Suppose that
is a 2-dimensional excellent local domain with quotient field
,
is a finite separable
extension of
and
is a 2-dimensional local domain with quotient field
such that
dominates
. Suppose
that
is a
valuation of
such that
dominates
.
Let
be the
restriction of
to
. The associated
graded ring
was
introduced by Bernard Teissier. It plays an important role in local uniformization. We show that
the extension
of valued fields is without defect if and only if there exist regular local rings
and
such that
is a local ring
of a blowup of
,
is a local ring
of a blowup of
,
dominates
,
dominates
and the associated
graded ring
is a finitely
generated
-algebra.
We also investigate the role of splitting of the valuation
in
in finite
generation of the extensions of associated graded rings along the valuation. We say that
does not
split in
if
is the unique
extension of
to
which
dominates
. We
show that if
and
are regular local rings,
has rational rank 1 and
is not discrete and
is a
finitely generated
-algebra,
then
is a localization of
the integral closure of
in
, the extension
is without
defect and
does
not split in
.
We give examples showing that such a strong statement is not true when
does not satisfy these assumptions. As a consequence, we deduce that if
has rational rank 1 and
is not discrete and if
is a nontrivial sequence of quadratic transforms along
, then
is not a finitely
generated
-algebra.
