j
hen
since
Sid(M) > n
j
hen
he e
exis s a
submodule
0
=,~
M'
C
M
such
ha
Id(M')
>n
j
(including
oo)
and
since
M'
is
also
a
n
;-coc i ical
we
a e
again
in
case
(a),
and
M'
is
now
he
equi ed
objec
.
1 .3
P oposi ion
.
Le
R
be a
le
noe he ian
ing,
G
nu
(j
=
0,
1,
...
)
-n,
:,A
R-mod
and
m
=
min{Id(C)
1C
is
cyclic
Qnj
-coc i ical
wi h
Id(C)
>
nj}
.
Then
m
=
nj+l
.
P oo
..
By
hypo hesis
and
Lemma
1
.2
i is
clea
ha
G

,,
always
exis s
and
ha
G
nu
CG

, .
Suppose
ha
he e
exis s
k
E
N1
such
ha
Cni
C
Gk
C_
G

,, .
Since
R
is
noe he ian
[2]
he e
exis s a
Q
nj
-coc i ical
Qk
-
o sion
le
R-module
M,
and
hence
nj
<
Sid(M)
<_
k
.
So,
by
he
pa
(a)
in
he
p oo
o
Lemma
1 .2,
he e
exis s
a
cyclic
submodule
C
C_
M
such
ha
Id(C)
=
Sid(M)
.
Since
C
is
also
Qn
j
-coc i ical
and
Id(C)
>n
;
.
Then,
by
he
de ini ion o
m
we
mus
ha e
Id(C)
>
m
.
Hence
k
>_
m
and
hus
k
=
m
.
No e
ha , in pa icula ,
i
m
=
min{Id(S)JS
E
R-mod
is
simple}
hen
m=nl
.
1 .4
Obse a ion
.
Since e e y
subchain
has
a
leas
wo
e ms,
i
is
na u al
o
analize
he
s ep
Qnj
<
Qnj+,
.
In
each
o
he e s eps
he e
exis s
a
Qn
j -
coc i ical
cyclic
le
R-module
C, such
ha
Id(C)
=
Id(C')
_
Sid(C)
>
nj+i
o
all
submodules
0
:,A
C
C
C
.
F om
he e,
Theo em
C
o
B
.
Oso sky
in
[5]
ollows
immedia ely
.
In
he
nex
heo em,
we
will
see
ha
we
can
o
ex ac
a
nice
subclass
o
he
class
o cyclic
le
R-modules,
o
compu e
he
le
global
di iension
.
1
.5
Theo em
.
Le
R
be a
le
noe he ian
ing,
such
ha
G
dim(R)
=
Then
lgl
dim(R)
=
sup{Id(C)
¡C
is
cyclic,
Id(C)
=
Id(C'),
o
all
0
jA
C'
C
C
and
- ,,-coc i ical,
wi h
<
/3}
.
P oo
.
Le
_1
<
,,
<
. . .
<
-
p
be
he
Gab iel
il a ion
in
R- o s
and
le lgl
dim(R)
=
nk
(o
oo)
.
Fo any
gi en
j
<
k
we
ha e
a
s ep
Qn
;
<
Qn
;
+1
and by
[2],
he e
exis s
an
o dinal
a
<_
~3
which
is
leas
wi h
he
p ope y
ha
,,
:9
Qn
i
.
No e
ha
a
is
a
successo
.
Then
by
Obse a ion
1
.4
and he
ac
ha
,,,
:9
a
no
he e
exis s
a
T,,_
I
-coc i ical
Q
ni
-coc i ical
le
R-module
C, such
ha
Id(C)
=
Id(C')
>
nj+i
o
all
192

