Note 2
Polydisk
Bridge between Ring and Brane
1
For algebraic approach toward language, analytic space is prepared
Polydisk Dn that is useful* to identification of some different figures is defined by the following.
Dn = { ( x1, …, xn ) ∈ For all i, xi < 1 }
2
Dn has real topology that is stronger than Zariski topology.
On real topology, refer to the next.
On Zariski topology, refer to the next.
3
Dn has structure sheaf OhDn that has complex analytic function.
On sheaf and structure sheaf, refer to the next.
4
Finite complex analytic function h1, …, hm ∈ ( Dn, OhDn )
5
Sheaf of ideal I = ( h1, …, hm ) OhDn
On ideal, refer to the next.
6
Subset M = V ( I ) = { P ∈ Dn ; All of j, hj ( P ) = 0 }
7
Sheaf OhM = OhDn / I
8
Local ringed space ( M, OhM ) is the model of complex analytic space.
On local ringed space, refer to the next.
On complex analytic space, refer to the next.
9
Neighborhood ( U, OhM | U ) is isomorphism of the model.
On isomorphism, refer to the next.
[Note]
*Usefulness of polydisk will be developed at Distance Theory Algebraically Supplemented Brane Simplified Model on extension of circle’s identification as line segment.
On circle, refer to the next.
Tokyo October 31, 2007
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