231
Internat. J. Math. & Math. Sci.
Vol. 8 No. 2 (1985) 231-240
SEMINORMAL GRADED RINGS AND WEAKLY NORMAL
PROJECTIVE VARIETIES
JOHN V. LEAHY
Institute of Theoretical Sciences
and
Department of Mathematics, University of Oregon
Eugene, Oregon 97403
and
MARIE A. VITULLI
Department of Mathematics, University of Oregon
Eugene, Oregon 97403
(Received March 4, 1983)
ABSTRACT.
is paper is concerned with the seminormality of reduced graded rings
and the weak normality of projective varieties.
One motivation for this investi-
gation is the study of the procedure of blowing up a non-weakly normal variety
along its conductor ideal.
KEY WORDS AND PHRASES.
Seminormal Graded Rings, Normal Projective Varieties.
1980 ,THE4ATICS SUBJECT CLASSIFICATION CODE. 16A03.
In this paper we investigate the seminormality of reduced graded rings for the
purpose of studying the weak normality of projective varieties and the procedure of
blowing up a non-weakly normal variety along its conductor ideal.
A commutative
ring with identity is seminormal if and only if it contains every element of its
total ring of quotients whose square and cube are in the ring.
A variety (over an
algebraically closed field of characteristic zero) is weakly normal if and only if
every globally defined continuous function which is regular outside the singular
locus is in fact globally regular.
In [i] it is proven that a variety is weakly
normal if and only if all of its local rings are seminormal.
In section 2 of this paper the relevant commutative algebra is developed to
handle questions of weak normality for projective varieties and in particular to
show that the weak normalization of a projective variety is again projective.
We
have tried to work in as general an algebraic setting as possible, but since we
wanted the normalization of a graded ring to again be graded we considered only
reduced graded rings (indexed by the integers) having only a finite number of minimal prime ideals. Given an integral extension of graded rings we give an explicit
description of the relative seminormalization by adapting a construction of Swan
[2].
We then show that the seminormalization of a graded ring is a graded subring
of its normalization and that the operations of seminormalization and homogeneous
localization commute.
J. V. LEAHY AND M. A. VITULLI
232
In section 3 we consider reduced (but not necessarily irreducible) projective
varieties defined over an algebraically closed field of characteristic zero. In
particular we show that such a variety is weakly normal if and only if its veronese
subrings of order
2.
d.
are seminormal for sufficiently large
d
SEMINORMAL GRADED RINGS.
A graded ring is a ring
All rings are assumed to be commutative with identity.
R
such that RmR n-- R
that admits a decomposition (as abelian groups) R
n
m+n
h
of R. Notice that
of
elements
set
homogeneous
denote the
for all m,n. We let R
@ne_
R
In particular
O.
is homogeneous of degree
R
the identity element of
R
is a sub-
O
R.
ring of
A
If
A
is a ring we let
A
closure of
A
denote its normalization, i.e.
is the integral
Q(A).
in its total ring of quotients
is a reduced graded ring having only a finite number of
i
i
i
for
minimal primes 0 I,’’’,0. Let R
R/0 i,
0i and S
xS
S
Since the minimal primes of R are homogeneous
i
i,-..,. Set S
i
i
is graded where
are again graded. Therefore each S
the quotient rings R
1
h
i
i
n}.
S
S
Hence
deg(t)
{/t
and deg()
S
t e T
n
n
is a graded ring.
(2.0)
R
Suppose that
TIR
Rh
Ti
Ix’--
neon
Ri)
Then there exist homogeneous elements
+
r
r
for
s
i
i
ponent of s
of degree
Thus
i
e
I
Now
r
we see that
n
s
s.1 (n)
R
1
LEMLA 2.1.
for all
0
r
i
r
and
and consequently
i
S
Consequently
If
R
R
We know that
R
I--- .__R
R
Again by the domain case
R
for all
i
S
S
n < O.
s
si(n)
i
s
s
i
for all
I
R
n
R
I..- XR
.
R
of
then
R
is a
S
S
is a graded subring of
Ix’..S
(R i) n
Now assume that
for all
0
n
(i
0
is a graded
for all
0
n
By the domain
with notation as in (2.0).
R
Let
i
for each
R
n
0
l,--’,g) so that
_
Assume that there exists a homogeneous R-regular element of
positive degree. Let T denote the set of homogeneous R-regular elements of
is a graded subring of
i
PROOF.
primes of
R
Let
(i
i,--’,).
Pi’
R
Ti
indexed so that
Now suppose
i.
for all
be a reduced graded ring having only a finite number of
minimal primes.
Then
i.
