##
**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.

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.

Reviewer: Hans Schoutens (Piscataway)

### 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 |