On the module of effective relations of a standard algebra
Full text
On he mo dule o eec i e ela ions o a s anda d algeb a
By F ancesc Planas-Vilano a
Dep . Ma ema ica Aplicada I. ETSEIB-UPC. Diagonal 647, E-08028 Ba celona
.
1 In o duc ion
Le
A
be a commu a i e ing. We deno e by a s anda d
A
-algeb a a commu a i e g aded
A
-algeb a
U
=
n
0
U
n
wi h
U
0
=
A
and such ha
U
is gene a ed as an
A
-algeb a by
he elemen s o
U
1
. Take
x
ase o (p ossibly inni e) gene a o s o he
A
-mo dule
U
1
. Le
V
=
A
] b e he p olynomial ing wi h as many a iables
(o deg ee one) as
x
has elemen s
and le
:
V
!
U
be he g aded ee p esen a ion o
U
induced by he
x
. Fo
n
2,
we will call
module o eec i e
n
-
ela ions
he
A
-mo dule
E
(
U
)
n
=ke
n
=V
1
ke
n
;
1
.The
minimum p osi i ein ege
1such ha he eec i e
n
- ela ions a e ze o o all
n
+1
is known o be an in a ian o
U
.I is called he ela ion yp e o
U
and is deno ed by
(
U
). Fo an ideal
I
o
A
, we dene
E
(
I
)
n
=
E
(
R
(
I
))
n
and (
I
) = (
R
(
I
)), whe e
R
(
I
)=
n
0
I
n
n
A
] is he Rees algeb a o
I
.
In his pap e , wegi e wo desc ip ions o he
A
-mo dule o eec i e
n
- ela ions. In e ms
o And e-Quillen homology weha e ha
E
(
U
)
n
=
H
1
(
A U A
)
n
(see 2.3). I u ns ou ha
his mo dule do es no dep end on he chosen
x
. In e ms o Koszul homology we p o e ha
E
(
U
)
n
=
H
1
(
x
U
)
n
(see 2.4). Using hese cha ac e iza ions, weshow la e some p op e ies
on he mo dule o eec i e
n
- ela ions and he ela ion yp e o a g aded algeb a. Meanwhile,
ou line o disquisi ion app oaches us o se e al ea lie wo ks on he sub jec (see 2], 5 ], 6],
7], 9], 10 ], 13] and 14]).
Sec ion 2 is de o ed o s a e he ab o e men ioned (co)homological cha ac e iza ions o
he
A
-mo dule o eec i e
n
- ela ions and compa e hem wi h some al eady known esul s.
In sec ion 3, we gi e some applica ions. The in e es is sp ecially cen e ed on he mo dule
o
n
- ela ions o p owe s o an ideal and he mo dule o
n
- ela ions o Ve onese sub ings. In
pa icula , one concludes ha (
U
(
p
)
)
(
U
p
+
) bu , in gene al, (
U
(
p
)
)
6
= (
U
p
+
), whe e
U
+
=
n>
0
U
n
is he i ele an ideal o
U
and
U
(
p
)
=
n
0
U
np
is he
p
- h Ve onese sub ing
o
U
(see 3.12). Finally, in sec ion 4 we cha ac e ize, in e ms o a sys em o gene a o s,
which ideals ha e mo dule o eec i e
n
- ela ions ze o. In pa icula , a new cha ac e iza ion
o sequences o linea yp e is ob ained.
e-mail: [email p o ec ed]c.es
1
2 Homological desc ip ion o eec i e ela ions
Le
U
=
n
0
U
n
be a s anda d
A
-algeb a. Pu
U
+
=
n>
0
U
n
i s i ele an ideal. I
E
=
n
1
E
n
is a g aded
U
-mo dule and
1, we deno e by
F
(
E
) he submo dule o
E
gene a ed by he elemen s o deg ee a mos
. Pu (p ossibly inni e)
s(
E
) = min
1
j
E
n
= 0 o all
n
+1
g
:
Since (
E=U
+
E
)
n
=
E
n
=U
1
E
n
;
1
, hen he ollowing h ee condi ions a e equi alen :
F
(
E
)=
E
,s(
E=U
+
E
)
, and,
E
n
=
U
1
E
n
;
1
o all
n
+1.
Gi en
h
:
W
!
U
, a su jec i e g aded mo phism o s anda d
A
-algeb as, we a e in e -
es ed in he g aded
A
-mo dule
E
(
h
)=ke
h=W
+
ke
h
. The ollowing is an elemen a y, bu
use ul lemma:
Lemma 2.1
Le
:
V
!
U
and
g
:
W
!
V
be wo su jec i e g aded mo phisms o s anda d
A
-algeb as. Then, he e exis s a g adedexac sequenceo
A
-modules:
E
(
g
)
!
E
(
g
)
g
!
E
(
)
!
