8 <image of sheaf>
Homomorphism fo sheaf f : F → G
Image of sheaf Im ( f ) = associated sheaf F / Ker ( f )
9 < exact sequence of sheaf>
Sheaf F
Exact sequence of sheaf string of homomorphism Im ( f i-1 ) = Ker ( i )
10 <aberlian category>
Aberian category ( Shx ) ; all the sheafs in aberian group over topological space X
11 <ringed space>
Topological space X
Ring’s sheaf over X OX
Ringed space ( X, OX )
Local ring ring that has only one maximum ideal
Point of ringed space P
Local ringed space stalk at the point being local ring
12 <closed algebraic subset>
n-dimensional affine space Cn
n-dimensional polynomial ring C [ x1, …, xn ]
Ideal I ∈ C [ x1, …, xn ]
Subset of Cn
Common zero poin set V0 ( I ) = { P ∈ Cn ; h ( P ) = 0, ∀h ∈ I }
The subset is closed algebraic subset that satisfies closed set axiom.
V0 ( 1 ) = 0
V0 ( 0 ) = Cn
V0 ( IJ ) = V0 ( I ) ⋃ V0 ( J )
V0 ( ∑λIλ ) = ∩λV0 (Iλ)
Cn that is defined by the upper conditions is Zariski topology.
Usual Cn is real topology.
13 <Hilbert basis theorem>
Polynomial ring C [ x1, …, xn ]
Ideal of the polynomial ring I
Finite polynomial h1, …, hm ∈ C [ x1, …, xn ]
Generated I I = (h1, …, hm )
14 <Hilbert zero point theorem>
Set of all the points in affine space Cn P = ( a1, …, an )
Polynomial ring C [ x1, …, xn ]
Maximu ideal of the polynomial ring mP = (x1-a1, …, xn- an )
There is one to one correspondence between P and mP.
Tokyo September 29, 2007
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