2 NON-ARCHIMEDEAN ANALYTIC GEOMETRY: FIRST STEPS
over k with the property that their resolvents are analytic on the complement of
the spectrum defined in the usual way as a subset of k (the field was assumed
to be algebraically closed). When I understood that, a very natural idea came
to me. In the classical situation, the spectrum of an operator coincides with the
complement of the analyticity set of its resolvent. Can one find out what the
spectrum of a non-archimedean operator is by investigating a similar analyticity
set of its resolvent? That such a resolvent takes values in the Banach algebra of
bounded linear operators was not a problem. It was the notion of analyticity set
that was not clear. But at least, one could try to investigate sets from a reasonable
class at which the resolvent is analytic. For example, the resolvent is analytic at the
complement of a closed (or open) disc with center at zero of a big enough radius.
At the beginning, I considered the so-called quasiconnected (and infracon-
nected) sets introduced by M. Krasner in 1940s, and I found a curious phenomenon
whose slightly weakened form states the following. If the resolvent of a bounded
linear operator is analytic at a standard set (i.e., the complement of a nonempty
finite disjoint union of open discs in the projective line), then it is analytic at
a strictly bigger standard set (i.e., all of the radii of the corresponding discs are
strictly bigger or smaller). Of course, in the light of our present knowledge this
phenomenon is completely clear since the standard sets being defined by non-strict
inequalities are closed subsets of the compact projective line. But the analyticity
set of the resolvent being the complement of the compact spectrum is an open set.
At that time I was not so smart to see the above. I considered the analyticity
sets as strictly increasing families of finite disjoint unions of standard sets, and
the spectra as strictly decreasing families of complementary sets of the same type.
(A precise definition of complementary sets is given on p. 141 of my book.) The
latter families can be viewed as filters of finite unions of standard sets. It turned
out that one can easily describe the maximal elements in the family of filters, i.e.,
ultrafilters, and there are four types of them.
First of all, every element a k defines an ultrafilter which is formed by the
sets (finite unions of standard sets) that contain the point a, i.e., a base of this
ultrafilter is formed by closed discs with center at a. Furthermore, every closed
disc E(a; r) of radius r 0 with center at a k defines an ultrafilter. If r |k∗|,
a base of the corresponding ultrafilter p(E(a; r)) is formed by standard sets of the
form E(a; r )\
n
i=1
D(ai; ri) with ri r r , |ai a| r and |ai aj | = r for
1 i = j n, where D(ai; ri) is the open disc of radius ri with center at ai. If
r
|k∗|,
a base of the corresponding ultrafilter p(E(a; r)) is formed by the closed
annuli E(a; r )\D(a; r ) with r r r . Finally, if the field k is not maximally
complete, then every family of nested discs with empty intersection is a base of an
ultrafilter. (By the way, it is easy to see that there is a natural bijection between
the set of ultrafilters and the set of nested families of closed discs.) The above four
types of ultrafilters correspond to what are now known as points of types (1)-(4) of
the affine line, and elements of the ultrafilters are precisely affinoid neighborhoods
of those points.
In fact, as soon as I found the above description, I knew that the space of all
ultrafilters must be considered as the affine line
A1
over k. This space is endowed
with a natural topology with respect to which it is locally compact: its basis consists
of sets of ultrafilters which contain a given standard set. It is also endowed with a
natural sheaf of local rings, the sheaf of analytic functions OA1 . But my main reason
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