0
:
(1)
In pa icula ,
s(
E
(
))
s(
E
(
g
))
max(s(
E
(
))
s(
E
(
g
)))
. Mo eo e , i
V
and
W
a e
wo symme ic algeb as, hen
E
(
g
)
n
=0
and
E
(
g
)
n
=
E
(
)
n
o al l
n
2
.
P oo
. Exac sequence (1) ollows om he snake lemma applied o he commu a i e dia-
g am:
W
1
ke
g
n
;
1
W
1
ke (
g
)
n
;
1
W
1
ke
n
;
1
0
- - -
- - - -
0
ke
g
n
ke (
g
)
n
ke
n
0
???
1
g
n
;
1
g
n
Mo eo e , i
W
=
S
(
W
1
) and
V
=
S
(
V
1
), hen ke
g
=
F
1
(ke
g
).
Deni ion 2.2
Le
U
be a s anda d
A
-algeb a and le
:
S
(
U
1
)
!
U
be he g aded
mo phism o s anda d
A
-algeb as induced by he iden i yon
U
1
.Gi en
n
2, he
module
o eec i e
n
-
ela ions
o
U
is dened o be
E
(
U
)
n
= ke
n
=U
1
ke
n
;
1
. Pu
E
(
U
) =
n
2
E
(
U
)
n
=ke
=
S
+
(
U
1
)
ke
. Then, he
ela ion ype
o
U
is dened o be (
U
) =
s(
E
(
U
)). Rema k ha i
h
:
W
!
U
is any
symme ic p esen a ion o
U
, ha is,
W
is
a symme ic algeb a and
h
is a su jec i e g aded mo phism o s anda d
A
-algeb as, hen
h
can be ac o ized in o
h
=
g
, whe e
g
:
S
(
W
1
)
!
S
(
U
1
) is he induced mo phism
by
h
1
:
W
1
!
U
1
and
=
. Thus, applying Lemma 2.1,
E
(
U
)
n
=
E
(
h
)
n
o all
n
2
and s(
E
(
U
)) = s(
E
(
h
)). I
I
is an ideal o
A
, he
module o eec i e
n
-
ela ions
o
I
is
E
(
I
)
n
=
E
(
R
(
I
))
n
and he
ela ion ype
o
I
is (
I
)= (
R
(
I
)), whe e
R
(
I
)=
n
0
I
n
n
is he Rees algeb a o
I
. An ideal wi h mo dule o eec i e 2- ela ions ze o is called
syzyge ic
.
An ideal o ela ion yp e 1 is called o
linea ype
(see, e.g., 8]).
2
Rema k 2.3
In ac , sequence (1) is pa o along exac sequence o And e-Quillen ho-
mology. Indeed, he Jacobi-Za iski sequence asso cia ed o he mo phisms
g
:
W
!
V
and
:
V
!
U
, wi h esp ec o he
U
-mo dule
A
=
U=U
+
, gi es ise o
:::
!
H
1
(
WVA
)
!
H
1
(
WUA
)
!
H
1
(
V U A
)
!
H
0
(
WV A
)
!
::: :
Using
H
1
(
A A=I M
) =
I=I
2
M
and
H
0
(
A A=I M
) = 0 o any ideal
I
o
A
and any
A=I
-mo dule
M
,we ge (1) (see 1]).
On he o he hand, he Jacobi-Za iski sequence asso cia ed o he mo phisms
A
!
S
(
U
1
)
and
:
S
(
U
1
)
!
U
, wi h esp ec o he
U
-mo dule
A
,is
:::
!
H
1
(
A
S
(
U
1
)
A
)
!
H
1
(
A U A
)
!
H
1
(
S
(
U
1
)
UA
)
!
H
0
(
A
S
(
U
1
)
A
)
!
::: :
Using
H
1
(
A
S
(
U
1
)
A
) = 0 and
H
0
(
A
S
(
U
1
)
A
) =
H
0
(
A U A
), we ge he g aded iso-
mo phism o
A
-mo dules
H
1
(
A U A
) =
H
1
(
S
(
U
1
)
UA
) = ke
=
S
+
(
U
1
)
ke
. Thus,
H
1
(
A U A
)
n
=
E
(
U
)
n
is he mo dule o eec i e
n
- ela ions o
U
. In pa icula , (
U
) =
s(
H
1
(
A U A
)).
The e is also a desc ip ion o he mo dule o eec i e
n
- ela ions in e ms o Koszul
(co)homology. Le
:
V
!
U
be a su jec i e g aded mo phism o s anda d
A
-algeb as.
Fo each
p
1, conside he map
V
p
U
!
U
sending
x
y
o
p
(
x
)
y
and le
K
(
p
)
be he Koszul complex asso cia ed o his
U
-linea o m (see 1.6.1 o 3]). Since i is an
homogeneous o m o deg ee ze o,
K
(
p
)is a complex o g aded
U
-mo dules ha ing die -
en ials homogeneous mo phisms o deg ee ze o. Conc e ely,
K
(
p
)=
n
0
K
(
p
)
n
whe e
K
(
p
)
n
is he ollowing sub complex (
U
n
=0 o
n<
0):
:::
;!
