SIMULTANEOUS RESOLUTION
3
Let L be a splitting field ofF over K, and let S be the integral closure of R in L.
LEMMA 1. For 1 :::; j :::; n let
L1
be the splitting field of
F1
over
K
in
L,
and
let
S1
be the integral closure of
R
in
L1.
Then we have the following.
(1.0) Assume that char(K)
"1-
2 and for every j
E
J we have
c1
=
n1xrjysj
with D1
E
R\M(R) and nonnegative integers r1, s1. For any integer r, let
r
denote
the residue of
r
modulo 2, i.e.,
r
is the unique integer in {0, 1} such that
r-
r
is
even.
Then for every j
E
J there exists H1
E
L with
HJ
=
DjXTJYSJ
and, for any such H1, upon letting
we have the following:
{
J:
1
=
{j_E
J: (rj,Sj)
=
(0,
1)}
J
=
{J E J :
{rj,
s
j)
= (
1, 0)}
J'"
=
{j
E
J:
(r1,
Sj)
=
(1, 1)}
(i) If J' U J" U J"'
=
0
then
S
is a two dimensional regular semilocal domain
and for its localization Tat any maximal ideal in it we have
M(T)
=(X,
Y)T.
(i') For every j
E
J' we have S1
=
R[H1]
=
a two dimensional regular local
domain with
M(S1)
=(X,
H1).
(i") For every l
E
J"
we have 81
=
R[Hl]
=
a two dimensional regular local
domain with
M(S1)
=
(Hl, Y).
(ii') If J'
"1-
0 =
(J"UJ'") then Sis a two dimensional regular semilocal domain
and for its localization Tat any maximal in it we have with
M(T)
=(X, H1) where
j is any element of
J'.
(ii") If J"
"1-
0
=
(J'UJ"') then Sis a two dimensional regular semilocal domain
and for its localization Tat any maximal ideal in it we have with
M(T)
=
(Hl, Y)
where
l
is any element of
J".
(iii)
If
J'
"1- 0 "1-
J"
then
S
is a two dimensional regular semilocal domain and
for its localization Tat any maximal ideal in it we have with
M(T)
=
(H1, H1)
where j and l are any elements of
J'
and
J"
respectively.
(iii') If J' -j.
0
-j. J
111
then S is a two dimensional regular semilocal domain and
for its localization
T
at any maximal ideal in it we have with
M (T)
= (
Hu/ H1, H1)
where j and
u
are any elements of
J'
and
J'"
respectively.
(iii") If J"
"1- 0 "1-
J
111
then
S
is a two dimensional regular semilocal domain and
for its localization Tat any maximal ideal in it we have with
M(T)
=
(Hl,Hu/Hl)
where l and
u
are any elements of J" and J
111
respectively.
(iv) If (J' U J")
"1- 0
or J"'
=
0
then Sis a two dimensional regular semilocal
domain.
(v) If (J' U J")
=
0
and J"'
"1-
0
and
R'
is any two dimensional quadratic
transform of
R,
then the integral closure
S'
of
R'
in
L
is a two dimensional regular
semilocal domain.
(1.1) Assume that char(K)
"1-
2 and for every j
E
J we have
Cj
=
DjXTJYSJ
with D
1
E
R \ M ( R) and nonnegative integers r
1,
s
1.
Then either S is a two dimen-
sional regular semilocal domain, or for every two dimensional quadratic transform
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