Abstract
2 Fontaine’s sheaves on Faltings’ site 6 2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 The classical period rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Continuous sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Faltings’ topos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 The Kummer etale site of X . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 The finite Kummer etale sites U fket L . . . . . . . . . . . . . . . . . . . . . 13 2.2.3 Faltings’ site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.4 Continuous Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.5 Geometric points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.6 The localization functors. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.7 The computation of Rv ∗ . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Fontaine’s sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 The sheaves OX and OX . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 The morphism Θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.3 The sheaf Alog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.4 The sheaf Alog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.5 Properties of Alog and Alog . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.6 The sheaves Blog and Blog . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.7 The sheaves Blog,K and Blog,K . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.8 The monodromy diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.9 The fundamental exact diagram . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.10 Cohomology of Blog and Blog . . . . . . . . . . . . . . . . . . . . . . . . 33
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