Pub
.
Ma
.
UAB
Vol
.
30
Ng
1
Maiq
1986
A
GENERALIZATIONOF WRIGHT'S
INEQUALITY
J-L
.
Ga cíaRoig
Le
R
be
a
commu a i e ingwi h
iden i y,
E
an
R-module
and
x
1
,
. . .
,x
ER
a
mul iplici y
sys em
on
E
(
see
[1],
p
.295
) .
Then
he
leng h
R(E/(xn1~
. .
.,xn )E) is
ini e
o
any
posi i e
in ege s
n
1
,
.
.
.,n
,
and
W igh 'sinequali y
(
see
[1]
p
.296
)
says
E
/(x
n

n
R(

1,
.
.
.,x )E)

`

n1
. .
.n
. R(
E
/
(x
1
>
. . . .
x- )E),
o
a bi a yn
1
,
.
.
.,n
.
This
inequali y
can
be
w i en
as
RH
0
K(x
1
1
,
.
.
.,x
¡E) 5
n
1
. .
.n
"
RH
0
K(x
1
, .
.
.,x
JE),
whe e

K(x
11
,.
. . .
x
¡E)
deno es
he
Koszul
.
complex
de ined
by
E
and
he
elemen s
x11,
.
.
.,x
.
In
his
pape
we
es ablish
ha
o
a
Noe he ian
modulesimila
inequali ies
hold
o
he
highe
Koszul
homo-
logy
modules,
i
.e
.,
o
i>,O,
H
i
K(x
11
,
. .
.,x
JE)
.~
n1
. .
.n * HiK(xl,
. . . .
x
¡E)
.
Mo eo e ,
he
same is ue
.
o he
highe Eule -Poinca écha ac e is ics
o
he
Koszulcomplexes
K(x
1
1
,
.
.
. .
x
¡E),

i
.e
.,

o
1i0,

we
ha e

xi(x11,
.
. .
.
x ¡E)
n1
.
.
.n *xi(x1,
. . . .
x
JE),
whe e,
by
de ini ion,
xi
(
x
19
.
.
.,x
¡E) =
B
.(-1)
j-1
LHjK(x1,
. .
.,x ¡E)
.
j
>,i
Ac ually
he
inequali y
o
xi
is an
equali y
i
i=0
(see
[1]
p
.311
) .
We also
p o e
ha
he
unc ions

R
H
i
K(x
1
1
, .
.
.,x
JE)
and

X
i(x11,
.
.
.,x ¡E)

inc ease
wi h
he
exponen s

n
l
,
. . . .
n
.
In
wha ollows
R
deno es
a
commu a i e ingwi h
iden i y,
E
is
a
Noe he ian
module
o e
R
and
x
1
, . .
.,x
ER
is
a
sys em
o
múl iplici y
on
E
.
This will
ensu e
ha
all
he
leng hs
which
appea
a e
indeed
ini e,
hough
some
o
he
esúl s
would
also hold
wi hou
assuming
he
leng hs
o be
ini e
.
We
deno e
he
leng h
by
R
o
R,
R
.
Lemma
1
.
Le
E
be
an
R-module
and
x
1
,
. . .
.
x
,y
be elemen s
o
R
.
Then,
o any
i>,O,
p
.
IV-2

)
~
RH
i
K(x
1
,
. .
.,x
JE)

:

R
R
H
i
K(x
l
y,x
2
9
.
.
. .
x
JE)
.
P oo
.
The
inequali y
ollows
om
he
exac
sequence
(c
.
[2]
H
i
K(x
2
,
. .
.,x
¡
E)
0-~

Hi
K(x
1,x
29
. .
.,x
¡E)
--~
x
1
H
i
K(x
2
"
.
.,x
¡E)
--
. (0
:x1)

-
H
i-l
K(x
29
. .
.,x
¡E)
and he
co esponding
one o
H
i
K(x
1
y,x
2
,
.
.
.,x
¡E),
by
obse -
ing
ha
bo h

(O :x
)

c
(O
:x
y)
1
Hi-1K(x23
. .
.yx ¡E)

1

H
i-1
K(x
2
,"
,x
JE)
and

x
1
H
i
K(x
21
.
.
. y
x
JE)
;?x
l
yH
i
K(x
2
, . . .
.x
1E)
.
Bea ing
in
mind
ha
he
Koszul
homology
modules
do
no
depend
on
he
o de
o
he
elemen s
de ining
i ,
w
.e
ge
he
ollowing
92
P oposi ion
2
.
Fo any
i30,
he
mapping
om
QV
o
U
de ined
by
(ni
,
.
.
.,n )~---!LHiK(x11,
. .
.,x ¡E)
is
inc easing
,
i
.e
.,
n
1
I<m
l
,..
.,n
.<m
imply
R,H
i
K(x
11
,
.
.
.,x
~E)
:
LHiK(x11,
. .
.,x JE)
.
Lemma
3
.
I aER,
hen
we
ha e
:
i)
a(O
:a
n
)
<
n-L(O
:a),
and
E

E
ii)

a(
E
l
anE
)
6n
.9,(E/aE)
.
P oo
.
By induc ionon
n
.
F om
he
exac
sequence
n-1
R/aR
.a

