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Vladimir G. Berkovich: Integration of One-forms on P-adic Analytic Space
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Contents
Introduction
1. Naive Analytic Functions and Formulation of the Main Result
1.1
1.2
1.3
1.4
1.5
1.6
Preliminary remarks and notation
The sheaf of naive analytic functions
DX -modules and DX -modules
Logarithms
Logarithmic Poincaré lemma
Formulation of the main results
2. Étale Neighborhoods of a Point in a Smooth Analytic Space
2.1
2.2
2.3
2.4
Étale neighborhoods of a point with s(x) = dim(X)
The local structure of a smooth analytic curve
Étale neighborhoods of a point with s(x) < dim(X)
Basic curves
3. Properties of Strictly Poly-stable and Marked Formal Schemes
3.1
3.2
3.3
3.4
3.5
Strictly poly-stable formal schemes
Open neighborhoods of the generic point of an irreducible component
A property of strata
A tubular neighborhood of the diagonal of a stratum closure
The same for proper marked formal schemes
4. Properties of the Sheaves 1,cl
X /dOX
4.1
4.2
4.3
4.4
4.5
Analytic curve connectedness of closed analytic spaces
c
1
v
The sheaves OX
, OX
, and OX
1
Structure of the sheaves X /d OX for smooth analytic curves
v
Injectivity of the homomorphism dLog : OX
⊗Z cX → 1,cl
X /d OX
1,cl
A subsheaf X ⊂ X /d OX and a subspace VX,x ⊂ 1,cl
X,x /d OX,x
5. Isocrystals
5.1
5.2
5.3
5.4
5.5
Wide germs of analytic spaces and of formal schemes
D-modules on smooth strictly k-affinoid germs and isocrystals
A construction of isocrystals
The filtered isocrystals EB and the shuffle algebras
Unipotent isocrystals E i (X, Z)
1
7
7
9
10
14
17
20
23
23
28
32
35
39
39
43
47
48
51
55
55
57
59
63
65
71
71
74
78
81
83
vi
CONTENTS
6. F-isocrystals
6.1
6.2
6.3
6.4
6.5
Frobenius liftings
A Frobenius structure on the isocrystals E i (X, Z)
A uniqueness property of certain F -isocrystals
Structure of a commutative filtered DB -algebra on E(X, Z)
Filtered F -isocrystals E K (X, Z) and F λ (X, Z)
7. Construction of the Sheaves SXλ
7.1
7.2
7.3
7.4
7.5
7.6
Induction hypotheses
Split one-forms
Marked and weakly marked one-forms
Construction of a primitive of a weakly marked one-form
Construction of the DX -modules SXλ,n+1
End of the proof
8. Properties of the sheaves SXλ
8.1
8.2
8.3
8.4
8.5
Filtered D(X,Y ) -algebras E λ (X, Y ) for germs with good reduction
Filtered DXη -algebras E λ (X) for proper marked formal schemes
λ
λ
A filtered DOX,x-subalgebra EX,x
⊂ SX,x
and the space VX,x
More uniqueness properties
A filtered DX -subalgebra sX ⊂ SX and the sheaf X
9. Integration and Parallel Transport along a Path
9.1
9.2
9.3
9.4
9.5
Integration of closed one-forms along a path
Nontrivial dependence on the homotopy class of a path
Locally unipotent and quasi-unipotent DX -modules
Parallel transport along a path
Parallel transport along an étale path
87
87
88
89
91
92
95
95
98
99
102
105
108
113
114
117
119
124
127
131
131
134
136
139
144
References
149
Index of Notation
153
Index of Terminology
155