Completion of Language 2011

  

Stable and Unstable of Language

For the Supposition of KARCEVSKIJ Sergej

Completion of Language

TANAKA Akio

September 23, 2011

[Preparation]

1.

n dimensional complex space Cⁿ

Open set 

Whole holomorphic function over U 

Ring sheaf for 

U →O^an(U)

Complex analytic manifold C_anⁿ

Algebraic manifold Aⁿ Multinomial of C_anⁿ

Ideal of multinomial ring a  [x₁, x₂, ..., x_n]

V(a) = {(a₁, a₂, ..., a_n)  Cⁿ f (a₁, a₂, ..., a_n) = 0,  a }

Whole closed set of V(a) 

Fundamental open set D(f) = {(a₁, a₂, ..., a_n)  Cⁿ | f (a₁, a₂, ..., a_n) ≠ 0}

Arbitrary family of open set {Ui} 

Easy sheaf F 

Zariski topological space 

Ring sheaf O

Affine space Aⁿ = (  , O)

Ring R

Set of whole maximum ideal Spm R1

Spm R Spectrum of R

<Proposition>

Spm R is Norther- like.

◊

<Proposition>

R is integral domain.

Whole of open sets without null set U_x

Quotient field K

Mapping from U_x to whole partial set of K O

O(V(a)^c) =  R_f

^c expresses complimentary set.

O is easy sheaf of ring over Spm R that is whole set K.

◊

<Definition>

R is finite generative integral domain over k.

Triple (i) (ii) (iii) is called affine algebraic variety.

(i) Set Spm R

(ii) Zariski topology

(iii) Ring's sheaf O

O is called structure sheaf of affine algebraic variety.

◊

Ring homomorphism between definite generative integral domains 

Upper is expressed by  .

Ring holomorphism O_X(U) → O_Y((^t )^-1U)

Morphism from affine algebraic variety Y to X ( O_X(U) → O_Y((^t )^-1U), X→Y )

When  is surjection, ^t is isomorphism overclosed partial set defined by p= Ker  .

Upper is called to closed immersion.

2.

Ring holomorphism 

Morphism between affine algebraic varieties 

Kernel of  p

Image of  

<Definition>

It is called that when  is injection is dominant.

◊

<Definition>

R is medium ring between S and qi's quotient field K.

When  that is given by natural injection  is isomorphism over open set,  is called open immersion.

◊

<Definition>

When X is algebraic variety, longitude of maximum chain is equal to transcendental dimension of function field k(X).

It is called dimension of algebraic variety X, expressed by dim X.

◊

<Definition>

Defined generative field over k K

Space ( X, O_x )added ring that is whole sets of K that has open covers {U_i} satisfies next condition is called algebraic variety.

(i) Each U_i is affine algebraic variety that has quotient K .

(ii) For each i, j  I, intersection  is open partita; set of  .

◊

3.

<Definition>

Tensor product between ring and itself becomes ring by each elements products.

Elements  that defines surjective homomorphism is expressed by  .

Image  of closed embedding defined by  is called diagonal.

◊

<Definition>

Field K

Ringed space that have common whole set K (A, O_A) (B, O_B)

Topological space C

Open embedding 

A and B have common partial set C.

Topological space glued A and B by C 

Easy sheaf over W O_W

adhere, arbitrary open set Ø ≠ 

Ringed space  is called glue of A and B by C.

◊

<Definition>

Integral domains that have common quotient field K R, S

Element R a_m ≠ 0

Element S b_n ≠ 0

Spm T  Spm R, Spm T  Spm S

Glue defined by the upper is called simple.

◊

<Definition>

Affine algebraic varieties U¹, U²

Common open set of U¹, U² U^C

Diagonal embedding 

When the upper is closed set, glue is called separated.

◊

<Proposition>

For simple glue , next is equivalent.

(*) It is separated.

(**) Ring  is generated by R and S.

◊

<Definition>

R and S are integral domains that have common quotient field K.

For partila ring T=RS generated by R and S, when <Definition> simple is satisfied, it is called "Spm R ad Spm S are simple glue."

◊

<Sample>

Projective space Pⁿ is simple glue.

◊

<Definition>

Algebraic Variety's morphism is glue of affine algebraic variety's ring homomorphism image.

Algebraic direct product is direct product of affine algebraic variety.

◊

4.

Affine algebraic variety X

Ring over k R

 is called R value point of X.

Whole  is called set of R value point of X, expressed by X(R).

Ring homomorphism over k 

X(f) := X(R)X(S)

Ring homomorphism , 

<Definition>

 is function from ring category over k to category of set.

◊

<Definition>

Functors from ring category to set category F, G

Ring R

Family of  over ring R {}

{} has functional morphism.

Functors F,G have isomorphism ( or natural transformation).

Functor from ring's category to set's category that is isomorphic to algebraic variety, is called representable or represent by X, or fine moduli.

◊

<Definition>

Functor from ring's category to set's category F

When  satisfies the next conditions, X is called corse moduli.

(i) There is natural transformation  :  .

(ii) Natural transformation  ,

Morphism that satisfies  is existent uniquely.

(iii) For algebraic close field k'  k, (k') is always bijection.

◊

<Definition>

Algebraic variety G that  is functor to group's category is called algebraic group.

◊

<Definition>

Finite generative ring over k A

When G = Spm A satisfies 3 conditions on the next triad is called affine algebraic group.

Triad

Conditions

(i)  are commutative for .

(ii)There is identity map for A.

(iii) There is coincident with  for A.

◊

5.

Projective space over C Pⁿ

(2n+1) dimensional spherical surface {}

Pⁿ has continuous surjection from .

Pⁿ is compact.

<Definition>

Map  is called closed map when  is closed set image  becomes closed set.

◊

<Definition>

Algebraic variety X is called complete when projection  is closed map for arbitrary manifold Y.

◊

<Definition>

Morphism from complete algebraic manifold X to separated algebraic manifold Y,  is closed map.

◊

<Proposition>

Projective space Pⁿ is complete.

◊

<System>

Algebraic manifold that has closed embedding at Pⁿ is complete.

This algebraic manifold is called projective algebraic manifold.

[Interpretation]

Here language is expressed by P^n.

Word is expressed by projective algebraic manifold.

Meaning of word is expressed by closed embedding.

This paper has been published by Sekinan Research Field of Language.
2011 by The Sekinan Research Field of Language