A
2
(
V
p
)
A
U
n
;
2
p
@
2
;!
V
p
A
U
n
;
p
@
1
;!
U
n
;!
0
whe e
@
q
((
x
1
^
:::
^
x
q
)
y
)=
P
q
i
=1
(
;
1)
i
;
1
x
1
^
:::
^
b
x
i
^
:::
^
x
q
p
(
x
i
)
y
, o all
x
i
2
V
p
and
y
2
U
n
;
qp
. In pa icula , o e e y
q
0,
H
q
(
K
(
p
)) is a g aded
A
-mo dule wi h
H
q
(
K
(
p
))
n
=
H
q
(
K
(
p
)
n
).
Theo em 2.4
Le
:
V
!
U
and
g
:
W
!
V
be wo su jec i e g aded mo phisms o
s anda d
A
-algeb as. Le
:
S
(
U
1
)
!
U
be he canonical mo phism and suppose
W
is a
symme ic algeb a. Gi en
(
n
2
p
=1)
o
(
n
2
p
+1
p
2)
, he e a e isomo phisms o
A
-modules
H
1
(
K
(
p
)
n
)=
ke (
g
)
n
W
p
ke (
g
)
n
;
p
=
ke
n
S
p
(
U
1
)
ke
n
;
p
:
In pa icula , he module o eec i e
n
- ela ions o
U
is
E
(
U
)
n
=
H
1
(
K
(
1)
n
)
and he
ela ion ype o
U
is
(
U
)=s(
H
1
(
K
(
1) ))
.
P oo
.Pu
h
=
g
. Since
n
;
p
p
, hen
W
n
;
p
ke
g
p
W
p
ke
g
n
;
p
W
p
ke
h
n
;
p
.
Applying he snake lemma o he commu a i e diag am o exac ows
3
ke
g
p
W
n
;
p
W
p
ke
h
n
;
p
W
p
W
n
;
p
V
p
U
n
;
p
0
- - -
- - - -
0
ke
h
n
W
n
U
n
0
? ?
?
?
?
h
n
g
p
h
n
;
p
we ge he exac sequence o
A
-mo dules
0
!
(
g
p
h
n
;
p
)(
Z
1
(1
W
p
)
n
)
!Z
1
(
p
)
n
!
ke
h
n
=W
p
ke
h
n
;
p
!
0
whe e
Z
1
(1
W
p
)
n
,
Z
1
(
p
)
n
s and o he
n
- h comp onen o he 1-cycles mo dule o
K
(1
W
p
),
K
(
p
). I
Z
1
(1
W
p
)
n
=
B
1
(1
W
p
)
n
( he
n
- h comp onen o he 1-b ounda ies mo dule o
K
(1
W
p
)), hen (
g
p
h
n
;
p
)(
Z
1
(1
W
p
)
n
)=
B
1
(
p
)
n
Thus, he s isomo phism is demon-
s a ed p o ided wep o e
H
1
(
K
(1
W
p
))
n
= 0 o a symme ic algeb a
W
(see nex lemma).
In pa icula , i we ake
V
=
U
and
=1
U
, hen
h
=
g
=
g
and one o he p ossible
choices o
h
is he canonical mo phism
. Hence, applying wice he s equali y o
and
o any
h
:
W
!
U
a ising om a symme ic algeb a
W
,weha e
H
1
(
K
(1
U
p
)
n
)=
ke
n
S
p
(
U
1
)
ke
n
;
p
=
ke
h
n
W
p
ke
h
n
;
p
:
Lemma 2.5
Le
M
be an
A
-module and
W
=
S
(
M
)
he symme ic algeb a o
M
. Then,
o
(
n
1
p
=1)
o
(
n
2
p
+1
p
2)
,
H
1
(
K
(1
W
p
))
n
=0
.
P oo
. Pu
T
(
M
) he enso ial algeb a o
M
and
q
=
n
;
p
. Applying he snake lemma o
he commu a i e diag am o exac ows
T
p
(
M
)
T
q
(
M
)
T
n
(
M
)
0
-
- - - -
0
ke
!
W
p
W
q
W
n
0
?
?
?
?
"
!
we ge he exac sequence 0
!
ke
!
ke
"
!
ke
!
!
0. Thus,
Z
1
(1
W
p
)
n
= ke
!
