In mathematics, a Berkovich space, introduced by Berkovich (1990), is an analogue of an analytic space for p-adic geometry, refining Tate's notion of a rigid analytic space.
A seminorm on a ring
A
is a non-constant function
f
→
|
f
|
from
A
to the non-negative reals such that |0| = 0, |1| = 1, |f + g| ≤ |f| + |g|, |fg| ≤ |f||g|. It is called multiplicative if |fg| = |f||g| and is called a norm if |f| = 0 implies f = 0.
If A is a normed ring with norm
f
→
|
|
f
|
|
then the Berkovich spectrum of A is the set of multiplicative seminorms || on A that are bounded by the norm of A. The Berkovich spectrum is topologized with the weakest topology such that for any f in A the map taking || to |f| is continuous..
The Berkovich spectrum of a normed ring A is non-empty if A is non-zero and is compact if A is complete.
The spectral radius ρ(f) = lim |fn|1/n of f is equal to supx|f|x
If A is a commutative C*-algebra then the Berkovich spectrum is the same as the Gelfand spectrum. A point of the Gelfand spectrum is essentially a homomorphism to C, and its absolute value is the corresponding seminorm in the Berkovich spectrum.
Ostrowski's theorem shows that the Berkovich spectrum of the integers (with the usual norm) consists of the powers |f|ε
p of the usual valuation, for p a prime or ∞. If p is a prime then 0≤ε≤∞, and if p=∞ then 0≤ε≤1. When ε=0 these all coincide with the trivial valuation that is 1 on all non-zero elements.
If k is a field with a multiplicative seminorm, then the Berkovich affine line over k is the set of multiplicative seminorms on k[x] extending the norm on k. This is not a Berkovich spectrum, but is an increasing union of the Berkovich spectrums of rings of power series that converge in some ball.
If x is a point of the spectrum of A then the elements f with |f|x=0 form a prime ideal of A. The quotient field of the quotient by this prime ideal is a normed field, whose completion is a complete field with a multiplicative norm generated by the image of A. Conversely a bounded map from A to a complete normed field with a multiplicative norm that is generated by the image of A gives a point in the spectrum of A.
