nLab Berkovich space

Idea

Berkovich analytic spaces are a version of analytic spaces over nonarchimedean fields. Unlike the rigid analytic spaces (see rigid analytic geometry) of Tate, which are locally defined via maximal spectra of Tate algebras glued via the Grothendieck G-topology, the Berkovich analytic spaces are actual topological space equipped with a cover by affinoid domains via the analytic spectrum construction, due to Vladimir Berkovich. This spectrum can be viewed as consisting of the data of prime ideal plus the extension of the norm to the residue field; thus the Berkovich spectrum has far more points (though fewer than, say, Huber's adic spaces which may also contain valuations of higher order).

For more background see analytic geometry.

Definition of Berkovich analytic spaces

Let $k$ be a non-archimedean field.

Definition

Given $n \in \mathbb{N}$ and positive elements $\{r_1, \cdots, r_n \in k\}$, consider the sub-power series algebra over $k$ of those series which converge inside the radii $k_i$, i.e. the algebra defined by

$$\{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\} := \left\{ \sum_\nu a_\nu T^\nu | \lim_{{\vert \nu\vert} \to \infty} {\vert a_\nu \vert} r^\nu = 0 \right\} \,.$$

This is a commutative Banach algebra over $k$ with norm ${\Vert f \Vert} = max {\vert a_\nu\vert} r^\nu$.

A $k$-affinoid algebra is a commutative Banach $k$-algebra $A$ for which there exists $n$ and $\{r_i\}$ as above and an epimorphism

$$\{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\} \to A$$

such that the norm on $A$ is the quotient norm.

If one can choose here $r_i = 1$ for all $i$ then $A$ is called strictly $k$-affinoid.

The category of $k$-affinoid spaces is the opposite category of the category of $k$-affinoid algebras and bounded homomorphisms between them.

Via the analytic spectrum $Spec_{an}$ there is a topological space associated with any $k$-affinoid space. Often this underlying topological space is referred to as the analytic space.

Definition

An affinoid domain in an affinoid space $X = Spec_{an} A$ is a closed subset $V \subset X$ such that there is a homomorphism of $k$-affinoid spaces

$$\phi : Spec_{an} A_V \to X$$

for some $A_V$, whose image is $V$, and such that every other morphism of $k$-affinoid spaces into $X$ whose image is contained in $V$ uniquely factors through this morphism.

Definition

A $k$-analytic space is a locally Hausdorff topological space $X$ equipped with an atlas by $k$-affinoid domains and affinoid domain embeddings, such that their underlying analytic spectra topological spaces form a net of compact subsets on $X$.

(Berkovich 09, def. 3.1.4)

Properties

Cohomology

Under some mild conditions, the algebraic and the analytic étale cohomology of Berkovich spaces agree. (Berkovich 95)

The underlying topological space $X^{an}$ given by the Berkovich analytic spectrum has as singular cohomology the weight 0-cohomology of $X$ (Berkovich 09).

See also MO discussion here.

Local contractibility

A complex analytic manifold and a smooth complex analytic space is locally isomorphic to a polydisk and hence is trivially a locally contractible space. But over a non-archimedean field analytic spaces no longer need to be locally isomorphic to polydisks (but $p$-adic polydisks are still contractible (Berkovich 90)). The following result establishes, under mild conditions, that general analytic spaces are nevertheless locally contractible.

Assume that the valuation on the ground field $k$ is nontrivial.

Definition

A $k$-analytic space $X$ is called locally embeddable in a smooth space if each point of $X$ has an open neighbourhood isomorphic to a strictly $k$-analytic domain in smooth $k$-analytic space.

Theorem

Every $k$-analytic space which is locally embeddable in a smooth space, def. , is a locally contractible space.

More precisely, every point of a locally smooth $k$-analytic space has an open neighbourhood $U$ which is contractible, and which is a union $U = \cup_{i = 1}^\infty U_i$ of analytic domains.

The local contractibility is Berkovich (1999), theorem 9.1. The refined statment in terms of inductive systems of analytic domains is in Berkovich (2004).

Examples

• analytic affine line

Applications

• The proof of the local Langlands conjecture for $GL_n$ by Harris–Taylor uses étale cohomology on non-archimedean analytic spaces (in the sense of Berkovich) to construct the required Galois representations over local fields.

Related concepts

• non-commutative analytic space

• p-adic geometry

• G-topology

• Huber space, perfectoid space

• global analytic geometry

References

Introductions and reviews

A nice survey is in

• Bernard Le Stum, One century of $p$-adic geometry – From Hensel to Berkovich and beyond, talk notes, June 2012 (pdf)

A good introduction to the general idea is at the beginning of

• Sam Payne, Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry (arXiv:1309.4403)

Basic notions are listed in

• M. Temkin, Non-archimedean analytic spaces (pdf slides)

A review of basic definitions and facts about affinoid and rigid $k$-analytic spaces can be found in

• Gaëtan Chenevier, lecture 5 (pdf)

See also the references at rigid analytic geometry.

A review of definitions and results on $k$-analytic spaces is in

• Vladimir Berkovich, $p$-Adic analytic spaces, in Proceedings of the International Congress of Mathematicians, Berlin, August 1998, Doc. Math. J. DMV, Extra Volume ICM II (1998), 141-151 (pdf)

A more detailed set of lecture notes along these lines is

• Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

Introductory exposition of the Berkovich analytic spectrum is

• Jérôme Poineau, Global analytic geometry, pages 20-23 in EMS newsletter September 2007 (pdf)

• Frédéric Paugam, section 2.1.4 of_Global analytic geometry and the functional equation_ (2010) (pdf)

A exposition of examples of Berkovich spectra is in

• Scott Carnahan, Berkovich spaces I (web)

Original articles

• Vladimir Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.

• Vladimir Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. IHES 78 (1993), 5-161.

Discussion of Berkovich09cohomology of Berkovich analytic spaces includes

• Vladimir Berkovich, On the comparison theorem for étale cohomology of non-Archimedean analytic spaces. Israel Journal of Mathematics 92.1-3 (1995): 45-59.

• Vladimir Berkovich, A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures, Algebra, Arithmetic, and Geometry. Birkhäuser Boston, 2009. 49-67.

Discussion of local contractibility of smooth $k$-analytic spaces is in

• Vladimir Berkovich, Smooth $p$-adic analytic spaces are locally contractible, Invent. Math. 137 1-84 (1999) (pdf)

• Vladimir Berkovich, Smooth p-adic analytic spaces are locally contractible. II, in Geometric Aspects of Dwork Theory, Walter de Gruyter & Co., Berlin, (2004), 293-370. (pdf)

and more generally in

• Ehud Hrushovski, François Loeser, Non-archimedean tame topology and stably dominated types (arXiv:1009.0252)

• Ehud Hrushovski, François Loeser, Bjorn Poonen, Berkovich spaces embed in Euclidean spaces (arXiv:1210.6485)

Relation to other topics

On the relation to buildings:

• Annette Werner, Buildings and Berkovich Spaces (pdf)

Relation to integration theory

• Vladimir Berkovich, Integration of 1-forms on $p$-adic analytic spaces, Princeton University Press,

Aspects of the homotopy theory/étale homotopy of analytic spaces are discussed in

• Aise Johan de Jong, Étale fundamental groups of non-archimedean analytic spaces, Mathematica, 97 no. 1-2 (1995), p. 89-118 (numdam)

Relation to formal schemes:

• Jérôme Poineau, MO comment

Discussion of Berkovich analytic geometry as algebraic geometry in the general sense of Bertrand Toën and Gabriele Vezzosi is in

• Oren Ben-Bassat, Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)