=
(ke
"
)is he
A
-mo dule gene a ed by he elemen s
(
x
1
x
p
;
1
x
p
)
(
y
1
y
2
y
q
)
;
(
x
1
x
p
;
1
y
1
)
(
x
p
y
2
y
q
)
whe e
x
i
y
j
2
M
and
x
1
x
p
s ands o he p o duc in
W
=
S
(
M
). Clea ly, i (
n
1,
p
= 1), hen
Z
1
(1
W
p
)
n
=
B
1
(1
W
p
)
n
. Supp ose (
n
2
p
+1,
p
2), i.e.,
q > p
. Then,
H
1
(
K
(1
W
p
)
n
) = 0 ollows om he equali y:
(
x
1
x
p
)
(
y
1
y
q
)
;
(
x
1
x
p
;
1
y
1
)
(
x
p
y
2
y
q
)=
(
x
1
x
p
)
(
y
1
y
q
)
;
(
y
2
y
p
+1
)
(
x
1
x
p
y
1
y
p
+2
y
q
)+
(
y
2
y
p
+1
)
(
x
1
x
p
;
1
y
1
x
p
y
p
+2
y
q
)
;
(
x
1
x
p
;
1
y
1
)
(
x
p
y
2
y
q
)
:
4
Rema k 2.6
Le
:
S
(
F
)
!
S
(
M
) b e he induced mo phism on he symme ic algeb as
by an epimo phism
:
F
!
M
o
A
-mo dules. Then, he las h ee nonze o e ms o
K
(
p
)
p
+
q
,
q
p
1, dene he sequence:
A
2
(
S
p
(
F
))
A
S
q
;
p
(
M
)
@
2
!
S
p
(
F
)
A
S
q
(
M
)
@
1
!
S
p
+
q
(
M
)
!
0
(2)
wi h
@
2
((
x
1
x
p
)
^
(
y
1
y
p
)
z
)=(
y
1
y
p
)
(
x
1
x
p
)
z
;
(
x
1
x
p
)
(
y
1
y
p
)
z
and
@
1
((
x
1
x
p
)
)=
(
x
1
x
p
)
,
x
i
y
j
2
F
,
z
2
S
q
;
p
(
M
)and
2
S
q
(
M
).
On he o he hand, Micali and Roby dened (in 10]) he sequence o
A
-mo dules
T
A
p
+
q
(
F
)
!
S
p
(
F
)
A
S
q
(
M
)
!
S
p
+
q
(
M
)
!
0
(3)
wi h
(
x
1
:::
x
p
+
q
)=(
x
1
x
p
)
(
x
p
+1
x
p
+
q
)
;
(
x
1
x
p
;
1
x
p
+1
)
(
x
p
x
p
+2
x
p
+
q
)
and
=
@
1
. By a simila a gumen o ha one o he end o Lemma 2.5, one can p o e
ha Im
@
2
is always con ained in Im
and ha i
q>p
, hen b o h mo dules a e equal. Thus,
he exac ness o (2) (se led by Theo em 2.4 ei he o
q
p
=1 o ei he o
q>p
2)
assu es he exac ness o (3). Ne e heless, i
q
=
p
2, hen (2) migh no be exac (see
p o o o Lemma 3.8) while (3) is always exac (see 10]).
Co olla y 2.7
Le
U
be a s anda d
A
-algeb a and le
x
bea(possibly inni e) se o o ms
o deg ee one gene a ing
U
+
.I
H
1
(
x
U
)
deno es he s Koszul homology g oup associa ed
o
x
, hen
E
(
U
)
n
=
H
1
(
x
U
)
n
o al l
n
2
. In pa icula ,
(
U
)=s(
H
1
(
x
U
))
.
P oo
. Take in Theo em 2.4,
:
S
(
F
)
!
U
induced by a ee p esen a ion
F
!
U
1
asso cia ed o
x
. Then,
K
(
1) =
K
(
x
U
)is he usual Koszul complex asso cia ed o he
elemen s
x
.
Rema k 2.8
Using duali y be ween Koszul homology and cohomology (see 1.6.10 o 3])
we eco e Schenzel's esul (
U
)=s(
H
d
;
1
(
x
U
)) +
d
,when
x
is ni e o ca dinal
d
(see
13]).
Rema k 2.9
Le
I
b e an ideal o
A
and
R
(
I
)=
n
0
I
n
n
i s Rees algeb a. Take
=1
R
,
he iden i yon
R
(
I
), in Theo em 2.4. Then,
Z
1
(
p
)
n
=ke
;
I
p
I
n
;
p
!
I
n
=To
A
1
(
A=I
p
I
n
;
p
)
whichisknown o b e isomo phic o
Z
1
I
n
;
p
F=I
n
;
p
Z
1
, whe e 0
!
Z
1
!
F
!
I
p
!
0 is
a p esen a ion o
I
p
wi h
F
ee (see, e.g., 2.5 o 8]). Mo eo e , ia he same isomo phism
B
1
(
p
)
n
=Im
A
2
(
I
p
)
I
n
;
2
p
!
I
p
I
n
;
p
=
I
n
;
2
p
B
1
=I
n
;
p
Z
1
:
Thus, by Theo em 2.4, weha e
H
1
(
p
)
n
=
ke
n
S
p
(
I
)
ke
n
;
p
=
Z
1
I
n
;
p
F
I
n
;
2
p
B
1
which ep o es an ea lie esul o K uhl (see 1.2 o 9]).
5
3 Some applica ions
The pu p ose o his sec ion is o gi e some applica ions o Lemma 2.1 and Theo em 2.4.
