Foundations of Rigid Geometry I

摘要

In this research oriented manuscript, foundational aspects of rigid geometryare discussed, putting emphasis on birational side of formal schemes andtopological feature of rigid spaces. Besides the rigid geometry itself, topicsinclude the general theory of formal schemes and formal algebraic spaces, basedon a theory of complete rings which are not necessarily Noetherian (cf.introduction). The manuscript is encyclopedic and almost self-contained, andcontains plenty of new results. A discussion on relationship with J. Tate'srigid analytic geometry, V. Berkovich's analytic geometry and R. Huber's adicspaces is also included. As a model example of applications, a proof ofNagata's compactification theorem for schemes is given in the appendix. 5thversion (Feb. 28, 2017): minor changes.