J
.J
.
SIMÓN
PINERO
submodules 0
:~
C'
C_
C
.
Se ing
=
a
-
1
we
ha e
he
esul
in
iew
ha
he choice
o
j
was
a bi a y
.
The
nex
co olla y
is
o
pa icula
impo an e
inasmuch
as
he e
exis
a a
abundan e
o
examples
whe e
he
subchain
is
ini e
.
1
.6
Co olla y
.
Le ,
R
be
a
le
.noe he ian
ing
such
ha
Gdim(R)
=
,Q
.
Suppose
ha
ue
ha e
ini ely
many
e ms
in
he
subchain
.
Then
lgl
dim(R)
=
sup{Id(C)
¡C
is
cyclic,
Id(C)
=
Id(C')
o
all
0
=,4
C'
C_
C
and
,,-coc i ical
u)he e
p
<
Q
is
ixed}
.
P oo
.
.
Conside
he
las
s ep
in
he subchain,
Qnj_,
<
Qnj
=
X
.
Then
he e
exis s
a<
,3
such
ha
T
C
,
:1
unj_l
.
He e,
p
=
a
-
1
.
1
.7
Obse a ion
.
Le
R
be
a
commu a i e
noe he ian
ing
.
We
ake
in
his
case
he
o iginal
de ini ion
o
K ull
dimension
o e
he
p ime
ideals
o
R
.
Suppose
ha
now,
o
all
S
E
R-niod
simple
we
ha e
ha
S
E
G

,
o
some
n
E
IN
( ixed)
.
Le
Jbeany
maximal
ideal
o
R, hen
R/J
is
a
simple
le
R-module
and
Supp(R/J)
=
J
;
u he mo e,
R/J
E
G
n
.
Then, by
[11]
he
local
ing
Ri
is
egula
wi li
K
dim(Rj)
<_
n
.
By
[13]
we
lla e Sida,,
(RJ)
=
lgl
dim(Ri) <
n,
o
all
J
.
B,y
[13]
we
ha e
lgl
dim(R)
<
n,
lience
R
E

Tha
is,
we
ha e
jus
p o ed
ha
o
any
commu a i e
noe he ian
ing,
i
u,
j
:~
X
hen
To
:1
u
n
,
and by
Theo em
1 .5
we
ha e
he
well-known
esul
:
lgl
dim(R)
=
sup{Id(S)
¡S
is
simple
R-module
}
.
2
.
P ojec i e
dimension
When
we
compu e
he
le
global
dimension
as
he
sup emum
o
he
p ojec i e
dimensions
o
coc i ical
and
c i ical
le
R-modules
wc
lla e
analogous
esul s o
he
abo e
;
u he mo e,
we
can
elax
he
noe he ian
condi ion
o a
he
ing
R
.
The
S ong
P ojec i e
Dimension
[11]
is
de ined
as
Spd(M)
=
sup{Pd(N)
¡M
->
N
->
0
is
exac }
.
We
lla e
o
all
n
E
N,
he
classes
U,,
=
{M
E
R-mod
ISpd(M)
_<
n}
.
In
his
case
[11]
U
n
a e
o sion
classes
o
a a
a bi a y
ing
R,
and
lgl
dim(R)
=
Spd(RR)
.
We
deno e by
pn
he
o sion
heo y
co esponding
o
U

,
.
Again,
we
ha e
a
chai a
Uo
C_
U
C_
. . .
C_
U
n
C_
.
. .
and
adding
U_
.
a ad
U,,,
we
can
ake
a
s ic ly
ascending
subchain
Uno
C
. . .
C
Uni
C
. . .
wi h
no
=
-oo
.
F om
he e,
we
can
do
he
duali a ion
in a simila
way
o
he
i s
pa
and
we
can
emo e
he
noe he ian
condi ion
.
So,
we
will
w i e
o lly
he
p incipal
esul
.
Fo
p,,-coc i ical
le
R-modules
C, such
ha
Pd(C)
>_
nj+ ,
he
consequence
Pd(C)
=
Pd(C')
o
all
submodules
0
7~
C
C
C
will
be emo ed
in
iew
ha
all
p,,-coc i ical
sa is y
i
.
GLOBAL
DIMENSION

