Rings of separated power series and quasi-affinoid geometry

— 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" poly disc. The theory is based on the commutative algebra of power series rings 5 m , n that is developed in the first paper in this volume, Rings of Separated Power Series. Quasi-affinoid algebras (quotients Sm,n/I) share many properties with affinoid algebras (quotients Tm/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 Max A, the i?-subdomains. The second paper, Model Completeness and Subanalytic Sets, obtains a structure theory for images of analytic maps based on any subcollection of S = U 5 m > n that satisfies certain closure properties; for example T = UT m . 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 Max A in general may be covered by finitely many disjoint quasi-affinoid subdomains, just as the valuation ring K° is the union of its maximal ideal K°° and its multiplicative units. 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 5 m ? n -analytic varieties X = 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 i-» Ox(U) is a presheaf of A-algebras for i?-subdomains U of X. The quantifier elimination 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" i 7 m , n of a ring T m + n of strictly convergent power series and its "completion" 5 m , n . This links the algebraic properties of affinoid and quasi-affinoid algebras. © Astérisque 264, SMF 2000