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]
Tokyo July 28 2007
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