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,
, 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.

(
such that
).

(
).
DEFINITION 03.3
Let
be a projective system of
topological rings. Then we say that a
projective limit
of
is given if

is a topological ring.

is a continuous homomorphism.

for
such that
.)

is a universal object among objects which satisfy (1)(3).
The ``universal'' here means the following:
If
satisfies

is a topological ring.

is a continuous homomorphism.

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?)
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20080610