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.
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 .)