O.
n
LA 2.2.
by
s.1
is a graded subring of
R
for all negative integers
0
n
m.
is a graded subring of
0
ri + si(n)
be a reduced graded ring having only a finite number of
R
case ([e], vol. 2, Thm. Ii, p.157)
n
denote the homogeneous com-
With notation as above, the normalization
negative integers
Thus
s.
0.(RI’’’Rn ).
n
such that
Comparing homogeneous components of
i.
for all
I
by
i
p.
r
Ix---S
S
S
Let
minimal primes.
PROOF.
r
n
si(n)
let
i
r e R
Suppose
and
of degree n.
I is homogeneous
i
(i
i,---,) so that
is a graded subring of S
and
0
graded subring of
subring of
r.
Replacing
n.
XR
of degree
For each
i,’--,.
i
i
we may assume that
Jegree
Rlx
is a graded subring of
R
Notice that
R.
For each
T-IR.
be as in (2.0).
Let
qiR
i,-..).
Oi
(i
By 2.1 there exist a
i
i
Ti
ql’’’’’q
such that
choose a homogeneous element
b
i
R.
denote the minimal
ai B
q#i0j
R + qi
0 i.
Re-
SEMINORMAL GRADED RINGS AND WEAKLY NORMAL PROJECTIVE VARIETIES
aib i
deg(a i) + m
for
n
0
deg(a i) > 0
i
N
m
Let
a.1
i
j#i0
we may assume that a E
Pi and a.
i
j
R-regular element of positive degree and choose m
a
by
i
be a homogeneous
placing
1,’’-
i
where
ai
and let
nl...n
cB
is homogeneous and
m
R.
Pi
for
N > 0
for
ai am
g
i
R
a
so that
we may assume that
for
1
i
Replacing
i,...
I,.-.,4.
i
Let
R.
B1
by
i
a.B
1
and
N/n.l
i
deg(a i)
we may assume that
a
Replacing
[j#iPj
E
233
Then
c
a
by
i
al+.--+a
To finish the proof it suffices to see that
c
is
R-regular.
Suppose that c
+ Z a and
c
a.
j#i
I
R
Z
a
a
0
ji j
T- 1 R.
g
B
R-regular and
is a zero divisor in
R
i
__T-1R.
Hence
T-1R
One can easily check that
Since
implies
Oi,
i
c
for some
0.
c
is
TIRI’..TIRE.
S
R
by 2.1 we know that
Now
i.
Thus
a contradiction
is a graded subring of
S
is a graded subring of
so that
is a graded subring of
T-1R.
For a graded ring
R
the subring of
R(d)
n
On.ZR
R(d)
R
OnRnd"
COROLLARY 2.3.
Let
R(d)
d,
integer
PROOF.
for
ideal.
Let
R
Since
R
Suppose
0
Bi
R
R(d)
for some
Bde
i
0 #
But
i
such that
[.=lqi O. In
qi
(O:8i) for
R(d)-regular.
is
R
If
R(d)
qi(d)
R(d)
qi
plies
Let
ql(d)
Then
a e
Since
qj.
must have
be a homogeneous element of
a
i
oi(d),’’’,o(d)
Q(R(d))
Q(R/Ol(d))x.--xQ(R/%(d))
and
(R/q E(d)). So
integral domain and R O # R.
K
Q(R)
and let
r
is an additive subgroup of
icm(d,m)
and set
f
--
adc
R(d).
Q(R(d))
in its total quotient ring
Q(R/q l(d))
Let
Then
qi"
qi
and that
R(d)
qi(d)
qj (d)
Then
im-
and we
qj
R(d).
Similarly,
is the integral closure
--Q(R(d)/01(d))x.-.(R(d)/0E(d))
Q(R(d)/ql(d))x---Q(R(d)/qE(d))
it suffices to establish the result when
h
R
We
O.
Q(R(d))
Now
Q(R(d))
{0}.
and hence
e/m.
qi(d)
are the minimal primes of
are the minimal primes of
We wish to see that
R(D)
ql(d),...,qE(d)
then
Nq(d)
For suppose
is generated by homogeneous elements
qi
Hence
j.
R
Bi O, a contradiction Hence
Q(R). Similarly Q(R(d))
Q(R(d))
claim that there are no containments amongst these primes.
qj (d).
l,---,g
i
d
and
I,--.,E.
i
qi
is a zero divisor in
R-regular and
for
N R
q
0i
i
is a homogeneous
particular there exist
Q(R).