Example 3.1
Cyclic s anda d algeb as
Le
U
b e a cyclic s anda d
A
-algeb a gene -
a ed by a deg ee one o m
x
2
U
1
. Pu
:
A
]
!
U
wi h
(
)=
x
in Theo em 2.4. Then,
E
(
U
)
n
=
H
1
(
K
(
1)
n
) = (0 :
x
)
U
n
;
1
and (
U
) = min
1
j
(0 :
x
+1
) = (0 :
x
)
g
.
Example 3.2
Change o base ing
Le
U
be a s anda d
A
-algeb a and le
'
:
A
!
B
be a homomo phism o ings. Take
:
V
!
U
any su jec i e g aded mo phism o
s anda d
A
-algeb as in Theo em 2.4. I induces
1 :
V
A
B
!
U
A
B
. Since
K
(
1
p
)
n
=
K
(
p
)
n
A
B
, one can deduce (
U
A
B
)
(
U
). I
'
is a , hen
H
1
(
K
(
1
p
)
n
)=
H
1
(
K
(
p
)
n
)
A
B
. In pa icula , (
U
) = sup
(
U
p
)
j
p
2
Sp ec(
A
)
g
.
I
'
is ai h ully a , hen (
U
A
B
) = (
U
). In pa icula , ia he Naga a mo phism
A
!
A
]
m
]
=
B
,
m
a maximal ideal o
A
, one can always supp ose, when calcula ing he
ela ion yp e o
U
, ha
A
is a lo cal ing o maximal
m
and esidual eld
A=
m
=
k
inni e.
Le
I
be an ideal o
A
and
G
(
I
) =
n
0
I
n
=I
n
+1
i s asso cia ed g aded ing. Since
G
(
I
)=
R
(
I
)
A
A=I
, hen (by3.2) (
G
(
I
))
(
R
(
I
)) = (
I
). In 14 ], Valla showed ha
i (
G
(
I
)) = 1, hen (
I
)=1 oo. Nex p op osi ion is a gene aliza ion o ha esul .
P op osi ion 3.3
I
I
is an ideal, he e exis s
E
(
I
)
n
+1
!
E
(
I
)
n
!
E
(
G
(
I
))
n
!
0
, exac
sequence o
A
-modules, o al l
n
2
.In pa icula , i
(
I
)
<
1
, hen
(
G
(
I
)) = (
I
)
.
P oo
.I 1
R
, 1
G
, deno e he iden i y on
R
(
I
),
G
(
I
), esp ec i ely, hen o each
n
1,
he e is an exac sequence o complexes
K
(1
R
1)
n
+1
!K
(1
R
1)
n
!K
(1
G
1)
n
!
0. Since
he 0- h comp onen o he s mo phism is injec i eand
H
0
(
K
(1
R
1)
n
+1
) = 0, we ha e
enough o deduce he exac sequence
E
(
I
)
n
+1
!
E
(
I
)
n
!
E
(
G
(
I
))
n
!
0. In pa icula , i
(
I
)
<
1
, one can p o ceed by dec easing induc ion.
Rema k 3.4
I (
I
)=
1
, hen 3.3 migh b e alse as Example 4.4 o 11] shows. No e ha ,
as a consequence o nex p op osi ion, we will see ha o he i ele an ideal o a s anda d
algeb a hyp o hesis (
I
)
<
1
can b e emo ed.
P op osi ion 3.5
Le
U
be a s anda d
A
-algeb a and le
U
+
=
n>
0
U
n
deno e i s i ele an
ideal. Take
:
W
!
U
a su jec i e g aded mo phism o s anda d
A
-algeb as wi h
W
a symme ic algeb a. Gi en
(
n
2
p
= 1)
o
(
n
3
p
2)
, he module o eec i e
n
- ela ions o
U
p
+
is
E
(
U
p
+
)
n
=
M
q
np
ke
q
W
p
ke
q
;
p
:
In pa icula ,
E
(
U
p
+
)
n
= 0
i , and only i ,
(
U
p
+
)
n
;
1
. Fo
p
= 1
,
(
U
) = (
U
+
)
.
Mo eo e ,
U
+
is a syzyge ic ideal i , and only i ,
U
is a symme ic algeb a.
6
P oo
.Le
g
:
S
U
(
U
p
A
U
)
! R
(
U
p
+
)be induced by he na u al epimo phism o
A
-
mo dules
U
p
A
U
!
U
p
+
.I is no ha d o see
K
(
g
1)
n
=
i
0
K
(1
U
p
)
np
+
i
. Mo eo e , i
(
n
2
p
=1), hen
np
+
i
2 and i (
n
3
p
2), hen
np
+
i
2
p
+1. The e o e, by
Theo em 2.4,
E
(
U
p
+
)
n
=
H
1
(
K
(
g
1)
n
)=
M
i
0
H
1
(
K
(1
U
p
)
np
+
i
)=
M
i
0
ke
np
+
i
W
p
ke
(
n
;
1)
p
+
i
=
M
q
np
ke
q
W
p
ke
q
;
p
:
In pa icula ,
E
(
U
p
+
)
n
E
(
U
p
+
)
n
+1
.Thus,
E
(
U
p
+
)
n
= 0 is equi alen o (
U
p
+
)
n
;
1.
