Linguistic Note 13 Zariski Topology / August 24, 2007

Linguistic Note

13

Zariski Topology

     TANAKA Akio

1

Algebraically closed field     k

Positive integer     n

n-variable polynomial     f ( T₁, …..,T_n ) ∈ k [ T₁,…..,T_n ]

Ideal generated by equation system  f₁,….., f_s      I = ( f₁, ….., f_s ) ∈ k [ T₁,     ,T_n ]

Residue ring     A = k [ T₁,     ,T_n ] / I

All the prime ideals of A     Spec A

Affine algebraic scheme     X = ( Spec A, A )

Dimension of X      dim X

0 ≦ dim X ≦ n

I = { f₁,….., f_s } = 0

A = k [ T₁,     ,T_n ]

Affine algebraic scheme     A_n = ( Spec k [ T₁,     ,T_n ], k [ T₁,     ,T_n ] )

n- dimension affine space     A_n

Quotient field of integral domain A     Q

Transcendence degree of Q(A)     dim X

Affine algebraic variant     X

2

Affine algebraic scheme     X = ( Spec A, A )     Y = ( Spec B, B)

Isomorphism of k-algebra     φ : B → A

Prime ideal     P ∈ Spec A

Prime ideal of B     φ’ = φ^-1 ( P )

Regular map from X to Y     Φ = ( φ’, φ )

Line on k     A¹ = ( Spec k [T₁], k [T₂] )

Regular function on X     Regular map from X to A¹

3

Affine algebraic scheme     X = ( Spec A, A )

Element of Spec A     x, y, …

Prime ideal of A     P_x, P_y,…

u ∈ A

Subset of Spec A     D ( u )

Topology of Spec A     Zariski topology

Topological space that has arbitrary open subset gives quasi-compact.     Noetherian space

[Note]

Affine algebraic scheme or Zariski topology may be useful to describe language on the whole or symbolic world expressed by language.

Tokyo August 24, 2007

Sekinan Research Field of Language