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The p-adic Simpson Correspondence (AM-193)$
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Ahmed Abbes, Michel Gros, and Takeshi Tsuji

Print publication date: 2016

Print ISBN-13: 9780691170282

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691170282.001.0001

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Cohomology of Higgs isocrystals

Cohomology of Higgs isocrystals

Chapter:
(p.307) Chapter IV Cohomology of Higgs isocrystals
Source:
The p-adic Simpson Correspondence (AM-193)
Author(s):

Takeshi Tsuji

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691170282.003.0004

This chapter describes the cohomology of Higgs isocrystals, which are introduced to replace the notion of Higgs bundles. The link between these two notions uses Higgs envelopes and calls to mind the link between classical crystals and modules with integrable connections. After discussing Higgs isocrystals and Higgs crystals, cohomology of Higgs isocrystals, and representations of the fundamental group, the chapter presents the main result: the construction of a fully faithful functor from the category of Higgs (iso)crystals satisfying an overconvergence condition to that of small generalized representations. It also proves the compatibility of this functor with the natural cohomologies and concludes by comparing the cohomology of Higgs isocrystals with Faltings cohomology.

Keywords:   cohomology, Higgs isocrystal, Higgs bundles, Higgs envelopes, Higgs crystals, fundamental group, overconvergence, Faltings cohomology

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