Arithemetic Geometry 1 Dimension of Word

Arithemetic Geometry 1

Dimension of Word

TANAKA Akio
24 December 2012

ifbetrue / Tokyo

1. 

Theorem by C. Soulé, Lectures on Arakelov Geometry, 1992. 

 is morphism between regular arithmetic varieties.

 is pullback.

When  are morphism between regular arithmetic varieties, the next is concluded.

 .

2.

Interpretation of the upper theorem.

Word : .

Decomposition of word : , named pullback.

Decomposed meaning unit in word :  and .

3.

Pullback is defined by the next from Efton Park, Complex Topological K-Theory, 2008.

Let  and be topological spaces, let  be a vector bundle over , and suppose  is a continuous map,

Define

4.

Interpretation of the upper theorem 2.

Word can be deposed to meaning units by pullback.

Meaning unit also become pullback.

5.

Pullback contains order function ord.

ord contains length.

Here length is longitude of composition series.

Here composition series is defined by the next.

A is commutative ring.

M is A module.

If A be field,  is M's dimension over A.

6.

From 5. leads the next supposition.

When word is decomposed to meaning unit, each unit has dimension that determines word's dimension.

(Here ends the paper.)