4 NON-ARCHIMEDEAN ANALYTIC GEOMETRY: FIRST STEPS
After the above question had been formulated, I already knew the answer: every
such character goes through a character k[T ] H(x) for a unique point x A1.
The proof is very easy. Indeed, given a character χ : k[T ] K, consider the family
of closed discs of the form E(a; |χ(T a)|) with a k and χ(T a) = 0. It is easy
to see that it is a nested family of discs and, if x is the corresponding point of A1,
the character χ goes through the character k[T ] H(x).
Thus, the affine line A1 can be defined as the set of equivalence classes of
characters k[T ] K to non-archimedean fields K over k or, equivalently, as the
set of all multiplicative seminorms on k[T ] that extend the valuation on k. Wow,
this definition was so simple and easily seen to be applicable in a much more
general setting (e.g., for defining affine spaces of higher dimension). It also gave a
clear idea how to define the non-archimedean analog of the Gelfand spectrum of a
complex commutative Banach algebra. But the main thing I was struck by was the
fact that this definition was also applicable in the classical situation and gave the
corresponding classical objects. In this way, I was thrown into the new (for me)
area of analytic geometry. It took me several days to calm down and to quietly
look at what all that meant.
The above observation made it clear how to define analytic spaces over an arbi-
trary field k complete with respect to a nontrivial valuation (archimedean or not).
First of all, one should start one step earlier and define the affine space
An
as the
set of multiplicative seminorms on the ring of polynomials k[T1, . . . , Tn] that extend
the valuation on k. The space
An
is endowed with the evident topology, and each
point x
An
defines a character k[T1, . . . , Tn] H(x) : f f(x) to a complete
valuation field H(x) over k so that the corresponding seminorm is the function
f |f(x)|. Furthermore, as we were taught by Krasner, an analytic function f on
an open subset U An should be defined as a local limit of rational functions. The
latter means that f is a map that takes each point x U to an element f(x) H(x)
with the following property: one can find an open neighborhood x U U such
that, for every ε 0, there exist polynomials g, h k[T1, . . . , Tn] with h(x ) = 0
and f(x )
g(x )
h(x )
ε for all points x U . Finally, arbitrary analytic spaces
are those locally ringed spaces which are locally isomorphic to a local model of the
form (X, OX ), where X is the set of common zeros of a finite system of analytic
functions f1, . . . , fm on an open subset U
An
and OX is the restriction of the
quotient OU/J by the subsheaf of ideals J OU generated by f1, . . . , fm. By the
way, the spectrum M(A) of a commutative Banach k-algebra A should be defined
as the space of all bounded multiplicative seminorms on A.
If k = C, the affine space An is the maximal spectrum of the ring of polyno-
mials C[T1, . . . , Tn] (i.e., the vector space Cn), analytic functions are local limits
of polynomials, and the spectrum of a complex commutative Banach algebra is
the Gelfand space of its maximal ideals. If k = R, the above construction gives
a new object: the real analytic affine space An is the maximal spectrum of the
ring of polynomials R[T1, . . . , Tn] (i.e., the quotient of Cn by the complex conjuga-
tion), and local limits of polynomials with real coefficients are not enough to define
analytic functions.
The above definition of an analytic space was a lodestar in my journey, but I was
unable to work with it directly. The difficulty was in establishing functional analytic
properties of the analytic spaces, whereas establishing their geometric properties
was much easier. Fortunately, the fundamental paper by John Tate on rigid analytic
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