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Resolutions of singularities.
Yoshifumi Tsuchimoto
DEFINITION 03.1
For any commutative ring

, we define its
spectrum as

is a prime ideal of $A$
For any subset

of

we define
Then we may topologize

in such a way that the closed
sets are sets of the form

for some

. Namely,

closed
We refer to the topology as the
Zariski topology.
EXERCISE 03.1
Prove that Zariski topology is indeed a topolgy.
That means, the collection

satisfies the axiom of closed sets.
PROPOSITION 03.3
For any ring homomorphism
, we have a map
It is continuous with respect to the Zariski topology.
DEFINITION 03.5
Let

be a topological space. A closed set

of

is said to be
reducible if there exist closed sets

and

such that
holds.

is said to be
irreducible if it is not reducible.
DEFINITION 03.6
Let

be an ideal of a ring

. Then we define its
radical to be

such that
PROPOSITION 03.7
Let
be a ring. Then;
- For any ideal
of
, we have
.
- For two ideals
,
of
,
holds if and only if
.
- For an ideal
of
,
is irreducible if and only if
is a prime ideal.
It is knwon that
has a structure of ``locally ringed space''.
A locally ringed space which locally lookes like an affine spectrum of a ring
is called a scheme.
DEFINITION 03.8
Let

be a

-graded ring.
We put

.
We define

It is known that
carries a ringed space strucure on it and
that it is a scheme.
DEFINITION 03.9
Let

be a ring. Let

be an ideal of

.
The scheme

associated to the
graded ring

is called
the blowing up of

with respect to

.
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2014-04-25