19
3
2
.1
Theo em
.
Le ,
R
be
a,
ing
wi h
Gab iel,
dimension,
suppose
ha
G
dim(R)
=
0
.
Then
lgl
dim(R)
=
sup{Pd(C)1C
is
cyclic
and
T,,-coc i ical,
wi h
h<
,(3}
.
2
.2
Examples
.
(i)
In
a
non-noe he ian
ing
R,
wi h
Gab iel
di nen-
sion,
he
classes
G
n
,
a e no
in
gene al
o sion
classes,
bu
hey
a e
:
Se e
subca ego ies
(see
[5])
.
E en
i
subchains can
be
ound, he
esul s
ha
we
ha e
seen
do
no
hold
.
Fo
example,
le
S
be
(Z2)
Nand
R
C
S
he
sub ing gene a ed
by
(7Z2) '>
oge he
wi h
1
E
S
.
Then
R
is
a
commu a i e
boolean
se nia inian
he edi a y
V- ing,
ha ing
as
chai a
G_,,
C
LO
C
G1
=
R-mod
.
No e
ha
he
o sion
class
gene a ed
by C
o
is
he
same
ha
G
.
(ii)
Le
R
be
a ing
wi h
Gab iel
dimension
and
nonsingula
as
le
R-module
(Z(j?R)
=
0)
.
Then
i
R
is
no
se ni
simple
we
ha e
lgl
dim(R)
=
sup{Pd(C)
¡C
is
cyclic
singula }
.
P oo
.
We
shall
p o e
ha
e e y
C
o
Theo em
2
.1
adini s
ano he
singula
module
D
such ha
Pd(D)
>_
Pd(C)
.
By
[4],
in
e e y
non-
singula
ing, cyclic
uni 'o m
nodules
a e
ei hc-
"
singula
o
nonsingula
.
Since
i is
clea
when
C
is
singula ,
we
assu ne ha
C
is
nonsingula
.
So ake
a
le
ideal
I
o
R
such
ha
R/I
=
C
.
Because
I
is
no
la ge
in
R,
he e
is
a
le
ideal
0
~
J
o
R
wi h
I
(D
J
la go
in
R
.
By
aking
D
=
R/I
®
.I
we
ha e
Pd(D)
=
1
+
Pd(I
®
J)
>_
1
+
Pd(I)
_>
Pd(C)
and
R/I
is
singula
.
(i )
I
R
is
le
se nia inian,
lgl
dim(R)
=
sup{Pd(S)JS
is
simple}
(see-,
"
[8])
.
Semi
pe 'ec
ings a e
:
se nia inian,
o
ins ance
.
( )
Finally
we
e e
o
i kjec i e
and
p ojec i e
dimension
in
le
Fully
Bounded
Noe he ian
(FBN)
ings
wi hou
any
o he
assu ip ion
(like
he
commonly
used
igh
cohe ence
o
[12],
[15])
.
In
his
ings,
he
( wo
sided)
p ime
ideals
o en
ha e
no
hand
desc ip io is
a,nd
we
can
see-
:
lnow
ou
classes
wo k
.
Le
R
be
a
le
FBN
ing
and
ake
G

; z,~
R-mod
(U,
L
,
=,
A
R-mod)
.
By
he
esul s
abo e,
he e
exis s
a
cyclic
o,
Lj
-coc i ical
(P,,
;-coc i ical)
le
R-module
C
.
Take
[14]
he associa ed
p ime
ideal
ass(C)
and
no e
ha
by
[14]
i
x
E
C
is
such
ha
R
-
x
=C
hen
ass(C)
C_
l(x)
( he
le
annihila o
o x)
and
hence,
R/ass(C)
1
G,,,
(R/ass(C)
0
U,,,,)
and
by
[14]
we
ha e
R/ass(C)
is
o,
EJ
- o sion ' ee
(P,,,- o sio i ee)
.
Since
R
is
le
FBN
he i,
hc
;
injec i e
hulls
E(R/ass(C))
=
E(C)
a nd
hence
he e
exis s a
copy
o
C,
say
C
again,
C
C_
E(R/ass(C))
.
Take
K
=
C
n
R/ass(C)
a ad
no e ha
lince
K
C
C
hen
K
has
he p ope ies

