Noncommutative Distance Theory Note 1 Groupoid

Noncommutative Distance Theory

Note 1

Groupoid

TANAKA Akio

1

<Open covering>

Topological space     X

Family of open sets     U_i ⊂ X ( i ∈ I )

Open covering is sum of the family　     X = ∪_i ∈ I U_i

On topological space, refer to the next.

Distance Theory Algebraically Supplemented / 1 Distance Preparatory Consideration / Tokyo October 8, 2007

On open covering, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 1-3 / Tokyo September 17, 2007

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.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 2-1 / Tokyo September 20, 2007

3

<Affine open covering>

X = { U_i = Spec A_i }

On affine space, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 3-4 / Tokyo September 24, 2007

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.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 3-1 / Tokyo September 23, 2007

5

<Locally Noetherian scheme>

Each A_i 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)

Tokyo November 30, 2007

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