Monday 5 January 2015

Linguistic Note 9 Homomorphism / July 28 2007

Linguistic Note

9

Homomorphism


    TANAKA Akio

1
Ring     R
Map from R-additive group M to R-additive group N     φ M   N
Condition that φ is R-additive group’s homomorphism
Against arbitrary u M and arbitrary a  R
φ u + v ) =φ u ) +  φ v )
φ au ) = aφ u )
When φ is bijective homomorphism, φ is isomorphism and M and are isomorphic.
Isomorphic M and N      M  N
Against homomorphism     φ M   N
Kernel     Ker ( φ ) = { u  M |  φ u ) = 0 }
Image      Im ( φ ) = { φ u ) | u  M }
Cokernel (Quotient module)    Coker ( φ ) = N / Im ( φ )

2
Topological space     X
Arbitrary opened set     U
Commutative group     F ( U )
Two opened sets      V
Homomorphism τ UV F ( V )    F ( U )
When homomorphism τ’ s condition is below, {  F ( U ), τ UV } becomes commutative group’s presheaf on X.
F ( 0 ) = { 0 }
τUU  = idU  ( identity map)
U V W 
τUW  = τUV   τVW

3
Presheaf      FGH
Sequence of homomorphism     F  f   G   g   H
When Im f = Ker g is made up, sequence is called exact sequence of presheaf.

4
Topological space     X
Ring’s sheaf on X    OX   
Ringed space     ( XOX )
Structure sheaf        OX
Free sheaf     direct sum of structure sheaf       OX

5
Scheme    X
O–additive sheaf      F
Next condition makes F as quasi-coherent sheaf.
Arbitrary point P  X
P’s neighborhood     U  X
Free sheaf   OUΛ  → F|  0
Next additional condition makes as coherent sheaf.
X is algebraic scheme.
Λs are finite set.

6
Complex analytic space becomes coherent sheaf by upper (5)’s alike operation.

[Note]
Coherent sheaf of complex analytic space is helpful to the solution for the present problem of language’s model.  

[References]
Language and Spacetime     Shift of Time     Tokyo April 20, 2007
Language and Spacetime     Stability of Language     Tokyo April 30, 2007
Linguistic Note     Complex Analytic Space     Tokyo July 21, 2007

Tokyo July 28 2007
Sekinan Research Field of Language

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