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