Berkovich space

In the complex case, algebraic geometry begins by defining the complex affine space to be ${\displaystyle \mathbb {C} ^{n}.}$ For each ${\displaystyle U\subset \mathbb {C} ^{n},}$ we define ${\displaystyle {\mathcal {O}}_{U},}$ the ring of analytic functions on ${\displaystyle U}$ to be the ring of holomorphic functions, i.e. functions on ${\displaystyle U}$ that can be written as a convergent power series in a neighborhood of each point.

We then define a local model space for ${\displaystyle f_{1},\ldots ,f_{n}\in {\mathcal {O}}_{U}}$ to be

${\displaystyle X:=\{x\in U:f_{1}(x)=\cdots =f_{n}(x)=0\}}$

with ${\displaystyle {\mathcal {O}}_{X}={\mathcal {O}}_{U}/(f_{1},\ldots ,f_{n}).}$ A complex analytic space is a locally ringed ${\displaystyle \mathbb {C} }$-space ${\displaystyle (Y,{\mathcal {O}}_{Y})}$ which is locally isomorphic to a local model space.

When ${\displaystyle k}$ is a complete non-Archimedean field, we have that ${\displaystyle k}$ is totally disconnected. In such a case, if we continue with the same definition as in the complex case, we wouldn't get a good analytic theory. Berkovich gave a definition which gives nice analytic spaces over such ${\displaystyle k}$, and also gives back the usual definition over ${\displaystyle \mathbb {C} .}$

In addition to defining analytic functions over non-Archimedean fields, Berkovich spaces also have a nice underlying topological space.

A seminorm on a ring ${\displaystyle A}$ is a non-constant function ${\displaystyle |\!-\!|:A\to \mathbb {R} _{\geq 0}}$ such that

${\displaystyle {\begin{aligned}|0|&=0\\|1|&=1\\|f+g|&\leqslant |f|+|g|\\|fg|&\leqslant |f||g|\end{aligned}}}$

for all ${\displaystyle f,g\in A}$. It is called multiplicative if ${\displaystyle |fg|=|f||g|}$ and is called a norm if ${\displaystyle |f|=0}$ implies ${\displaystyle f=0}$.

If ${\displaystyle A}$ is a normed ring with norm ${\displaystyle \|\!-\!\|}$ then the Berkovich spectrum of ${\displaystyle A}$, denoted ${\displaystyle {\mathcal {M}}(A)}$, is the set of multiplicative seminorms on ${\displaystyle A}$ that are bounded by the norm of ${\displaystyle A}$.

The Berkovich spectrum is equipped with the weakest topology such that for any ${\displaystyle f\in A}$ the map

${\displaystyle {\begin{cases}\varphi _{f}:{\mathcal {M}}(A)\to \mathbb {R} \\|\cdot |\mapsto |f|\end{cases}}}$

is continuous.

The Berkovich spectrum of a normed ring ${\displaystyle A}$ is non-empty if ${\displaystyle A}$ is non-zero and is compact if ${\displaystyle A}$ is complete.

If ${\displaystyle x}$ is a point of the spectrum of ${\displaystyle A}$ then the elements ${\displaystyle f}$ with ${\displaystyle |f|_{x}=0}$ form a prime ideal of ${\displaystyle A}$. The field of fractions of the quotient by this prime ideal is a normed field, whose completion is a complete field with a multiplicative norm; this field is denoted by ${\displaystyle {\mathcal {H}}(x)}$ and the image of an element ${\displaystyle f\in A}$ is denoted by ${\displaystyle f(x)}$. The field ${\displaystyle {\mathcal {H}}(x)}$ is generated by the image of ${\displaystyle A}$.

Conversely a bounded map from ${\displaystyle A}$ to a complete normed field with a multiplicative norm that is generated by the image of ${\displaystyle A}$ gives a point in the spectrum of ${\displaystyle A}$.

The spectral radius of ${\displaystyle f,}$

${\displaystyle \rho (f)=\lim _{n\to \infty }\left\|f^{n}\right\|^{\frac {1}{n}}}$

is equal to

${\displaystyle \sup _{x\in {\mathcal {M}}(A)}|f|_{x}.}$

If ${\displaystyle k}$ is a field with a valuation, then the n-dimensional Berkovich affine space over ${\displaystyle k}$, denoted ${\displaystyle \mathbb {A} _{k}^{n}}$, is the set of multiplicative seminorms on ${\displaystyle k[x_{1},\ldots ,x_{n}]}$ extending the norm on ${\displaystyle k}$.

The Berkovich affine space is equipped with the weakest topology such that for any ${\displaystyle f\in k}$ the map ${\displaystyle \varphi _{f}:\mathbb {A} ^{n}\to \mathbb {R} }$ taking ${\displaystyle |\cdot |\in \mathbb {A} ^{n}}$ to ${\displaystyle |f|}$ is continuous. This is not a Berkovich spectrum, but is an increasing union of the Berkovich spectra of rings of power series that converge in some ball (so it is locally compact).

We define an analytic function on an open subset ${\displaystyle U\subset \mathbb {A} ^{n}}$ as a map

${\displaystyle f:U\to \prod _{x\in U}{\mathcal {H}}(x)}$

with ${\displaystyle f(x)\in {\mathcal {H}}(x)}$, which is a local limit of rational functions, i.e., such that every point ${\displaystyle x\in U}$ has an open neighborhood ${\displaystyle U'\subset U}$ with the following property:

${\displaystyle \forall \varepsilon >0\,\exists g,h\in {\mathcal {k}}[x_{1},\ldots ,x_{n}]:\qquad \forall x'\in U'\left(h(x')\neq 0\ \,\land \ \left|f(x')-{\frac {g(x')}{h(x')}}\right|<\varepsilon \right).}$

Continuing with the same definitions as in the complex case, one can define the ring of analytic functions, local model space, and analytic spaces over any field with a valuation (one can also define similar objects over normed rings). This gives reasonable objects for fields complete with respect to a nontrivial valuation and the ring of integers ${\displaystyle \mathbb {Z} .}$

In the case where ${\displaystyle k=\mathbb {C} ,}$ this will give the same objects as described in the motivation section.

These analytic spaces are not all analytic spaces over non-Archimedean fields.

• Baker, Matthew; Conrad, Brian; Dasgupta, Samit; Kedlaya, Kiran S.; Teitelbaum, Jeremy (2008), Thakur, Dinesh S.; Savitt, David (eds.), p-adic geometry, University Lecture Series, vol. 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, vol. 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, vol. 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

• Berkovich space at the nLab
• Institut de Mathématiques de Jussieu Summer School «Berkovich spaces» 2010