R
/anR
-
.
R
/
a
n-1
R
-o
0
i
we
apply
Hom
R
(
,E),
we
ge
i),
o
Hom
R
(
R/
anR
,E)=0
:a
n
,
E
and
i we
apply
.&E,
we
ge
ii),
o
(
R
/
anR
)RE'z
E
/a
nE
.
#
P oposi ion
4
.
The
ollowing
inequali y
holds
o
all
i10,
RH
i
K(xi,x
2
,
.
.
.,x
JE)
<n- HiK(x1,x2,
. .
.,x ¡E)
.
P oo
.
F om
he
exac sequences
(see
[2j
p
.IV-2
)
0
-!H
0
K(ajH
i
K(x
2
,
. . . y
x
JE))-1-H
i
K(a,x
2
, . . . y
x
¡E)
wi h
a=x
1
o
x1
,
we
ge
-
i
H
1
K(ajH
i-l
K(x
2y
. . . y
x
¡E))--
0,
n

HiK(x2%
. .
.,x ¡E)
g,H
i
K(x
1 ,x2
, .
.
.,x
JE)=«
n
)+k(O
:xi)
x1HiK(x2,
.
.
.,x JE)

H
i-1
(x2
,"
,x
¡E)
HiK(x2*
.
.
.
.
x JE)
x
1
H
i
K(x
2
,
.
.
.
y
x
E)

i_l

2

n
.
H
i
K(x
1,x 2
, . . .
.x
/E),
he
inequali ies
being
by i ue
o
lemma
3
.
#
Again
by
he
independence
o
he
Koszul
homology
moduleswi h espec
o
he
o de
o
he
elemen s,
P oposi ion
4
yields
he
ollowing heo em
which
gene alizesW igh 's
inequali y
.
Theo em
5
.
Fo any
i,>O,
andany
n1,
. .
.,n ,O,
we
ha e
94
2H
i
K(x
11
,
. .
.,x
¡E) <
n1
. .
.n -RHiK(x1,
. .
.,x ¡E)
.
#
Le
>>s
conside
now he
highe
Eule -Poinca é
cha ac
e is ics
.
Obse e i s
ha
xo
(x
11
,
.
.
.,x
~E) _
n
1
-n
-x
0(x
1
, . .
.,x
IE)
and
ha
x
0
(x1
,
.
.
.,x JE)%0
(c
.
[1]
p
.311
) .
Fo he
highe
cha ac e is ics
we
ha e
P oposi ion
6
.
Fo
all
i,0,
he
mapping
(n1,
. .
.,n )
I--0
Xi (x11
,
. .
.,x
JE)
om
IN
o
IN
is
inc easing,
i
.e .,
n1<m1,
.
.
.,n <m ,
imply
X i
(x11
,
..
.,x
JE),<
Xi(x11,
. .
.,x ¡E)
.
P oo
.
By
[2]
p
.IV-56,
we
ha e
xi(a,x2*
. . .
.
x
JE)

=

RH1K(ajHi_lK(x21
. .
.,x ¡E))+
+
xo(aIxi(x29
. .
.,x ¡E))
.
Se inga
=
x
1
o
x
1
y
and
using
hemul iplica i i y
o£
(
see
[1]
p
.309
Thm
.7
),
we
deduce
xi (x1 ,x
21
. .
.,x
JE) :
xi(xly,x2,
.
.
.9x ¡E)
.
F om
his
we ge ,
o
n n,
ha
xi(x1,x2,
. . . .
x
¡E)

-<
xi(x1,x2,
. .
.,x ¡E)
.
P oceeding
equally
wi h
he
o he a iables
(
x
i
does
no
depend
on
he
o de
o
he
elemen s),
we
ge he
esul
.
#
We inish
wi h
a heo em
on
highe
Eule -Poinca é
cha ac e is ics
simila
o
heo em
5
.
Theo em
7
.
Fo
any
i30,
andany
ni,
. .
.,n >
.0,
we
ha e
xi(x
1
1
,. .
.,x
¡E)

:n
1
-
n
"
xi(x1
...
.,x ¡E)
.
P oo
.
I is
enough
o
p o e
n
xi(x1,x2%
. . . .
x
¡E)

,
n
"
xi(x1,x2,
. . . .
x ¡E),
and
his
can
be
done
by
conside ing
he
o mulaused
in
he
p oo
o
he
p eceding
p oposi ion
wi h
a=x
1
o
x1
.
We
ge
x
i
(x1
x2,
..
.,x /E)
_
-_
R(0
:x
n
)

+
x(xn1xi(x2,
. .
.,x ¡E))
Hi-lK(x2,
. .
.0x /E)
o
n
R(o
:x
i)

+
n-x(x
l lxi(x2
,
. .
.,
x
JE))
_
H1-1K(x2,
.
.
.,x ¡E)
0
o

=
n-xi(x1,x2,
.
.
. .
x ¡E),
he
inequali ybeingjus i iedby lemma
3
.
#
Re e ences
[1]
No hco ,
D
.G
.
Lessons
on
Rings,
Modules
and
Mul ipli
-
ci ies
,
Camb idge
U
.P
.(1968)
.
[2]
Se e,
J-P
.
Algéb e
locale
:
Mul iplici és
,
Lec u e
No es
in
Ma h
.
No
.11,
Sp inge ,
Be lin(1975)
.
Rebu
el 14 de
no embne
del
1985
C uzRoja,
26 3s 4a
.
Hospi ale
Ba celona
ESPANYA
9
6