Rigid Cohomology over Laurent Series Fields

1. Front Matter

   Pages i-x

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2. Introduction

   • Christopher Lazda, Ambrus Pál

   Pages 1-15

3. First Definitions and Basic Properties

   • Christopher Lazda, Ambrus Pál

   Pages 17-69

4. Finiteness with Coefficients via a Local Monodromy Theorem

   • Christopher Lazda, Ambrus Pál

   Pages 71-129

5. The Overconvergent Site, Descent, and Cohomology with Compact Support

   • Christopher Lazda, Ambrus Pál

   Pages 131-171

6. Absolute Coefficients and Arithmetic Applications

   • Christopher Lazda, Ambrus Pál

   Pages 173-225

7. Back Matter

   Pages 227-267

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In this monograph, the authors develop a new theory of p-adic cohomology for varieties over Laurent series fields in positive characteristic, based on Berthelot's theory of rigid cohomology. Many major fundamental properties of these cohomology groups are proven, such as finite dimensionality and cohomological descent, as well as interpretations in terms of Monsky-Washnitzer cohomology and Le Stum's overconvergent site. Applications of this new theory to arithmetic questions, such as l-independence and the weight monodromy conjecture, are also discussed.

The construction of these cohomology groups, analogous to the Galois representations associated to varieties over local fields in mixed characteristic, fills a major gap in the study of arithmetic cohomology theories over function fields. By extending the scope of existing methods, the results presented here also serve as a first step towards a more general theory of p-adic cohomology over non-perfect ground fields.

Rigid Cohomology over Laurent Series Fields will provide a useful tool for anyone interested in the arithmetic of varieties over local fields of positive characteristic. Appendices on important background material such as rigid cohomology and adic spaces make it as self-contained as possible, and an ideal starting point for graduate students looking to explore aspects of the classical theory of rigid cohomology and with an eye towards future research in the subject.

• p-adic cohomology
• rigid geometry
• local function fields
• weight-monodromy
• (φ,∇)-modules

“The book is thorough and very carefully written, with useful appendices on classical rigid cohomology, adic spaces and cohomological descent. Moreover, instead of deducing results from the known cases in classical rigid cohomology (when possible), the authors have the choice of writing down complete proofs in their setting. This makes the exposition clearer and the book self-contained. I believe that it will soon become a reference on the subject … .” (Jérôme Poineau, zbMATH 1400.14002, 2019)

• Università Degli Studi di Padova, Padova, Italy

  Christopher Lazda

• Imperial College London, London, United Kingdom

  Ambrus Pál

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