19
4

J
.J
.
SIMÓN
PINCRO
o
modules
in
Theo ems
1 .5
and
2
.2
.
Since
R/ass(C)
is
a
le
o de
in
a
simple
ing
[7],
[14]
hen
o
any x
E
K
we
ha e
(R/ass(C))
-
x
R/ass(C)
as
le
R/ass(C)-modules
and
hence
as
le
R-modules
.
This
implies ha
Id(R/ass(C))
=
Id(C)
(Pd(R/ass(C)
=
Pd(C))
.
So
we
ha e
ha
i
R
is
a
le
FBN
ing
hen
lgldim(R)=sup{Id(R/I)II
E
Spec(R)}=sup{Pd(R/I)II
E
Spec(R)}
.
a
No e
.
The
au ho
hanks
he
e e ee
o
epo ing
him
abou
he
exis ence
o
[6]
.
Re e ences
1
.

M
.F
.
ATIYAII
AND
I
.G
.
MACDONALD,
"
In oduc ion
o
commu a-
i e
Ageb a,"
Addison-Wesley,
Reading,
Mass
.,
1969
.
2
.

J
.S
.
GOLAN,
"To sion
heo ies,"
Longman
Scien i ic
&
Technical
-
John
Wiley
&
Sons,
1986
.
3
.

J
.S
.
GOZAN
AND
Z
.
PAPP,
Coc i ically
hice ings
and
Boyle's con-
jec u e,
Comm
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Algeb a
8(18)
(1980),
1775-1798
.
4
.

K
.R
.
GOODFARL,
"Ring
Theo y,"
Ma cel
Dekke ,
New
Yo k,
1976
.
5
.

R
.
GORDON
AND
J
.C
.
ROBSON,
K ull
dimension,
Mem
.
Ame
.
Ma h
.
Soc
.
133
(1973)
.
6
.

J
.
J
.
KOKBR,
Homological
dimension
o ings
wi h
K ull
and
Gab iel
dimension,
Ph
.D
.
Thesis,
Uni e si y
o
Wisconsin-Milwaukee, 1990
.
7
.

J
.C
.
MC
CONNELL
AND
J
.C
.
ROBSON,
"Noncommu a i e
noe he-
ia,n
ings,"
John Wiley
&
Sons,
Chiclies e
New
Yo k
B isbane
To on o
Singapo e,
1987
.
8
.

C
.
NÁSTÁSGSCU,
Dimension
globale
des
anneaux
semi-a iniens,
C
.R
.
Acad
.
Sci
.
Pa is,
Sé
.
A-B
268
(1969),
A685-A688
.
9
.

B
.L
.
OSOFSKY,
Global
dimension
o
alua ion
ings,
T uns
.
Ame
.
Ma h
.
Soc
.
12
7
(1967),
136-149
.
10
.

I
.
PALMCR
AND
J
.-E
.
ROOS,
Fo mules
explici es
pou
la
dimension
homologique
des
anneaux
de
ma ices
gene alisées,
C
.R
.
Acad
.
Sci
.
Pa is,
Sé
.
A-B
273
(1971),
A1026-A1029
.
11
.

Z
.
PAPP,
On
he s ong
injec i e
(p ojec i e)
dimension
o
modules,
A ch
.
Ma h
.
25
(1974),
354-360
.
12
.

J
.
RAINWATER,
Global
dimension
o
ully
bounded
noe he ian
ings,
Comm
.
Algeb a
15(10)
(1987),
2146-2156
.
13
.
J
.
ROTMAN,
"An
in oduc ion
o
Homological
Algeb a," O lando,
Academic
P ess,
1979
.
GLOBAL
DIMENSION

19
5
14
.

BO
STENSTRÓM,
"Rings o
quo ien s,"
Sp inge -Ve lag,
Be lin,
Hei-
delbe g,
New
Yo k,
1975
.
15
.

M
.
TEPLY,
Global
dimensions
o
igh
cohe en
ings
wi h
le
K ull
dimension, Ball
.
Aus al
.
Ma h
.
Soc
.
39,
no
2
(1989),
215-223
.
Depa amen o
de
Ma emá icas
Uni e sidad
de
Mu cia
30071
Mu cia
SPAIN
P ime a
ue sió
ebulla
el
8
de
Juliol
de
1991,
da e a
ue sió
ebl da
el
15
d'Oc ub e
de
1991