## Rings of separated power series and quasi-affinoid geometry.(English)Zbl 0957.32011

Astérisque. 264. Paris: Société Mathématique de France, iv, 171 p. (2000).
The papers in this volume present a theory of rigid analytic geometry over an ultrametric field $$K$$ that generalizes the classical affinoid theory to the setting of relative rigid analytic geometry over an open polydisc. The theory is based on the commutative algebra of power series rings $$S_{m,n}$$ that is developed in the first paper in this volume, “Rings of separated power series”. Quasi-affinoid algebras (quotients $$S_{m,n}/I$$) share many properties with affinoid algebras (quotients $$T_m/I$$ of a ring of strictly convergent power series). Among the principal results are the Nullstellensatz for quasi-affinoid algebras $$A$$ and the universal property for a broad class of open subdomains of $$\text{Max } A$$, the $$R$$-subdomains.
The second paper, “Model completeness and subanalytic sets”, obtains a structure theory for images of analytic maps based on any subcollection of $$S = \bigcup S_{m,n}$$ that satisfies certain closure properties. As a corollary, one obtains that the complement of a rigid subanalytic set is again subanalytic. The argument exploits the existential definability of the Weierstrass data as well as a difference between affinoid and quasi-affinoid rigid analytic geometry; namely, that a quasi-affinoid variety $$\text{Max} A$$ in general may be covered by finitely many disjoint quasi-affinoid subdomains. A crucial role is played by the theory of generalized rings of fractions developed in the first paper.
The third paper, “Quasi-affinoid varieties”, defines the category of $$S_{m,n}$$-analytic varieties $$X = \text{Max } A$$ and establishes the acyclicity of quasi-affinoid covers. The proofs employ results from the first paper; in particular, the fact that the assignment $$U\mapsto\mathcal O(U)$$ is a presheaf of $$A$$-algebras for $$R$$-subdomains $$U$$ of $$X$$. The quantifier elimination (over the rings $$S_{m,n}$$) of the second paper is used to relate quasi-affinoid and affinoid covers, a key step in the proof of the Acyclicity Theorem.
The fourth paper, “A rigid analytic approximation theorem”, gives a global Artin approximation theorem between a “Henselization” $$H_{m,n}$$ of a ring $$T_{m+n}$$ of strictly convergent power series and its “completion” $$S_{m,n}$$. This links the algebraic properties of affinoid and quasi-affinoid algebras.
Apart from the intrinsic interest in developing this relative rigid analytic geometry, the results, especially the theorem of the complement from the second paper, form a key ingredient in the rigid analytic quantifier elimination theorem of T. S. Gardener and the reviewer [Flattening and subanalytic sets in rigid analytic geometry, to appear in Proc. Lond. Math. Soc.; see also the forthcoming volume in this series by the reviewer.

### MSC:

 32P05 Non-Archimedean analysis 32B05 Analytic algebras and generalizations, preparation theorems 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 32B20 Semi-analytic sets, subanalytic sets, and generalizations 32C35 Analytic sheaves and cohomology groups