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Definition of <stalk and germ>
Topological space X
Point on X x
Presheaf of topological space X F
Set of all the opened neighborhood U
U, V ⊂ U U ⊂ V
Order of U U ≺ V
rUV : F ( U ) → F ( V )
Set of F ( U )
direct sum of the set ⎿⏌F ( U )
Elements of s, s’
∈ F ( U ) s’ ∈ F ( V )
W ⊂U ∩V
s | Ｗ = s’ | Ｗ
s ∼ s’
Equivalence class contained s ∈ F ( U ) sx
Homomorphism F ( U ) → F x
F x is stalk on x of F.
sx is germ on x of F.
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Definition of <sheaf space>
Presheaf of commutative ring F
F d = ⎿⏌x∈X F x
Map p : F d → X
F d ×XF d = { a, b | p(a) = p(b) }
F d ×X F d ⊂ F d × F d
Conditions of F d ×X F d
(1)
p is local isomorphism.
(2)
a) m : F d ×X F d ∋(a, b) ↦ a + b ∈ F d
b) l : F d ∋ - a ↦ F d
F d is sheaf space of commutative group on X.
Tokyo September 17, 2007
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