Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties
In this volume, the authors construct a theory of weights on the log crystalline cohomologies of families of open smooth varieties in characteristic p>0, by defining and constructing four filtered complexes. Fundamental properties of these filtered com
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1959
Yukiyoshi Nakkajima · Atsushi Shiho
Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties
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Yukiyoshi Nakkajima
Atsushi Shiho
Department of Mathematics Tokyo Denki University 2-2 Nishiki-cho Kanda Chiyoda-ku Tokyo 101-8457 Japan [email protected]
Graduate School of Mathematical Sciences University of Tokyo 3-8-1, Komaba Meguro-ku Tokyo 153-8914 Japan [email protected]
ISBN: 978-3-540-70564-2 e-ISBN: 978-3-540-70565-9 DOI: 10.1007/978-3-540-70565-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008932186 Mathematics Subject Classification (2000): 14F30 (Primary), 14F40, 13K05 (Secondary) c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com
Preface
The main goal of this book is to construct a theory of weights for the log crystalline cohomologies of families of open smooth varieties in characteristic p > 0. This is a p-adic analogue of the theory of the mixed Hodge structure on the cohomologies of open smooth varieties over C developed by Deligne in [23]. We also prove the fundamental properties of the weightfiltered log crystalline cohomologies such as the p-adic purity, the functoriality, the weight-filtered base change theorem, the weight-filtered K¨ unneth formula, the convergence of the weight filtration, the weight-filtered Poincar´e duality and the E2 -degeneration of p-adic weight spectral sequences. One can regard some of these results as the logarithmic and weight-filtered version of the corresponding results of Berthelot in [3] and K. Kato in [54]. Following the suggestion of one of the referees, we have decided to state some theorems on the weight filtration and the slope filtration on the rigid cohomology of separated schemes of finite type over a perfect field of characteristic p > 0. This is a p-adic analogue of the mixed Hodge structure on the cohomologies of separated schemes of finite type over C developped by Deligne in [24]. The detailed proof for them is given in another book [70] by the first-named author. We have to assume that the reader is familiar
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