9
Homomorphism
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, v ∈ 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 N 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 U ⊂ 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 F, G, H
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 ( X, OX )
Structure sheaf OX
Free sheaf direct sum of structure sheaf ⊕OX
5
Scheme X
OX –additive sheaf F
Next condition makes F as quasi-coherent sheaf.
Arbitrary point P ∈ X
P’s neighborhood U ⊂ X
Free sheaf OUΛ → F|U → 0
Next additional condition makes F 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
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