Rigid Cohomology over Laurent Series Fields: 21 by Christopher Lazda (PDF)

Description

In this monograph, the authors develop a new theory of p-adiccohomology for varieties over Laurent series fields in positive characteristic,based on Berthelot’s theory of rigid cohomology. Many major fundamentalproperties of these cohomology groups are proven, such as finite dimensionalityand cohomological descent, as well asinterpretations in terms of Monsky-Washnitzer cohomology and Le Stum’soverconvergent site.Applications of this new theory to arithmetic questions, such as l-independenceand the weight monodromy conjecture, are also discussed.The construction of these cohomology groups, analogous to theGalois representations associated tovarieties over local fields in mixed characteristic, fills a major gap in the study of arithmetic cohomology theoriesover function fields. By extending the scope of existing methods, the results presented here also serve as a firststep towards a more general theory of p-adic cohomology overnon-perfect ground fields.Rigid Cohomology over Laurent Series Fields will provide a useful tool for anyone interested in thearithmetic of varieties over local fields of positive characteristic.Appendices on important background material such as rigid cohomology and adicspaces make it as self-contained as possible, and an ideal starting point forgraduate students looking to explore aspects of the classical theory of rigidcohomology and with an eye towards future research in the subject.

User’s Reviews

Descrizione prodotto Recensione “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) Dalla quarta di copertina In this monograph, the authors develop a new theory of p-adiccohomology for varieties over Laurent series fields in positive characteristic,based on Berthelot’s theory of rigid cohomology. Many major fundamentalproperties of these cohomology groups are proven, such as finite dimensionalityand cohomological descent, as well asinterpretations in terms of Monsky-Washnitzer cohomology and Le Stum’soverconvergent site.Applications of this new theory to arithmetic questions, such as l-independenceand the weight monodromy conjecture, are also discussed.The construction of these cohomology groups, analogous to theGalois representations associated tovarieties over local fields in mixed characteristic, fills a major gap in the study of arithmetic cohomology theoriesover function fields. By extending the scope of existing methods, the results presented here also serve as a firststep towards a more general theory of p-adic cohomology overnon-perfect ground fields.Rigid Cohomology over Laurent Series Fields will provide a useful tool for anyone interested in thearithmetic of varieties over local fields of positive characteristic.Appendices on important background material such as rigid cohomology and adicspaces make it as self-contained as possible, and an ideal starting point forgraduate students looking to explore aspects of the classical theory of rigidcohomology and with an eye towards future research in the subject.

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