2
SHREERAM S. ABHYANKAR AND MANISH KUMAR
NOTATION. By
M(R)
we denote the maximal ideal in a (always noetherian)
local ring
R.
By a quadratic transform of a regular local domain
R,
with maximal
ideal
M(R)
and quotient field
K,
we mean a domain
R'
which, for some nonzero
element X in
M(R),
can be expressed as a localization of R[(Y/X)YEM(R)] at
some prime ideal containing
M ( R);
note that then
R'
is a regular local domain
with quotient field
K
and the dimension of
R'
is
~
the dimension of
R.
By a two
dimensional quadratic sequence we mean an infinite sequence (Ri)i=o,
1 ,2 , ...
of two
dimensional regular local domains such that
Ri
is a quadratic transform of Ri-l
for all i 0. By a two dimensional regular semilocal domain we mean a noetherian
domain
S
having at least one and most a finite number of maximal ideals such that
the localization of
S
at any maximal ideal in it is a two dimensional regular local
domain. For any element
A
in a local ring
R
we put ordRA = oo if
A
= 0, and if
A
=F
0 then
ordRA =the largest nonnegative integer e with A
E
M(R)e.
Moreover, for any polynomial
in an indeterminate Z with coefficients Ai in R we put ordRF(Z) = oo if Ai = 0
for all i, and if Ai
=F
0 for some i then
ordRF(Z) = min(i
+
ordRAi)
where the min is taken over all those i for which Ai
=F
0.
A ring (always commutative with 1) R is said to be pseudogeometric if for any
prime ideal
P
in
R,
the integral closure of
R/ P
in any finite algebraic field extension
of its quotient field is a finite ( = finitely generated)
(R/
F)-module. Obviously any
field is pseudogeometric and any homomorphic image of any pseudogeometric ring is
pseudogeometric. It is also well-known that: every affine ring over (=finitely gener-
ated ring extension of) a pseudogeometric ring is pseudogeometric, the localization
of any pseudogeometric ring at any multiplicative set in it is pseudogeometric, and
every complete local ring is pseudogeometric
2. Local Theory
NOTATION FOR LEMMA 1 AND THEOREM 1. Let
R
be a two dimensional
regular local domain with quotient field
K
and
M(R)
=
(X, Y)R.
For 1
~
j
~
n,
where
n
is a positive integer, let
Fj
=
Fj(Z)
=
Z 2
+
AjZ
+
Bj
be a monic quadratic polynomial in an indeterminate Z with Aj, Bj in R. Let
C3
=
AJ- 4B3
and
J = {j: 1
~
j
~
n
with
C3 =F
0}
with
and
F
=
F(Z)
=
IT
Fj(Z).
1$j$n
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