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Resolutions of singularities.
Yoshifumi Tsuchimoto
PROPOSITION 05.1
For any ring
, the map
is proper.
PROOF..
See http://amathew.wordpress.com/2010/10/23/a-projective-morphism-is-proper/.
COROLLARY 05.2
For any ring
and for any
-graded ring
which is generated by a finite subset of
over the ring
, the map
is proper.
DEFINITION 05.4
Let
be a commutative ring. Let
be its prime ideal. Then we define
the localization of
with respect to
by
DEFINITION 05.5
A commutative ring
is said to be a local ring if it has only one
maximal ideal.
LEMMA 05.7
- Let
be a local ring. Then the maximal ideal of
coincides with
.
- A commutative ring
is a local ring if and only if
the set
of non-units of
forms an ideal of
.
PROOF..
(1) Assume
is a local ring with the maximal ideal
.
Then for any element
,
an ideal
is an ideal of
.
By Zorn's lemma, we know that
is contained in a maximal ideal of
.
From the assumption, the maximal ideal should be
.
Therefore, we have
which shows that
The converse inclusion being obvious (why?), we have
(2) The ``only if'' part is an easy corollary of (1).
The ``if'' part is also easy.
COROLLARY 05.8
Let
be a commutative ring. Let
its prime ideal. Then
is
a local ring with the only maximal ideal
.
DEFINITION 05.9
Let
be local rings
with maximal ideals
respectively.
A local homomorphism
is a homomorphism which
preserves maximal ideals. That means, a homomorphism
is said to be loc
al
if
EXAMPLE 05.10 (of NOT being a local homomorphism)
is not a local homomorphism.
PROPOSITION 05.12
Let
be a commutative ring. let
be an ideal of
such that
. Then there exists a maximal ideal
of
which contains
.
PROOF..
Since
, there exists elements
such that
holds. In a matrix notation, this may be rewritten as
with
,
.
Using the unit matrix
one may also write :
Now let
be the adjugate matrix of
. In other words, it
is a matrix which satisfies
Then we have
On the other hand, since
modulo
, we have
for some
. This
clearly satisfies
We need a criterion for regularity. Instead of developing the
vast theory of regular rings, we site here the following theorem:
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2014-05-30