© 2014

Lectures on Formal and Rigid Geometry


Part of the Lecture Notes in Mathematics book series (LNM, volume 2105)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Siegfried Bosch
    Pages 1-5
  3. Classical Rigid Geometry

    1. Front Matter
      Pages 7-7
    2. Siegfried Bosch
      Pages 9-29
    3. Siegfried Bosch
      Pages 31-63
    4. Siegfried Bosch
      Pages 65-91
    5. Siegfried Bosch
      Pages 93-116
    6. Siegfried Bosch
      Pages 117-147
  4. Formal Geometry

    1. Front Matter
      Pages 149-149
    2. Siegfried Bosch
      Pages 151-173
    3. Siegfried Bosch
      Pages 175-214
    4. Siegfried Bosch
      Pages 215-227
  5. Back Matter
    Pages 229-256

About this book


A first version of this work appeared in 2005 as a Preprint of the Collaborative Research Center "Geometrical Structures in Mathematics" at the University of Münster. Its aim was to offer a concise and self-contained 'lecture-style' introduction to the theory of classical rigid geometry established by John Tate, together with the formal algebraic geometry approach launched by Michel Raynaud. These Lectures are now viewed commonly as an ideal means of learning advanced rigid geometry, regardless of the reader's level of background. Despite its parsimonious style, the presentation illustrates a number of key facts even more extensively than any other previous work.

This Lecture Notes Volume is a revised and slightly expanded version of the original preprint and has been published at the suggestion of several experts in the field.


Formal blowing-up Formal scheme Rigid analytic space Tate algebra Tate's Acyclicity Theorem

Authors and affiliations

  1. 1.Westfälische Wilhelms-Universität Mathematisches InstitutMünsterGermany

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“Its aim is to offer a rapid and mostly self-contained ‘lecture-style’ introduction to the theory of classical rigid geometry established by Tate, together with the formal algebraic geometry approach set up by Raynaud. Furthermore, the volume provides enlightening examples of rigid spaces and points out analogies with and differences from the theory of schemes. The book is suitable for a first course on formal and rigid geometry, but it can be used equally well for personal study.” (Alessandra Bertapelle, Mathematical Reviews, March, 2016)

“All notions introduced are discussed thoroughly, proofs are lucid and elegant, and the hypotheses made and their relevance are clear throughout the text. … The reader comes away from the text with a thorough understanding of the internal motivations of the theory of formal and rigid spaces. The book is an extremely readable introduction to its subject, as well as to the techniques of modern geometry in general.” (Jeroen Sijsling, zbMATH 1314.14002, 2015)