Let
and let
is a graded ring and hence each
R(d)-regular element is
every
R(d).
minimal primes of
is reduced we know that
a
For each positive
R.
denote the normalization of
ql’’’’’q% denote the
By 2.1
R(d)
be a reduced graded ring having only a finite number
R
homogeneous elements
e
denote
is a graded ring where
R.
is the normalization of
i,.--,.
i
R
Let
of minimal primes.
of
R(d)
we let
d
R(d)
Then
R
nd"
If R is the homogeneous coordinate ring of a projective variety then
is called a Veronese subring of
_
and a positive integer
defined by
Let
J
R(d)
J
n
R(e)
{n
I(T-I)
R
# 0}.
for some positive integer
R(fm)
R(m)(f)
and
is an
Then
J
m.
Let
R(d)
J. V. LEAHY AND M. A. VITULLI
234
R(e)
R(m) (f).
R(fm)
R
Thus replacing
R
and
Let
(T-IR)
L
R
a
n
L
scendental over
and hence
L[t]
0
and
b
and
Q(R(d))
then
R.
i
let
Bm + Cl
R(d).
in
ab
b
R(d)
8
Say
m
L[t,t
d
a
-I]
R(d) n
t
L
abd-I/b
a/b
and
R(d).
iBm-l+ ’’-+am_lS+am
0
8 e
Let
where
a
denote the homogeneous component of
c.
L.
i,
L[t
Since
t
-1]
is tran-
t
L[t,t
-I
is
it is integrally closed in
as a graded subring.
We claim that
is integral over
+
has degree
in
t
Q(R(d)).
d
Hence
For if
Lit
d
t
a,b e Rn
C Q(R(d))
-d]
L(td).
Q((d))
Clearly
over
d-I
t
with respect to powers of
L[td,t-d].
(d)
Since
is a polynomial ring over
is contained in
R(d)
Thus
L[t]
and
the localization of
K
(at-n)t n.
R(m), respec-
and
homogeneous element
with a graded subring of
R
Then we may identify
O
is identified with
_
i
T-IR.
where
R(m)
by
tively, we may assume there exists a nonzero degree
Bm-l+’’’+cm-I B+cm
0
and
c l,.--,c
me
R(d)n
R
ai
so that
(i
of degree
i
R(d)
ind.
B
so that
Thus to finish the proof it suffices to see that
d
L[t t
B
is integral
For each
l,’-’,m).
R(d)
Then
is integral over
is integrally closed
-d]
If
L[t d,t -d]
y
R.
hence over
-i
e[t,t
(d)
(d)
is integral over
[] L[t
y
d,t -d]
Thus
y
then
R(d)
is integral over
and
is a graded subring of
since
].
look at the question of seminormality for a reduced graded ring
We will now
For a ring
A
A.
+A
A
J(A)
let
the normalization of
The ring
R
We first recall the relevant definitions.
as above.
denote the Jacobson radical of
The seminormalization
{b
g
b
J(x
+A.
A
+
X
is said to be seminormal if
A
For an integral extension of rings
+B
{b
A
We say that
A
X
A
is seminormal in
B
V
of
A
A
A
if
+A.
B
and
A
let
denote
is defined by
x Spec(A)}
There is also a relative notion.
B, the seminormalization
bx a Ax + J(Bx)
B
+A
V
+A
B
is defined by
xe Spec(A)}
We refer the reader to [5] for
some of the fundamental results on seminormality.
Recall that Hamann’s criterion for seminormality asserts that A is seminormal in B if and only if A contains each element b of B such that b 2 b 3 e A
([i], Prop.l.4).
McCrimmon
In ([4], Thm. 2) Anderson adapts an argument of Brewer, Costa and
([5], Thm.l) to show that a graded integral domain R
is
Ro RI...
seminormal if and only if
2
R
contains each homogeneous element
s e
such that
3 e
R.
We note that the same argument applies to any integral extension of rings
R
S such that S is a graded ring and R is a graded subring of S (we say
that R__ S is an integral extension of graded rings).
s
s
PROPOSITION 2.4.
R
is seminormal in
S
such that
PROOF.
s
2
s
Let
S
R
S
be an integral extension of graded rings.
if and only if
R
contains each homogeneous element
Then
s
of
3 e
R.
One half of the assertion follows immediately from
Thus it suffices to show that if
square and cube is in
R
then
R
R
Hamann’s
contains each homogeneous element of
is seminormal in
S.
criterion.
