Les espaces de Berkovich sont ang\'eliques
By Jérôme Poineau. 2011
51 citations recorded by the source corpus.
Although Berkovich spaces may fail to be metrizable when defined over too big a field, we prove that a large part of their topology can be recovered through sequences: for instance, limit points of subsets are actual limits of sequences and compact subsets are sequentially compact. Our proof uses extension of scalars in an essential way and we need to investigate some of its properties. We show that a point in a disc may be defined over a subfield of countable type and that, over algebraically closed fields, every point is universal: in an extension of scalars, it may be canonically lifted.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1f5c7afc29138056592dd5edecd0059cef9f00db
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In Algebra and Number Theory
In Computational Theory and Mathematics
In Geometry and Topology
Citations in the corpus
- Openness of K-semistability for Fano varieties (2019, background): “ V := {v ∈ ValX,|v(mξ) ∈ [ε,1/c] and AX,∆(v) ≤ 1} ⊆ ValX is a compact subset of the valuation space by [BdFFU15, Thm 3.1]. Since any compact subset of the valuation space is also sequentially compact [Poi13] (see also the proof of [LX16, Prop 3.9]), there exists a subsequence (vk j ) so that the lim”
- Singular semipositive metrics in non-Archimedean geometry (2011, background): “xists a unique point x E2X’Xan K whose center of Xis the generic point of E. Such points will be called divisorial points.1 The set Xdiv of divisorial points is dense in X, see Corollary 2.4 and also [Poi11]. 2.3. Model functions. Let Xbe a model of X. Each vertical fractional ideal sheaf a on Xdene”
- A variational approach to the Yau–Tian–Donaldson conjecture (2015, background): “+ ˆ: Let Wbe the subset of the Berkovich analytication (X C)an consisting of semivaluations wthat are C -invariant and satisfy w(˝) = 1, see [Berk90]. Then W is compact and even sequentially compact [Poi13], so after passing to a subsequence we may assume that w j!w 12W. For each m1, the multiplier ”