# , , and the ring of Witt vectors

Yoshifumi Tsuchimoto

No.03:

DEFINITION 03.1   An ordered set is said to be directed if for all there exists such that and .

DEFINITION 03.2   Let be a directed set. Let be a family of topological rings. Assume we are given for each pair of elements such that , a continuous homomorphisms

We say that such a system is a projective system of topological rings if it satisfies the following axioms.
1.     ( such that ).
2.     ( ).

DEFINITION 03.3   Let be a projective system of topological rings. Then we say that a projective limit of is given if
1. is a topological ring.
2. is a continuous homomorphism.
3. for such that .)
4. is a universal object among objects which satisfy (1)-(3).

The universal'' here means the following: If satisfies

1. is a topological ring.
2. is a continuous homomorphism.
3. for such that .)
Then there exists a unique continuous homomorphism

such that

PROPOSITION 03.4   For any projective system of topological rings, a projective limit of the system exists. It is unique up to a unique isomorphism. (Hence we may call it the projective limit of the system.)

DEFINITION 03.5   For any projective system of topological rings, We denote the projective limit of it by

Note: projective limits of systems of topological spaces, rings, groups, modules, and so on, are defined in a similar manner.

THEOREM 03.6

as a topological ring.

COROLLARY 03.7   is a compact space.

Note: There are several ways to prove the result of the above corollary. For example, the ring with the metric is easily shown to be totally bounded.

PROPOSITION 03.8   Each element of is expressed uniquely as

EXERCISE 03.1   Is invertible in ? (Hint: use formal expansion

is it possible to write down a correct proof to see that the result is true?)