S
whose
SEMINORMAL GRADED RINGS AND WEAKLY NORMAL PROJECTIVE VARIETIES
R
Assume that this condition is valid but
2
s of S\R such that s
s
exists an element
For an element
R.
denote the number of nonzero homogeneous components of
(s)
s
element
S
of
si(1)+’’’+Si(m)
max{k
p(s)
let
si(1),’’’,Si(k_l)
initial nonzero homogeneous component of
Amongst those elements of
element
(s)
with
s
p(s)
s
as above so that
Sil+’’’+Si,mk
(higher degree homogeneous terms) is in
si(1)3 NowR
and
hence2
(2si(1)s)
si(1)
>
9(2si(1)s)
Now
s
R.
2s(1)s2
(3s. (1)s)
so that
(3si(1)s)
2
and
s3
(2i(I) + i (k) )-component of
3s2(1)Si(k)
3s2i(l’Sij)
If
3s.2
ll)S
hence
2
(s
R.
R.
Consider
s
si(1))3
s
S\R.
2
p(s)
R
are in
2Si(lSi(k
for some
then
0
R.
3
2
_
e
si(1)
R.
E(2si(1)s)
and
s
Since
2
+ (terms
R)
in
for all
(s).
R
R
Again
2_
R)
j
in2eith
s
R
but
Since the
we must have
(s)
E(3s,2(1)s)
then
Sil)
si(1)3
3Si(lS + 3Si(lS
%(s) and
(s) and
2si(1)s R.
case2
(3si(1)s) <_ L(s).
and
for all
(s
Then
Since the
we must have
(3s.2l)S)
#
Sitl+
Thus in either
are in
0
2
2
Similarly
E(2si(1)s)
then %(2si(1)s)
2
+ (terms in
s
is 3s.
(i) i(k)
0 for some j then
3SlSi(. )"
If
si(1)
3
is
(s).
m
we must have
k
2si(1)si(j) 0
If 2si(1)si(j. #
If
p(s)
2
Thus
3
(2si(1)s)
2
and
(i(1) + i (k) )-component of
2Si(lSik,,,, R.
hence
2si(13s..
R
R
choose an
has the same properties then
t
p(t) _< p(s).
Write
then the
1
R
S\R whose square and cube are in
minimal and such that if
let
R.
is not in
s
S
i(1)<--’<i(m)
and
Si(j)
In particular if
Thus there
of
s
For each nonzero
s.
# si(j)
0
such that
R}.
S.
is not seminormal in
3
235
and
%(s)
and
2
R and
+ si(1)
a
(s),
si(1))
2SilS
(s
contradiction.
Hence
R
is seminormal in
Let
COROLLARY 2.5.
primes.
R
S.
be a reduced graded ring with a finite number of minimal
+R
Then the seminormalization
R
of
is a graded ring and contains
R
as
a graded subring.
PROOF.
By 2.1 the normalization
Let
graded subring.
_
Ine-r+Ri
R’
smallest seminormal subring of
R.
subring of
ring of
+R.
Since
R
R
R
of
R
is graded and contains
R’
By 2.4
n
that contains
is a graded subring of
+R
R’
R
and hence
R
we know that
R
Let
R
subring of
S)
S
([2], Thm. 4.1)
to describe
is graded sub-
T
be a graded subring of
S
R
S
T
whose cube are in
T.
T
of
Ta --c T8
a
let
T
8
whenever
a<sTa.
R. Now let
containing
R*
We inductively define graded subrings
R
(0)
T_I
eATa
is also a graded subring of
T,
(n+l)
R, R
S
Ta+ I Ta[se]
T’
so that
R
Let
{s
R.
ale
e
whose square and
S
such that
and for a limit ordinal
is again a graded subring of
(n)’
containing
we adapt
is a graded
R.
containing
be a well ordering of the set of all homogeneous elements of
_< 8. Let
Set T’
R
+R.
be an integral extension of graded rings (so that
and let
is the
is a graded
In order to prove the analogue of 2.3 for the seminormalization of
a construction of Swan
as a
+R
But
is seminormal.
and set
R*
">0 R(n)
S
Note that
_
J. V. LEAHY AND M. A. VITULLI
236
PROPOSITION 2.6.
R
S
be an integral extension of graded rings.
+S L_v>nR(n)
,
__
seminormality of
R
S
in
R(O) c_ +/-R.
R
nO
_
follows from 2 4
Suppose
T
(n)
R
T.
+SR.
T
c+S R
Now
for all
T c *R
S
a + 1
T_l
T
S
a <
for all
--.,
induction
so
we must have
Let
PROPOSITION 27
R(d).
PROOF.
_
Then
R(d)
rings
R
by 2.4.
Let
R
S
d
denote the normalization of
R(d)
is the normalization of
+R
__
we also have
R(d)
a
T.
(+R) (d)
and consequently
R
and is semi-
R
is the seminormalization
and let
d
be a positive integer.
