For the Supposition of KARCEVSKIJ Sergej
Completion of Language
September 23, 2011
[Preparation]
1.
n dimensional complex space Cn
Open set 
Whole holomorphic function over U 
Ring sheaf for
U →Oan(U)
Complex analitic manifold Cann
Algebraic manifold An Multinominal of Cann
Ideal of multinominal ring a 
[x1, x2, ..., xn]
[x1, x2, ..., xn]
V(a) = {(a1, a2, ..., an) 
Cn f (a1, a2, ..., an) = 0, 
a }
Cn f (a1, a2, ..., an) = 0, a }
Whole closed set of V(a) 
Fundamental open set D(f) = {(a1, a2, ..., an) Cn | f (a1, a2, ..., an) ≠ 0}
Cn | f (a1, a2, ..., an) ≠ 0}
Arbitrary family of open set {Ui} 
Easy theaf F 
Zariski yopological space
Ring theaf O
Affine space An = ( , O)
, O)
Ring R
Set of whole maximum ideal Spm R1
Spm R Specrum of R
<Proposition>
Spm R is Noether- like.
◊
<Proposition>
R is integral domain.
Whole of open stes without nul set Ux
Quotient field K
Mapping from Ux to whole partial set of K O
O(V(a)c) = 
Rf
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
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)
)-1U)
Morphism from affine algebraic variety Y to X ( OX(U) → OY((t )-1U), X→Y )
)-1U), X→Y )
When 
is surjection, t is isomorphism overclosed partial set defined by p= Ker 
.
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
p
Image of 
<Definition>
It is called that when is injection 
is dominant.
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.
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.
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 
.
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 
.
that defines surjective homomorphism is expressed by .
Image 
of cloed embedding defined by is called diagonal.
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 W OW
ahere, arbitrary open set Ø ≠ 
Ringed space 
is called glue of A and B by C.
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
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 UC
Diagonal embedding 
When the upper is closed set, glue is called separated.
◊
<Proposition>
For simple glue 
, next is equivalent.
, next is equivalent.
(*) It is separated.
(**) Ring 
is generated by R and S.
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).
is called set of R value point of X, expressed by X(R).
Ring homomorphism over k 
X(f) := X(R)
X(S)
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 {
}
over ring R {}
{} has functional morphism.
} 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.
satisfies the next conditions, X is called corse moduli.
(i) There is natural trasformation : 
.
: .
(ii) Natural transformation 
,
,
Morphism that satisfies 
is existent uniquely.
is existent uniquely.
(iii) For algebraic close field k' 
k, 
(k') is always bijection.
k, (k') is always bijection.
◊
<Definition>
Algebaraic variety G that 
is functor to group's category is called algebraic group.
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 
.
are commutative for .
(ii)There is identity map for A.
(iii) There is coincident with 
for A.
for A.
◊
5.
Projective space over C 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.
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.
is closed map for arbitrary manifold Y.
◊
<Definition>
Morphism from complete algebraic manifold X to separated algebraic manifold Y, 
is closed map.
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.
This paper has been published by Sekinan Research Field of Language.
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