Stable and Unstable of Language For the Supposition of KARCEVSKIJ Sergej Completion of Language

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 

Ring sheaf for 

U →Oan(U)

Complex analitic manifold Cann

Algebraic manifold An Multinominal of Cann

Ideal of multinominal ring a  [x1x2, ..., xn]

V(a) = {(a1a2, ..., an Cn (a1a2, ..., an) = 0,  a }

Whole closed set of V(a

Fundamental open set D(f) = {(a1a2, ..., an Cn | (a1a2, ..., an) ≠ 0}

Arbitrary family of open set {Ui} 

Easy theaf F 

Zariski yopological space 

Ring theaf O

Affine space An = (  , O)

Ring R

Set of whole maximum ideal Spm R1

Spm R Specrum of R

<Proposition>

Spm is Noether- like.

<Proposition>

is integral domain.

Whole of open stes without nul set Ux

Quotient field K

Mapping from Uto whole partial set of O

O(V(a)c) =  Rf

c expresses complmentary 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 theaf O

is called structure sheaf of affine algebraic variety.

Ring homomorphism between definite generative integral domains 

Upper is expressed by  .

Ring holomorphism OX(U) → OY((t )-1U)

Morphism from affine algebraic variety Y to ( OX(U) → OY((t )-1U), X)

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 qits quotienft 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 muximum chain is equel to tarnscendental deminsion of function field k(X).

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

<Definition>

Defined generative field over k K

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

(i) Each Ui is affine algebraic variety tha has quotient K .

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

3.

<Definition>

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

Elements  that defines surjective homomorphism is expressed by  .

Image  of cloed embedding defined by  is called diagonal.

<Definition>

Field K

Ringed space that have common whole set K (A, OA) (B, OB)

Topological space C

Open embedding 

A and B have common partial set C.

Topological space glued A and B by C 

Easy sheaf over OW

ahere, arbitrary open set Ø ≠ 

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

<Definition>

Intedgral domains that have common quotent field K R, S

Element R am ≠ 0

Element S bn ≠ 0

Spm T  Spm R, Spm T  Spm S

Glue defined by the upper is called simple.

<Definition>

Affine algebraic varieties U1U2

Common open set of U1UUC

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 Variery'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>

Fuctors 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 toalgebraic 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 trasformation  :  .

(ii) Natural transformation  ,

Morphism that satisfies  is existent uniquely.

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

<Definition>

Algebaraic 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 Pn

(2n+1) dimensional spherical surface {}

Pn has continuous surjection from .

Pn is conpact.

<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 langauge is expressed by Pn.

Word is expressed by projective algebraic manifold.

Meaning of word is expressed by closed embedding.

 

 

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