  We use the classical Ax-Kochen-Ershov analysis of the model theory of
Henselian fields to bring out some model-theoretical aspects of the structure
sheaf of the spectrum of Z^ and the ring of finite ad\`eles over Q. We show
that various structures associated to a prime ideal, such as quotients and
localizations, are well understood model-theoretically, and they are closely
connected to ultrafilters on the set of standard primes.
