Fujiwara K., Kato F. Foundations of Rigid Geometry I: ArXiv version

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.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