Since we have inclusions of graded
(d)
(+R) (d)
and
+(R(d)).
oR (n). Thus
+R
is seminormal
_
_
_
.
_ _Using the notation of 2.6 we have
R(n)(d) +(R(d))
Now R(O)(d)
Let
R
T
(n)
R(d)
T_I
Ta(d)
suppose
we have
and
rksk
d
k.deg(s)
(d)
If
B
for all
LEMMA 2.8.
Let
O
C-algebra map
2
Let
Ta[s]. where
Irks
(rksk)3 T(d)
s
for each
k
+(R(d))
rksk
te R
I
and
a
a
and since
R
(n+l)
r
e
sa h
Ta
and hence
+(R(d))
aAT
and
(R(d))
we have
TB(d)
jnOR(n) (d)
(+R)(d)
by
morphism of graded ring such that
Then
is invertible
To simplify notation let
:C[X,X-I]
<T
B
all
T (d)
8
so that
+ r(d)).
R
is the seminormalization of
PROOF
(rksk)
t
TB(d)
by
we also
+(R(d))
as
be a reduced graded ring having only a finite number of min-
imal primes and suppose that
(+R)
B
Then
T
limit ordinal then
R
We
as above.
and suppose that
-i
T
then
was an arbitrary homogeneous element of
+ (R) (d)
Consequently
Ta
Define
B
Let
1
Now
k.
Ta(d) +(R(d)) for
R(n+l) (d) +(R(d)).
(n)
+(r(d)) for all n
Thus R
(d)
desired.
+
a
Hence
assumption.
and
+(R(d)).
T_l(d
T.
a.
(by 2.3) implies that
R(d)
+(R(d)). Since t
+(R(d)). If
is a
have
be a well ordering of the set of all homogeneous
for all
8.
R(n)(d) +(R(d)).
and assume that
is homogeneous of degree
B
deg(r k)
a <
for all
T (d)
t
and
t
T
+ (B(d))
+ (R(d))
Ta(d)
need to show that
Since
A}
a
s
n > 0
Let
whose square and whose cube are in
T
it suffices to see that
n.
+(R(d)).
and let
R
elements of
for all
Since
--JaCATa
T’
so that
n
contains
by 2.3.
(+R)(d)
(+R)(d)
to see that
Thus it suffices
s
and that
3
S
be a reduced graded ring having only a finite number
R
For each positive integer
of minimal primes.
of
where
all
S
2
is a limit ordinal then
all
is the smallest subring of
Since
normal in
cR
B > -1
Suppose
by assumption
we must have
S
S
be a well
whose square and whose
then T8
8. If
Ta[s]
R
and
hence
P"
sa "
TB S R If
T
for
B Ja<BTa +SR bY(n+lassumption.) Hence Ta ()
+
for
T’
R
But
R
by
sRo
s+R that
R* +SR.
R
+S
, +R.
S
R
T
c
A}
a e
{sa
Let
+R.
R*c
We show by transfinite induction that
as above.
T
Define
S.
and the
Thus it suffices to see that
ordering of the set of all homogeneous elements of
cube are in
With
In particular R is a graded subring of
R* U R (n) is a graded subring of S
As observed earlier
PROOF.
Now
Let
R
notation as above
q0(X)
A
O
RO
t.
cp(B[X,X-I])
(+R)
B
Then
+R
’p
O
(R)
and
is the normalization
C +
()o
Define a
is a degree preserving iso-
and
@(A[X,X-I])
R.
SEMINORMAL GRADED RINGS AND WEAKLY NORMAL PROJECTIVE VARIETIES
C[X,X
Thus
homogeneous
So an element of
R
elements of
Let
Then
+R
Q(B)
B
Thus
-i
__r where
s
].
A
and
a e
P2
then
B
and let
K(p) (X)
Hamann’s
and
C
and
a
+A
are homogeneous
s
C
Since
2
a
3
B[X,X-I]
B
E
is normal
then
Thus we need only show
criterion.
P2B[X,X
is the seminormaliza-
lying over the same
is an isomorphism.
be a
(P)
(0)
B
Hence
P
Let
P1 P2"
Then the canonical map
K(P) (X)
characterization of
and
r
are prime ideals in B
-i
so that
P Q A.
0
A
Q(B)
Q(A)
C
-i
PIB[X,X
A
prime ideal of
an isomorphism since
and hence
is seminormal by
PI
If
_.
denote the set of
is a graded subring
_- _
-n
B
T
R
s is R-regular and __r is integral over R.
rt-n Q(A). A direct computaA and __r
R0
S
-n
is seminormal if
prime ideal of
Traverso’s
st
We already have
A[X,X
tion of
-I] +R
B.
