Monday, 30 April 2018

Noncommutative Distance Theory Note 1 Groupoid



Note 1
Groupoid


1
<Open covering>
Topological space     X
Family of open sets     Ui  X ( 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 A}
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 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     UR
Arrow over S     s :  U     R  U     μ R  t, U, s R R
Groupoid    ( UR, s, t, μ)

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