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Zariski Topology
1
Algebraically closed field k
Positive integer n
n-variable polynomial f ( T1, …..,Tn ) ∈ k [ T1,…..,Tn ]
Ideal generated by equation system f1,….., fs I = ( f1, ….., fs ) ∈ k [ T1, ,Tn ]
Residue ring A = k [ T1, ,Tn ] / 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,….., fs } = 0
A = k [ T1, ,Tn ]
Affine algebraic scheme An = ( Spec k [ T1, ,Tn ], k [ T1, ,Tn ] )
n- dimension affine space An
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 Px, Py,…
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
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