Normal scheme
A scheme all local rings (cf. Local ring) of which are normal (that is, reduced and integrally closed in their ring of fractions). A normal scheme is locally irreducible; for such a scheme the concepts of a connected component and an irreducible component are the same. The set of singular points of a Noetherian normal scheme has codimension greater than 1. The following normality criterion holds [1]: A Noetherian scheme is normal if and only if two conditions are satisfied: 1) for any point
of codimension
the local ring
is regular (cf. Regular ring (in commutative algebra)); and 2) for any point
of codimension
the depth of the ring (cf. Depth of a module)
is greater than 1. Every reduced scheme
has a normal scheme
canonically connected with it (normalization). The
-scheme
is integral, but not always finite over
. However, if
is excellent (see Excellent ring), for example, if
is a scheme of finite type over a field, then
is finite over
.
References
[1] | J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1975) MR0201468 Zbl 0296.13018 |
Comments
A normalization of an irreducible algebraic variety is an irreducible normal variety
together with a regular mapping
that is finite and a birational isomorphism.
For an affine irreducible algebraic variety, is the integral closure of the ring
of regular functions on
in its field of fractions. The normalization has the following universality properties. Let
be an integral scheme (i.e.
is both reduced and irreducible, or, equivalently,
is an integral domain for all open
in
). For every normal integral scheme
and every dominant morphism
(i.e.
is dense in
),
factors uniquely through the normalization
. So also Normal analytic space.
Let be a curve and
a, possibly singular, point on
. Let
be the normalization of
and
the inverse images of
in
. These points are called the branches of
passing through
. The terminology derives from the fact that the
can be identified (in the case of varieties over
or
) with the "branches" of
passing through
. More precisely, if the
are sufficiently small complex or real neighbourhoods of the
, then some neighbourhood of
is the union of the branches
. Let
be the tangent space at
to
. Then
is some linear subspace of the tangent space to
at
. It will be either a line or a point. In the first case the branch
is called linear. The point
on
is an example of a point with two linear branches (with tangents
,
), and the point
on
gives an example of a two-fold non-linear branch.
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References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Zbl 0367.14001 |
[a2] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Sect. II.5 (Translated from Russian) MR0366917 Zbl 0284.14001 |
[a3] | H. Matsumura, "Commutative algebra" , Benjamin (1970) MR0266911 Zbl 0211.06501 |
Normal scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_scheme&oldid=49499