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