So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field with an absolute value
like the rationals with the usual absolute value.
Completions
Definition 1 The completion
of
is defined as the set of equivalence classes of Cauchy sequences:
- A Cauchy sequence
satisfies
as
.
- Two Cauchy sequences
are equivalent if
as
.
First off, is a field, since we can add or multiply Cauchy sequences termwise; division is also allowable if the sequence stays away from zero. There is a bit of justification to check here, but it is straighforward. Also,
had an absolute value, so we want to put on on
. If
, define
. Third, there is a natural map
and the image of
is dense.
There are several important examples of this, of which the most basic are:
Example 1 The completion of
with respect to the usual absolute value is the real numbers.
Example 2 The completion of
with respect to the
-adic absolute value is the
-adic numbers
.
The second case is more representative of what we care about: completions with respect to nonarchimedean (especially discrete) valuations. By the general criterion (testing that integers have absolute value at most 1), completions preserve nonarchimedeanness. Next, here is a frequently used lemma about nonarchimedean fields:
Lemma 2 Let
be a field with a nonarchimedean absolute value
. Then if
and
, then
.
(Two elements very close together have the same absolute value. Or, any disk in a nonarchimedean field has each interior point as a center.)
Indeed, . Similarly
, and since
we can write in the second
.
Corollary 3 If
is discrete on
, it is discrete on
.
Indeed, if is the value group (absolute values of nonzero elments) of
, then it is the value group of
since
is dense in
.
Completions of Rings
Now, time to connect this idea of completion with the previous one.
Take a field with a discrete valuation
and its completion
. We can take the ring of integers
and
, and their maximal ideals
.
I claim that is the completion of
with respect to the
-adic topology. This follows because
consists of equivalence classes of sequences
of elements of
, the limit of whose absolute values
is
. This means from some point on, the
by discreteness, so wlog all the
. This is just the definition of an element of the completion of
. I leave the remaining details to the reader.
(To avoid discreteness, for sequences with
that do not go into
, replace it by the equivalent
—this way one replaces it with a sequence that lies in
.)
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