+A.
B
that
-n,
A.
B[X,X
Since
rt
Let
is normal.
is of the form
is integral over
S
C
we must have
C
C
By 2.2 we know that
st
r_
tion shows that
a a
R.
of the same degree,
deg(r).
n
is normal and hence
R-regular elements of
T-IR.
of
-i
237
must be
+A
by
([ ], Prop. 1.3).
We can show that normalization (respectively, seminormalization) and homogeneous
localization commute.
Let
COROLLARY 2.9.
minimal primes and let
(+R) f(d)
be homogeneous of positive degree
is the seminormalization of
is the degree
R(f)
and is homogeneous of degree
and
be a reduced graded ring having only a finite number of
# 0
(+R)(f)
the normalization and
PROOF.
R
f
i.
0
subring of
(R) f(d)
is the seminormalization of
Rf(d)
and
d.
Then
by 2.7.
immediately from applying 2.8 to the reduced graded ring
is
R(f).
f
Rf(d) is
Rf(d)
is the normalization of
Rf(d)
Rf
invertible
by 2.3
So the assertion follows
(Note that
Rf(d).
Rf(d)
has only a finite number of minimal primes by the proof of 2.3.)
RE,LARKS.
I
and
D. F. Anderson [6] shows that if
A
is a seminormal integral domain
an invertible fractional ideal then the Rees algebra
R(1)
seminormal,
A
Let
A
on
n
is again
be a reduced ring with a finite number of minimal primes and assume that
Let
is seminormal.
I
I
nO
I
be an ideal of
guarantee us that the Rees algebra
A.
It is natural to ask what conditions
R(1) (or the scheme Proj(R(1)) is again
seminormal. We do not have any satisfactory answer to these questions at this time
but we would like to make some observations.
(2.10)
ity of
A
A
Since
is a reduced ring with finitely minimal primes the seminormal-
implies the seminormality of the polynomial ring
use 2.2 and 2.4).
Now
total quotient ring as
I
n.
2
a
Thus
n
(cf. [6] in case
(2.12)
a
A[t].
By 2.4 we see that
each positive integer
a
A[It]
3 e
J
A[t] ([5], Thin. 2 or
A[t] and has the same
seminormal if and only if A[It] is
is a graded subring of
R(1)
is
A[t].
seminormal in
(2.11)
R(1)
Say an ideal
then
ps eudo-reduced
a
J.
A[It]
A[t]
is seminormal in
and each element
a e A
if
a
2
12n
if and only if for
3
3n
and a e I
then
A
is an integral domain.)
J
of a ring
A
A
are as above when are all powers of
If
and
I
is pseudo-reduced if whenever
a e A
I
and
J. V. LEAHY AND M. A. VITULLI
238
A
If
I.
power of
A: (A:I n)
Hence
is Anderson’s argument in
If
I
is a seminormal domain and
I
and
is pseudo-reduced for all
n
>__
(this
i
[6]).
in/i n+l
gri(A
A
is an invertible ideal then so is each
n
is reduced then all powers of
I
are pseudo-reduced.
hi0
I
So, for example, if
A
is a reduced ideal of
sequence then the Rees algebra
R(1)
that is generated by an
out that an analagous result for normality was proven by Barshay
is a normal domain and
gri(A)
R(I)
is reduced then
A-regular
We would like to point
is again seminormal.
[7].
A
Namely, if
is again a normal domain.
IMPLICATIONS FOR PROJECTIVE VARIETIES.
3.
Let
be a fixed algebraically closed field of characteristic
k
0.
When we
use the term variety we assume that the underlying topological space is the set of
closed points of a reduced, separated scheme of finite type over
Let
U
(X,Ox).
be an open subset of a variety
A
k.
k-valued function on
U
is
said to be
Let
O
c-regular if it is continuous and regular on the nonsingular points of U.
denote the sheaf of c-regular functions on X. We say that X is weakly
c
x
x e X
normal at
OX, x ocX,x
if
and that
X
is weakly normal if
cX.
X
An
afflne variety is weakly normal if and only if its affine coordinate ring is seminormal ([i], Thm. 2.2 and (2.7)).
For the fundamental results concerning weakly normal
varieties the reader can consult [I].
Let
cpn
X
T-IR
Now if
X
by 2.2.
Let
n
1
R
and suppose that
s I,
R[s I
Si_l] for
PROOF. Since R
e R
> 0.
whose degree
Let
0
R
Hi=ik.
R
is a graded
by 2.6.