Fo
p
= 1 and
n
2,
E
(
U
+
)
n
=
i
0
ke
n
+
i
=W
1
ke
n
;
1+
i
=
i
0
E
(
U
)
n
+
i
=
q
n
E
(
U
)
q
.
In pa icula , (
U
)=s(
E
(
U
)) = s(
E
(
U
+
)) = (
U
+
). Mo eo e ,
E
(
U
+
)
2
=
q
2
E
(
U
)
q
=
E
(
U
). Thus,
U
+
b e syzyge ic is equi alen o
U
b e a symme ic algeb a.
Now, le us o cus ou a en ion in o he ela ion yp e o Ve onese sub ings. Le
U
be a s anda d
A
-algeb a. Recall ha he
p
- h Ve onese sub ing o
U
is dened o be
he s anda d
A
-algeb a
U
(
p
)
=
n
0
U
np
.Clea ly, i
:
V
!
U
is a (su jec i e) g aded
mo phism o s anda d
A
-algeb as, hen i induces
(
p
)
:
V
(
p
)
!
U
(
p
)
ano he (su jec i e)
g aded mo phism o s anda d
A
-algeb as.
Lemma 3.6
Le
:
V
!
U
be a su jec i e g aded mo phism o s anda d
A
-algeb as. Then,
o al l
p
1
,
s(
E
(
(
p
)
))
1 + (s(
E
(
))
;
1)
=p
]
(
a
]
is he in ege pa o
a
).
P oo
. W i e s(
E
(
))
;
1=
pa
+
b
wi h 0
b<p
. So (s(
E
(
))
;
1)
=p
]=
a
. Take
n
2+
a
.
Then (
n
;
1)
p
pa
+
p
s(
E
(
)). Thus, ke
np
=
V
1
ke
np
;
1
=
:::
=
V
p
ke
(
n
;
1)
p
and
hence s(
E
(
(
p
)
))
1+
a
.
Lemma 3.7
Le
U
be a s anda d
A
-algeb a and le
:
V
!
U
be a symme ic p esen a ion
o
U
.I
(
n
2
p
=1)
o
(
n
3
p
2)
, hen he module o eec i e
n
- ela ions o
U
(
p
)
is
E
(
U
(
p
)
)
n
=
ke
np
V
p
ke
(
n
;
1)
p
:
P oo
. Take
g
:
S
(
V
p
)
!
U
(
p
)
induced by
p
:
V
p
!
U
p
in deg ee one. We ha e
K
(
g
1)
n
=
K
(
p
)
np
. Mo eo e , i (
n
2
p
=1), hen
np
2, and i (
n
3
p
2), hen
np
2
p
+1. Thus, by Theo em 2.4,
E
(
U
(
p
)
)
n
=
H
1
(
K
(
g
1)
n
)=
H
1
(
K
(
p
)
np
)=(ke
np
)
=
(
V
p
ke
(
n
;
1)
p
).
Lemma 3.8
Le
M
bean
A
-module and
S
(
M
)
i s symme ic algeb a. Then, o al l
p
1
,
(
S
(
M
)
(
p
)
)
2
. Mo eo e , i
p
2
and
M
is ni ely gene a ed, hen
(
S
(
M
)
(
p
)
)=1
i ,
and only i ,
M
is local ly cyclic.
P oo
. By Lemma 3.7,
E
(
S
(
M
)
(
p
)
)
n
= 0 o all
n
3. Thus, (
S
(
M
)
(
p
)
)
2. Supp ose
p
2 and (
A
m
k
)is lo cal (see 3.2). I
M
is cyclic, hen
S
(
M
)
(
p
)
=
S
(
S
p
(
M
)) and
(
S
(
M
)
(
p
)
) = 1. Con e sely, supp ose
M
ni ely gene a ed, bu no cyclic. Take
x y
pa o a basis o
M
k
and
x
p
y
p
x
p
;
1
y
in
S
p
(
M
)
k
. Then,
z
=
x
p
y
p
;
x
p
;
1
y
7
xy
p
;
1
2Z
1
(
1
k
1)
2
. Mo eo e , lo oking a he comp onen s o an elemen ina
k
-basis o
B
1
(
1
k
1)
2
, one sees ha
z =
2 B
1
(
1
k
1)
2
.Thus,
H
1
(
K
(
1
k
1)
2
)
6
= 0, hence (by
3.2)
H
1
(
K
(
1)
2
)
6
= 0 and (
S
(
M
)
(
p
)
)=2.
Rema k 3.9
Le
I
b e an ideal o linea yp e ni ely gene a ed, bu no lo cally p incipal.
Then, by Lemma 3.8, (
I
p
) = 2 o all
p
2, which ep o es 2.6 o 7 ].
