Linguistic Premise Premise of Algebraic Linguistics 4-2 / September 29, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 4-2

　　  TANAKA Akio

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     ( Sh_x ) ; all the sheafs in aberian group over topological space X    

11 <ringed space>

Topological space     X

Ring’s sheaf over X     O_X

Ringed space     ( X, O_X )

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, …, x_n ]

Ideal     I ∈ C [ x₁, …, x_n ]

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 ( ∑_λI_λ ) = ∩_λV₀ (I_λ)

 Cⁿ that is defined by the upper conditions is Zariski topology.

Usual Cⁿ is real topology.

13 <Hilbert basis theorem>

Polynomial ring C [ x₁, …, x_n ]

Ideal of the polynomial ring     I

Finite polynomial      h₁, …, h_m ∈ C [ x₁, …, x_n ]

Generated I     I  = (h₁, …, h_m )

14 <Hilbert zero point theorem>

Set of all the points in affine space Cⁿ     P = ( a₁, …, a_n )

Polynomial ring    C [ x₁, …, x_n ]

Maximu ideal of the polynomial ring     m_P = (x₁-a₁, …, x_n- a_n )

There is one to one correspondence between P and m_P.

Tokyo September 29, 2007

Sekinan Research Field of Language