properties of non-archimedean valuations

properties of non-archimedean valuations

If

K

is a field, and

|

⋅

|

a nontrivial

non-archimedean valuation

(or

absolute value

) on

K

, then

|

⋅

|

has some properties that are counterintuitive (and that are false for

archimedean valuations

).

Theorem 1

.

Let

K

be a field with a non-archimedean absolute value

|

⋅

|

. For

r

>

0

a

real number

,

x

∈

K

, define

B

⁢

(

x

,

r

)

=

{

y

∈

K

∣

|

x

-

y

|

<

r

}

,

the open ball of radius

r

at

x

B

¯

⁢

(

x

,

r

)

=

{

y

∈

K

∣

|

x

-

y

|

≤

r

}

,

the closed ball of radius

r

at

x

Then

1.

B

⁢

(

x

,

r

)

is both open and closed;

2.

B

¯

⁢

(

x

,

r

)

is both open and closed;

3.

If

y

∈

B

⁢

(

x

,

r

)

(resp.

B

¯

⁢

(

x

,

r

)

) then

B

⁢

(

x

,

r

)

=

B

⁢

(

y

,

r

)

(resp.

B

¯

⁢

(

x

,

r

)

=

B

¯

⁢

(

y

,

r

)

);

4.

B

⁢

(

x

,

r

)

and

B

⁢

(

y

,

r

)

(resp.

B

¯

⁢

(

x

,

r

)

and

B

¯

⁢

(

y

,

r

)

) are either identical or disjoint;

5.

If

B

1

=

B

⁢

(

x

,

r

)

and

B

2

=

B

⁢

(

y

,

s

)

are not disjoint, then either

B

1

⊂

B

2

or

B

2

⊂

B

1

;

6.

If

(

x

n

)

is a

sequence

of elements of

K

with

lim

n

→

∞

⁡

x

n

=

0

, then

∑

n

=

1

∞

x

n

is Cauchy (and thus if

K

is

complete

, a sufficient condition for convergence of a series is that the terms tend to zero)

Proof.

We start by proving (3). Suppose

y

∈

B

⁢

(

x

,

r

)

. If

z

∈

B

⁢

(

x

,

r

)

, then since the absolute value is non-archimedean, we have

|

z

-

y

|

=

|

(

z

-

x

)

+

(

x

-

y

)

|

≤

max

⁡

(

|

z

-

x

|

,

|

x

-

y

|

)

<

r

so that

z

∈

B

⁢

(

y

,

r

)

. Clearly

x

∈

B

⁢

(

y

,

r

)

, so reversing the roles of

x

and

y

, we see that

B

⁢

(

x

,

r

)

=

B

⁢

(

y

,

r

)

. Finally, replacing

B

by

B

¯

and

<

by

≤

, we get equality of

closed balls

as well.

(4) is now trivial: If

B

⁢

(

x

,

r

)

∩

B

⁢

(

y

,

r

)

≠

∅

, choose

z

∈

B

⁢

(

x

,

r

)

∩

B

⁢

(

y

,

r

)

; then by (3),

B

⁢

(

x

,

r

)

=

B

⁢

(

z

,

r

)

=

B

⁢

(

y

,

r

)

. An identical

argument

proves the result for closed balls.

To prove (5), choose

z

∈

B

1

∩

B

2

. Assume first that

r

≤

s

; then

B

⁢

(

z

,

r

)

=

B

1

, and

B

⁢

(

z

,

r

)

⊂

B

⁢

(

z

,

s

)

=

B

2

, so that

B

1

⊂

B

2

. If

s

≤

r

, then we have identically that

B

2

⊂

B

1

. (Note that (4) is a special case when

r

=

s

).

(1) and (2) now follow: for (1), note that

B

⁢

(

x

,

r

)

is obviously open; its complement consists of a union of open balls of radius

r

disjoint with

B

⁢

(

x

,

r

)

and its complement is therefore open. Thus

B

⁢

(

x

,

r

)

is closed. For (2),

B

¯

⁢

(

x

,

r

)

is obviously closed; to see that it is open, take any

y

∈

B

¯

⁢

(

x

,

r

)

; then

B

¯

⁢

(

x

,

r

)

=

B

¯

⁢

(

y

,

r

)

and thus

B

⁢

(

y

,

s

)

⊂

B

¯

⁢

(

y

,

r

)

for

s

<

r

is an open neighborhood of

y

contained in

B

¯

⁢

(

x

,

r

)

, which is therefore open.

Finally, to prove (6), we must show that given

ϵ

, we can find

N

>

0

sufficiently large such that

|

∑

i

=

m

n

x

i

|

<

ϵ

whenever

m

,

n

>

N

. Simply choose

N

such that

|

x

i

|

<

ϵ

for

i

>

N

; then

|

∑

i

=

m

n

x

i

|

≤

max

⁡

(

|

x

m

|

,

…

,

|

x

n

|

)

<

ϵ

Title

properties of non-archimedean valuations

Canonical name

PropertiesOfNonarchimedeanValuations

Date of creation

2013-03-22 18:01:11

Last modified on

2013-03-22 18:01:11

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 13F30

Classification

msc 13A18

Classification

msc 12J20

Classification

msc 11R99

Related topic

CompleteUltrametricField

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