European Mathematical Society, Switzerland, 2017. — 787 p. — (EMS Monographs in Mathematics) — ISBN 3037190361.Rigid geometry is one of the modern branches of algebraic and arithmetic geometry. It has its historical origin in J. Tate’s rigid analytic geometry, which aimed at developing an analytic geometry over non-archimedean valued fields. Nowadays, rigid geometry is a discipline in its own right and has acquired vast and rich structures, based on discoveries of its relationship with birational and formal geometries. In this research monograph, foundational aspects of rigid geometry are discussed, putting emphasis on birational and topological features of rigid spaces. Besides the rigid geometry itself, topics include the general theory of formal schemes and formal algebraic spaces, based on a theory of complete rings which are not necessarily Noetherian. Also included is a discussion on the relationship with Tate‘s original rigid analytic geometry, V.G. Berkovich‘s analytic geometry and R. Huber‘s adic spaces. As a model example of applications, a proof of Nagata‘s compactification theorem for schemes is given in the appendix. The book is encyclopedic and almost self-contained.
Introduction Preliminaries Basic Languages General topology Homological algebra Ringed spaces Schemes and algebraic spaces Valuation rings Topological rings and modules Pairs Topological algebras of type (V) Formal geometry Formal schemes Universally rigid-Noetherian formal schemes Adically quasi-coherent sheaves Several properties of morphisms Differential calculus on formal Formal algebraic spaces Cohomology theory Finiteness theorem for proper algebraic spaces GFGA comparison theorem GFGA existence theorem Finiteness theorem and Stein factorization Rigid spaces Admissible blow-ups Rigid spaces Visualization Topological properties Coherent sheaves Affinoids Basic properties of morphisms of rigid spaces Classical points GAGA Dimension of rigid spaces Maximum modulus principle
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