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.
- Baker, Matthew; Conrad, Brian; Dasgupta, Samit; Kedlaya, Kiran S.; Teitelbaum, Jeremy (2008), Thakur, Dinesh S.; Savitt, David, eds., p-adic geometry, University Lecture Series 45, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4468-7, MR 2482343
- Baker, Matthew; Rumely, Robert (2010), Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs 159, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4924-8, MR 2599526
- Berkovich, Vladimir G. (1990), Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1534-2, MR 1070709
- Berkovich, Vladimir G. (1993), "Étale cohomology for non-Archimedean analytic spaces", Publications Mathématiques de l'IHÉS (78): 5–161, ISSN 1618-1913, MR 1259429