Noncommutative Distance Theory
Note 1
Groupoid
1
<Open covering>
Topological space X
Family of open sets Ui ⊂ X ( i ∈ I )
Open covering is sum of the family X = ∪i ∈ I Ui
On topological space, refer to the next.
On open covering, refer to the next.
2
<Spectrum>
Set of prime ideals (≠1) over commutative ring A that has unit element 1 is spectrum of A. The expression is Spec A.
On prime ideal, refer to the next.
3
<Affine open covering>
X = { Ui = Spec Ai }
On affine space, refer to the next.
4
<Noetherian ring>
All the ideals of commutative ring R are generated from finite elements. R is Noetherian ring.
On Noetherian ring, refer to the next.
5
<Locally Noetherian scheme>
Each Ai of affine open covering at X is Noetherian ring. X is locally Noetherian scheme.
6
<Noetherian scheme>
When locally Noetherian scheme X is compact as topological space, X is Noetherian scheme.
7
<Groupoid>
Noetherian scheme S
Locally finite scheme over S U, R
Arrow over S s : R → U t : R → U μ : R ╳ t, U, s R →R
Groupoid ( U, R, s, t, μ)
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