NON-ARCHIMEDEAN ANALYTIC GEOMETRY: FIRST STEPS 3
to view the space
A1
as the affine line was the fact that it provided an answer to
the question on the spectrum of a bounded linear operator I posed to myself at the
very beginning. Namely, the spectrum of such an operator is a nonempty compact
subset of A1, and it coincides with the complement of the analyticity set of its
resolvent. The field k is naturally embedded in the affine line as a dense subset,
and the operators studied by Misha Vishik were precisely those with spectrum
contained in k.
It was a pleasant exercise to extend Vishik’s results to arbitrary operators, and
it helped me to understand better the topological tree-like structure of the non-
archimedean affine and projective lines, to get used to them, and to accept them
as reality. During this work I met with Misha several times to tell him about the
progress. At that time, he was the only person (besides my wife) who shared my
excitement about all this. The usual reaction of my colleagues was simple indiffer-
ence at best, and the quite understandable reason for that was nicely expressed by
Professor Manin. When I told him about what I was doing, he observed that it is
worthwhile to develop a general theory only having in mind a concrete problem. Of
course, understanding what the spectrum of a non-archimedean operator should be
was not a concrete problem. Had I followed this wise advice, I would have turned
back to concrete problems I had in abundance in the area of computers, and would
probably have become rich during the present age of the high tech boom since I
was a really good programmer. Fortunately, I was already stupid enough to miss
such an attractive opportunity, and I continued my exploration of the unknown
new world revealed to me by a fluke.
My job occupied me five days per week from 8am till 5pm. It took me several
years to learn to devote an hour or two to mathematics during working hours.
Time free from my job belonged to my family, and when I was completely hooked
on mathematics and an hour or two per day was not enough for it, I discovered an
additional source of time. I learned to get up every day very early (often as early
as at 2am), and thus extended the time for doing mathematics. At this time of the
day, the world around me was quiet and fresh, nobody and nothing disturbed me,
my head was clear, and I could plunge into another world to explore and describe
it.
When I had finished writing everything I had in mind, I could look at it quietly
and listen to an inner feeling that something was not satisfactory. I thought about
this from time to time more and more often, but could not even express what
tormented me. One day at the very end of 1985 all this obsessed me. I could not
stop thinking about it at my job, and later at home. I did not go to bed early
as usual. The right question and an immediate answer to it came early the next
morning.
As I mentioned above, the affine line
A1
is provided with a sheaf of rings OA1 .
Its stalk OA1,x at a point x is a local ring with residue field κ(x) provided with a
valuation that extends that on k. If x is of type (1) (i.e., corresponds to an element
of k), then OA1,x is the algebra of convergent power series at that point, and so
κ(x) = k. Otherwise, it is a field of infinite degree over k, and so it coincides with
the non-complete field κ(x). If H(x) denotes the completion of κ(x), one gets a
character k[T ] H(x) over k. The question that came to me on that early morning
was the following. What are all possible characters k[T ] K to non-archimedean
fields K over k?
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