This is not the case
i
Then there exist positive degree
3 e
but s
R[s i ,’’" ’si-i
in R such that
,s
i
m
In particular
s
l,’’’,m and
0
is not seminormal.
si2,si
+R
(+R)
R[Sl,’’" m]
R)
R[Sl].
1
Then
homogeneous piece is
R
1
k.
R.
Since
k
Ro c Ro Hi=ik
(where
is the diagonal mapping we must have
is a graded subring of
If
R1
(and of
+R)
is not seminormal there exists
whose square and whose cube is in R I. As above
R
s e R
2
1
>
then R
is a graded subrlng
0 so that if R
we must have deg(s
2
2
of R whose degree 0 homogeneous piece is k.
a homogeneous element
Rl[S2]
2)
Continuing in this fashion we obtain an increasing chain of graded subrings of
R:R < R
1
a finite
<
each of whose degree 0 homogeneous piece is k. Since R is
2
R-module ([3], vol. i, Thm. 9, p. 267) this process must end in a finite
R
number of steps.
2.4.
Clearly
R
m
If
c
R
+R
k
is not seminormal by 2.4 there exists a homogeneous element
is the number of minimal primes of
deg(s I)
Ro
T-IR
T
Thus letting
R we know that
is a graded subring of
whose square and whose cube is in
R
denote the homoThen no minimal
be the homogeneous coordinate ring of a projective variety
R
homogeneous elements
S
R
irreducible components then
has
LEMA 3.1.
Xc
+
Hence
+R.
with
(Xo,’’-,Xn).
R-regular elements in
denote the set of homogeneous
subring of
,xn]
X with respect to the given embedding.
contains the irrelevant maximal ideal
R
prime of
k[Xo,
R
be a projective variety and let
geneous coordinate ring of
m
R
is seminormal by
m
of
subring
semlnormal
smallest
is the
is the last ring in the chain then
and since
+R
SEMINORMAL GRADED RINGS AND WEAKLY NORMAL PROJECTIVE VARIETIES
R
that contains
+R.
R
we must have
239
m
It is well known that the normalization of a projective variety is again projecMost proofs assume that the original variety is irreducible (e.g. [8] Them. 4,
tive.
p. 400).
The general case follows quickly from the irreducible case.
We first take
the normalization of each irreducible component and then construct the disjoint union
of these normal projective
>
components and
X
varieties
observing that this union can be embedded in
I
pn
XC
We would like to point out that if
some projective space.
has
irreducible
then the homogeneous coordinate ring of the normalization of
cannot be realized as the normalization of some Veronese subring of the homogeneous
k[Xo,
R
coordinate ring
R)
Veronese subring of
"’,x
n]
of
For the normalization of
X.
has as its degree
0
homogeneous piece
homogeneous coordinate ring of any projective variety has
k
(and of every
R
I[=lk
whereas the
as its degree
ILomo-
0
geneous piece.
We now show
that the weak normalization of a projective variety is again proWe include a complete proof for the convenience of the reader.
PROPOSITION 3.2. The weak normalization of a projective variety is a projective
variety.
jective.
PROOF.
Let
n
x
R
k[Xo,’-’,Xn]/I
S
is a graded ring
d
such that
be a projective variety with homogeneous coordinate ring
Let S denote the seminormalization of R. Then
k[Xo’""x hI.
S(d)
(2.5) and
S
k
o
is generated as a
([8], Lemma, p. 403).
Now
S(d)
by 3.1.
Hence there exists a positive integer
k-algebra, by degree 1 homogeneous elements
is the seminormalization of
is the homogeneous coordinate ring of the Segre transform
X
d(X)
by
we may assume that
S
R(d)
by 2.7 and
R(d)
d(X)
of X. Replacing
k-algebra by degree 1 homo-
is generated as a
geneous elements.
Let
be a
Yo’’’’’YN
S
k-basis for
for
Yi
J.
defined by
S
L
and let
p:N
L
R
Since
F ,---,F
o
Let
I.
O,’’’,N and let J
i
ker
C]PN
is a graded subring of
(Y)
R
-IX
V
i
coordinate
0
Y
by
"(Yi)
the elements
x ,-.-,x
are in
o
n
respectively
n
xo ,’--,x
(Fo-’’,Fn). Let
(Fo(a),.--,Fn(a)). Since R
p(a)
projection defined by
c S
p
L
I
ization of
R-regular element.
