Last Updated:2025/12/20
Sentence
Let
V
be
an
irreducible
algebraic
variety.
Then
P
=
J(V)
is
a
prime
ideal
of
ℂ[X_1,...,X_n]
and
so
ℂ[V]=ℂ[X_1,...,X_n]/J(V)
is
an
integral
domain.
The
localization
ℂ[X_1,...,X_n]_P
is
a
subring
of
ℂ(X_1,...,X_n)
consisting
of
rational
functions
f/g:f,g∈ℂ[X_1,...,X_n],g∉P
which
are
defined
on
a
nonempty
subset
of
V.
If
V
=
{x}
is
a
point,
then
P
is
maximal
and
ℂ[X_1,...,X_n]_P=f/g:f,g∈ℂ[X_1,...,X_n],g(x)
ne
0
consists
of
rational
functions
which
are
defined
at
x.
Quizzes for review
3) Geometric interpretation of the localization. Let V be an irreducible algebraic variety. Then P = J(V) is a prime ideal of ℂ[X_1,...,X_n] and so ℂ[V]=ℂ[X_1,...,X_n]/J(V) is an integral domain. The localization ℂ[X_1,...,X_n]_P is a subring of ℂ(X_1,...,X_n) consisting of rational functions f/g:f,g∈ℂ[X_1,...,X_n],g∉P which are defined on a nonempty subset of V. If V = {x} is a point, then P is maximal and ℂ[X_1,...,X_n]_P=f/g:f,g∈ℂ[X_1,...,X_n],g(x) ne 0 consists of rational functions which are defined at x.
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