Theo em 3.10
Le
U
be a s anda d
A
-algeb a. Then,
(
U
(
p
)
)
max(1 + ( (
U
)
;
1)
=p
]
2)
o al l
p
1
. Mo oe e , i
U
is ni ely gene a ed and
p
2
, hen
(
U
(
p
)
)=1
i , and only
i ,
U
p
is local ly gene a ed by a
d
-sequenceo leng h 1.
P oo
. Le
:
S
(
U
1
)
!
U
be he canonical mo phism. Pu
g
:
S
(
S
p
(
U
1
))
!
S
(
U
1
)
(
p
)
and
=
(
p
)
. Then, by Lemma 2.1, (
U
(
p
)
)
max(s(
E
(
))
s(
E
(
g
))) and, by Lemmas
3.6 and 3.8, we p o e he inequali y. Supp ose
p
2and
U
ni ely gene a ed. By 3.2, one
can supp ose ha (
A
m
k
)is a lo cal ing o inni e esidual eld
k
. I
U
p
is gene a ed by
a
d
-sequence o leng h 1, hen (by 3.1) (
U
(
p
)
) = 1. Con e sely, supp ose (
U
(
p
)
) = 1.
Take
V
=
U
k
, so
V
(
p
)
=
U
(
p
)
k
and (
V
(
p
)
)
(
U
(
p
)
) = 1. The e o e,
V
(
p
)
is
a p olynomial ing o K ull dimension
l
=
(
V
p
) = dim
V
(
p
)
=dim
V
(since
V
(
p
)
V
is
an in eg al ex ension). Take
W
V
a g aded No e he no maliza ion (i exis s since
k
is
inni e, see 1.5.17 o 3]). Thus, dim
W
= dim
V
=
l
and so
;
l
+
p
;
1
p
=
(
W
p
)
(
V
p
)=
l
,
which o ces
l
=1. Hence,
(
U
p
)=
(
V
p
)=1,
U
p
=
Ax
is cyclic and, by 3.1 again,
x
is a
d
-sequence.
Rema k 3.11
The inequali y o 3.10 was s ly p o ed byBackelin and F ob e g o ni ely
gene a ed
k
-algeb as (see 2]). Recen ly, Johns on and Ka z showed a e y simila s a emen
o ha o 3.10, bu o
U
=
R
(
I
) he Rees algeb a o an ideal
I
(see 7]). Since
G
(
I
)
(
p
)
=
R
(
I
)
(
p
)
A=I
=
R
(
I
p
)
A=I
, hen (by 3.2) (
G
(
I
)
(
p
)
)
(
I
p
). In pa icula , o
I
=
U
+
he i ele an ideal o a s anda d algeb a
U
,
G
(
I
) =
U
and (
U
(
p
)
)
(
U
p
+
). Thus,
Johns on-Ka z's esul implies Backelin-F ob e g's esul and he inequali y o 3.10, when
U
is a No e he ian ing. Ne e heless, he whole Theo em 3.10 can no b e deduced di ec ly
om ea lie esul s since, in gene al, (
U
(
p
)
)
6
= (
U
p
+
) as nex example shows.
Example 3.12
Pu
U
=
k
x y z
]
=J
wi h
J
= (
x
3
y xy
3
z
4
x
2
y
2
z
3
). Then, (
U
) = 7,
(
U
(2)
)=2 and (
U
2
+
)=3 ( ema k ha max(1 + ( (
U
)
;
1)
=
2]
2) =4). Indeed, since
E
(
U
)
n
= ke
n
=U
1
ke
n
;
1
,
:
S
(
U
1
)
!
U
he canonical mo phism, hen
E
(
U
)
n
= 0
o all
n
2,
n
6
= 4
7and
E
(
U
)
4
=
k
3
and
E
(
U
)
7
=
k
.Thus, (
U
) = s(
E
(
U
)) = 7.
Since ke
8
F
4
(ke
), hen, by Lemma 3.7,
E
(
U
(2)
)
n
=ke
2
n
=
S
2
(
U
1
)
ke
2(
n
;
1)
=0
o all
n
3. Thus, (
U
(2)
)
2. Mo eo e , (
U
(2)
)=2 since
U
2
is no lo cally cyclic (see
Theo em 3.10). Besides, using P op osi ion 3.5,
E
(
U
2
+
)
4
=
q
8
(ke
q
=
S
2
(
U
1
)ke
q
;
2
)=0,
so (
U
2
+
)
3. Bu , since ke
7
6
=
S
2
(
U
1
)
ke
5
,
E
(
U
2
+
)
3
=
q
6
(ke
q
=
S
2
(
U
1
)ke
q
;
2
)
6
=
0. Hence, (
U
2
+
)=3.