Then
Q(R(f))
Q(S(f))
are the function
Y, respectively, so that
(U i)
for
rings
,n
is a finite morphism
X.
be an
and
R(xi)and
,n
:y
P
f
fields of
i
S
S
be the projective variety
(see [8], Proof of Thin. 4.1,
([8],
Prop. 6, p. 246). Then
y
the projective variety defined by the homogeneous ideal J
X
k[Xo’
n
Let
tion of
be indeterminates
’P:k[Yo,’’’,YN
k-algebras
Let y --]pN
be linear forms whose images in
.]pn be the
I, i.e.,
and
YN
o
S are
n
be the linear subspace defined by the ideal
p. 405) and
is
Y
P.
is an integral extension of graded rings
(Y)
and let
I
Define a degree preserving map of graded
by
i
0,’’" n
R(xi)
and
2.9.
Thus
hence
Y
Then
S(xi),
(Vi,
Li=oVi
U
i
and
V
i
are affine with affine
respectively, and
Vi
is
is the
S(xi)
the weak normalization
is weakly normal and
(Y,n)
seminormaliza-
of
U
for
i
is the weak normal-
X.
DEFINITION 3.3. We say that a projective variety X-pn is arithmetically
weakly normal if its homogeneous coordinate ring with respect to this embedding is
J. V. LEAHY AND M. A. VITULLI
240
seminormal.
REMARK 3.4.
n
X
Suppose
Then
its homogeneous coordinate ring with respect to this embedding.
normal for each
# f
0
the affine open subset
elements of
R
l)
g
by 2.8.
P’l
Xf
X
of
X
we see that
PROPOSITION 3.5.
X
Let
C
R
is arithmetically weakly normal and let
R(f)
Since
Rf
denote
is semi-
is the affine coordinate ring of
and these cover
X
(as
ranges through nonzero
f
is weakly normal.
pn
if and only if its Segre transforms
X
Then
be a projective variety.
is weakly normal
are arithmetically weakly normal for
Pd(X)
d
sufficiently large.
PROOF.
X
that
that
0
#
Let
R
R(f)
R
f
1
we know that
this implies
Then
is weakly normal for each
Hence
Supp(+R/R)
that (+R/R)
m
sufficiently large.
Now
R
Rf
for
0
+Ro
o
R+
+R/R
m
sufficiently large, i.e.
k
by 3.1 so that
R(d)
ficiently large and the latter is seminormal by 2.2 and 2.4.
homogeneous coordinate ring of the Segre transform
#d(X)
are weakly normal for
d
Ro
R(d)
d(X)
k
For if
X
for
R(d)
for
m
d
suf-
is the
X.
n
is a reducible normal projective var-
R
is its homogeneous coordinate ring then
whereas
R(d)
R(d)o
Ro
Since R(d) is the
0
for each d
0.
R(d) is not normal for all d
R(d) is the affine coordinate ring of the
i=l k
by 2.2 we know that
Geometrically this is easy to see because
union of
(+R)(d)
Since
+Rm
the Segre transforms
is isomorphic to
irreducible components and
normalization of
m
We again remind the reader that the analogous statement for a reduc-
ible normal variety is false.
iable with
d(X),
R
sufficiently large.
The converse is clear by 3.4 as
REMARK 3.6.
Suppose
a R
Rf
(+R)f
{mj.
# f
0
so
1
Thus
0 # f e R
is seminormal for each
I.
for each 0 # f e R
and letting m
1
Since
is a finitely generated graded R-module
Xf
is seminormal for each
by 2.8.
n.
X
denote the homogeneous coordinate ring of
is weakly normal.
cones through the origin in
An+l
and hence is not normal.
REFERENCES
i.
LEAHY, J. V. and VIULLI, M. A. Seminormal Rings and Weakly Normal Varieties,
N.agoya Math. J., 82 (1981), 27-56.
2.
SWAN, R.
3.
ZARISKI, O. and SAMUEL, P.
New Jersey 1960.
4.
ANDERSON, D. F.
5.
BREWER, J. W., COSTA, D. L. and MC CRIMMON, K. Seminormality and Root Closure
in Polynomial Rings and Algebraic Curves, J. Alg. 58 (1979) 217-226.
6.
ANDERSON, D. F.
7.
BARSHAY, J. Graded Algebras of Powers of Ideals Generated by A-Sequences,
J. Alg. 25 (1973) 90-99.
8.
MUMFORD, D.
On Seminormality, J. Alg. 67 (1980), 210-229.
Commutative
Algebra, Vol. I, II, Van Nostrand,
Seminormal Graded Rings, J.P.A.A.A. 21 (1981) 1-8.
Seminormal Graded Rings II, J.P.A.A.A. 23 (1981) 221-226.
I_ntroduction to Algebraic_Geometry Harvard Lecture Notes.