Berkovich spaces are angelic.
(Les espaces de Berkovich sont angéliques.)

*(French. English summary)*Zbl 1314.14046Berkovich spaces exhibit a large number of remarkable topological properties. However, they may fail to be metrizable when defined over too big a field. In the present paper, the author shows that even in non-metrizable situations, a large part of the topological properties of a Berkovich space can be recovered in terms of sequences (in place of nets). More precisely speaking, he shows that Berkovich spaces are Fréchet-Urysohn spaces: every point in the closure of a subset of a Berkovich space is the limit of a sequence of points in that subset. This main result of the present paper plays a key role for the development of a general theory of nonarchimedean dynamical systems. The author also proves that certain Berkovich cases (like, for example, curves or algebraizations of algebraic varieties) are angelic, meaning that their relatively \(\omega\)-compact subsets are relatively compact.

The proofs rely on an extension of scalar techniques: the author shows that every point in a disc can be defined over a subfield of countable type. Moreover, he shows that over algebraically closed fields, every point of a Berkovich space may be canonically lifted to spaces obtained via extension of scalars. These two results combined allow to descend a given Berkovich space to a metrizable space and then lift the Fréchet-Urysohn property back to the original space. To obtain them, the author generalizes a result of Berkovich on Shilov boundaries from a strictly affinoid to a general affinoid setting, using graded reduction techniques as they have been developed by M. Temkin [Isr. J. Math. 140, 1–27 (2004; Zbl 1066.32025)]. Along these lines, he generalizes some classical results from commutative algebra (such as the Nullstellensatz) to the graded setting.

The proofs rely on an extension of scalar techniques: the author shows that every point in a disc can be defined over a subfield of countable type. Moreover, he shows that over algebraically closed fields, every point of a Berkovich space may be canonically lifted to spaces obtained via extension of scalars. These two results combined allow to descend a given Berkovich space to a metrizable space and then lift the Fréchet-Urysohn property back to the original space. To obtain them, the author generalizes a result of Berkovich on Shilov boundaries from a strictly affinoid to a general affinoid setting, using graded reduction techniques as they have been developed by M. Temkin [Isr. J. Math. 140, 1–27 (2004; Zbl 1066.32025)]. Along these lines, he generalizes some classical results from commutative algebra (such as the Nullstellensatz) to the graded setting.

Reviewer: Christian Kappen (Essen)