8
4 Condi ions on he gene a o s
In his sec ion wecha ac e ize, in e ms o a sys em o gene a o s, which ideals ha e mo dule
o eec i e
n
- ela ions ze o. Ou wo k he e is inspi ed in p e ious esul s by Cos a, see
5] and 6]. Conc e ely, in 6], i was dened a
sequence o linea ype
as a sequence o
elemen s
x
1
::: x
d
such ha he ideals (
x
1
::: x
i
) a e o linea yp e o
i
=1
::: d
. As
a consequence o he main esul o his sec ion (see 4.7), we ge a new cha ac e iza ion
o sequences o linea yp e in ol ing annihila o ideals (see 4.9). Fo an ideal
I
gene a ed
by
d
elemen s
x
1
:::x
d
, we will deno e by
I
i
1
:::i
s
he ideal gene a ed by he
x
j
, whe e
j =
2
i
1
::: i
s
g
. Fo an
A
-mo dule
M
,we will deno e by
A
d
(
M
) he se o al e na ing
d
d
ma ices wi h co ecien s in
M
.
Lemma 4.1
Le
I
be gene a edby
d
elemen s
x
1
:::x
d
and ake
n
2
. Then,
E
(
I
)
n
=0
i , and only i , o al l
(
a
1
::: a
d
)
2
(
I
n
;
1
)
d
wi h
a
1
x
1
+
:::
+
a
d
x
d
= 0
, he e exis s
(
b
ij
)
2A
d
(
I
n
;
2
)
such ha
0
B
B
B
@
a
1
a
2
.
.
.
a
d
1
C
C
C
A
=
0
B
B
B
@
0
b
1
2
::: b
1
d
;
b
1
2
0
::: b
2
d
.
.
.
.
.
.
.
.
.
.
.
.
;
b
1
d
;
b
2
d
:::
0
1
C
C
C
A
0
B
B
B
@
x
1
x
2
.
.
.
x
d
1
C
C
C
A
:
P oo
. By Co olla y 2.7,
E
(
I
)
n
=
H
1
(
x
R
(
I
))
n
, whe e
K
(
x
R
(
I
))
n
is he
n
- h comp onen
o he Koszul complex asso cia ed o he elemen s
x
1
: : : x
d
in
R
(
I
) =
n
0
I
n
n
. Tha
is,
!
(
I
n
;
2
)
(
d
2
)
@
2
;!
(
I
n
;
1
)
d
@
1
;!
I
n
!
0, wi h
@
2
(
b
1
2
::: b
1
d
b
2
3
:::b
d
;
1
d
) =
(
a
1
:::a
d
) dened by
0
B
B
B
@
a
1
a
2
.
.
.
a
d
1
C
C
C
A
=
0
B
B
B
@
0
b
1
2
::: b
1
d
;
b
1
2
0
::: b
2
d
.
.
.
.
.
.
.
.
.
.
.
.
;
b
1
d
;
b
2
d
:::
0
1
C
C
C
A
0
B
B
B
@
x
1
x
2
.
.
.
x
d
1
C
C
C
A
,and
@
1
(
a
1
::: a
d
) =
a
1
x
1
+
+
a
d
x
d
.
Lemma 4.2
Le
I
be gene a ed by
d
elemen s
x
1
::: x
d
and ake
n
2
.I
E
(
I
)
n
= 0
,
hen
I
1
I
n
;
1
:
x
n
1
=
I
1
I
n
;
2
:
x
n
;
1
1
.
P oo
.I
a
2
I
1
I
n
;
1
:
x
n
1
, hen
ax
n
1
=
a
2
x
2
+
+
a
d
x
d
,
a
i
2
I
n
;
1
. In pa icula , (by 4.1)
0
B
B
B
@
ax
n
;
1
1
;
a
2
.
.
.
;
a
d
1
C
C
C
A
=
0
B
B
B
@
0
b
1
2
::: b
1
d
;
b
1
2
0
::: b
2
d
.
.
.
.
.
.
.
.
.
.
.
.
;
b
1
d
;
b
2
d
:::
0
1
C
C
C
A
0
B
B
B
@
x
1
x
2
.
.
.
x
d
1
C
C
C
A
,
b
ij
2
I
n
;
2
. Thus
ax
n
;
1
1
2
I
1
I
n
;
2
.
Rema k 4.3
I
d
= 1, hen he necessa y condi ion o Lemma 4.2 becomes 0 :
x
n
1
=0:
x
n
;
1
1
, whichisknown o b e sucien o assu e
E
(
I
)
n
= 0 (see Example 3.1).
Lemma 4.4
Le
I
begene a ed by
d
elemen s
x
1
:::x
d
(
d
2)
and
n
2
. I
E
(
I
)
n
=0
,
hen
(0 :
x
1
)
I
n
;
1
=
8
>
<
>
:
d
X
i
=2
a
i
x
i
j
a
i
2
I
n
;
2
x
1
0
B
@
a
2
.
.
.
a
d
1
C
A
=(
b
ij
)
0
B
@
x
2
.
.
.
x
d
1
C
A
o (
b
ij
)
2A
d
;
1
(
I
n
;
2
1
)
9
>
=
>
:
9