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