# , , and the ring of Witt vectors

Yoshifumi Tsuchimoto

No.02:

Let be a prime (base''). We would like to introduce a metric on such that

:small     is divisible by powers of

Namely:

DEFINITION 02.1   Let be a prime number.
1. We define a -adic norm on as follows.

2. We define a -adic distance on as follows.

LEMMA 02.2
1. enjoys the following properties.
1. .
2. is a metric space.

DEFINITION 02.3   A metric space is said to be complete if every Cauchy sequence of converges to an element of .

THEOREM 02.4   Let be a metric space. There exists a complete metric space with an isometry such that is dense in . Furthermore, is unique up to a unique isometry.

DEFINITION 02.5   Let be a metric space. We call as in the above theorem the completion of .

DEFINITION 02.6   Let be a prime number. We denote the completion of by and call it the ring of -addic integers. Thus elements of are -addic integers.

THEOREM 02.7   has a unique structure of a topological ring. Namely,
1. There exists unique continuous maps

(multiplication) which are extensions of the usual addition and multiplication of .
2. is a commutative associative ring.

DEFINITION 02.8   Let be a prime number. For any sequence such that , we consider a sequence defined by

Then the sequence is a Cauchy sequence in . We denote the limit of the sequence as

EXERCISE 02.1   compute