Integration of One-forms on P-adic Analytic Spaces (Annals of Mathematics Studies, Band 162) - Softcover

Berkovich, Vladimir G.

Softcover

ISBN 10: 0691128626 ISBN 13: 9780691128627

Verlag: Princeton University Press, 2006

Inhaltsangabe

Among the many differences between classical and p-adic objects, those related to differential equations occupy a special place. For example, a closed p-adic analytic one-form defined on a simply-connected domain does not necessarily have a primitive in the class of analytic functions. In the early 1980s, Robert Coleman discovered a way to construct primitives of analytic one-forms on certain smooth p-adic analytic curves in a bigger class of functions. Since then, there have been several attempts to generalize his ideas to smooth p-adic analytic spaces of higher dimension, but the spaces considered were invariably associated with algebraic varieties. This book aims to show that every smooth p-adic analytic space is provided with a sheaf of functions that includes all analytic ones and satisfies a uniqueness property. It also contains local primitives of all closed one-forms with coefficients in the sheaf that, in the case considered by Coleman, coincide with those he constructed. In consequence, one constructs a parallel transport of local solutions of a unipotent differential equation and an integral of a closed one-form along a path so that both depend nontrivially on the homotopy class of the path. Both the author's previous results on geometric properties of smooth p-adic analytic spaces and the theory of isocrystals are further developed in this book, which is aimed at graduate students and mathematicians working in the areas of non-Archimedean analytic geometry, number theory, and algebraic geometry.

Über diesen Titel

Über die Autorin bzw. den Autor

Vladimir G. Berkovich is Matthew B. Rosenhaus Professor of Mathematics at the Weizmann Institute of Science in Rehovot, Israel. He is the author of Spectral Theory and Analytic Geometry over Non-Archimedean Fields.

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Integration of One-Forms on P-adic Analytic Spaces

By Vladimir G. Berkovich

PRINCETON UNIVERSITY PRESS

Copyright © 2007 Princeton University Press
All right reserved.
ISBN: 978-0-691-12862-7

Contents

Introduction......................................................................................11. Naive Analytic Functions and Formulation of the Main Result....................................72. Étale Neighborhoods of a Point in a Smooth Analytic Space.................................233. Properties of Strictly Poly-stable and Marked Formal Schemes...................................394. Properties of the Sheaves Ω1,clX/dOx.....................555. Isocrystals....................................................................................716. F-isocrystals..................................................................................877. Construction of the Sheaves Sλx.....................................959. Integration and Parallel Transport along a Path................................................131References........................................................................................149Index of Notation.................................................................................153Index of Terminology..............................................................................155

Chapter One

Naive Analytic Functions and Formulation of the Main Result

After recalling some notions and notation, we give a precise definition of the sheaves of naive analytic functions N^K_X. We then recall the definition of a D_X-modules, introduce a related notion of a D_X-modules, and establish a simple relation between the de Rham complexes of a D_X-modules and those of its pullback under a so-called discoid morphism Y -> X. In §1.4, we introduce the logarithmic function Logλ(T) [member of] N^K,1(G_m) and a filtered D_X-modules L^λ(X) which is generated over O(X) by the logarithms Log^λ(f) of invertible analytic functions on X. Furthermore, given a so-called semi-annular morphism Y -> X, we establish a relation between the de Rham complexes of certain D_X-modules and those of the D_Y-modules which are generated by the pullbacks of the latter and the logarithms of invertible analytic functions on Y. It implies the exactness of the de Rham complex of the spaces of differential forms with coefficients in the D_X-modules L^λ(X) on a semi-annular analytic space X. In §1.6, we formulate the main result on existence and uniqueness of the sheaves S^λ_X and list their basic properties.

1.1 PRELIMINARY REMARKS AND NOTATION

In this book we work in the framework of non-Archimedean analytic spaces in the sense of [Ber1] and [Ber2]. A detailed definition of these spaces is given in [Ber2, §1], and an abbreviated one is given in [Ber6, §1]. We only recall that the affinoid space associated with an affinoid algebra A is the set of all bounded multiplicative seminorms on A. It is a compact space with respect to the evident topology, and it is denoted by M(A).

Let k be a non-Archimedean field with a nontrivial valuation. All of the k-analytic spaces considered here are assumed to be Hausdorff. For example, any separated k-analytic space is Hausdorff and, for the class of the spaces which are good in the sense of [Ber2, §1.2], the converse is also true.

Although in this book we are mostly interested in smooth k-analytic spaces (in the sense of [Ber2, §3.5]), we have to consider more general strictly k-analytic spaces. Among them, of special interest are strictly k-analytic spaces smooth in the sense of rigid geometry. For brevity we call them rig-smooth. Namely, a strictly k-analytic space X is rig-smooth if for any connected strictly affinoid domain V the sheaf of differentials Ω¹_V is locally free of rank dim(V). Such a space is smooth at all points of its interior (see [Ber4, §5]). In particular, a k-analytic space X is smooth if and only if it is rig-smooth and has no boundary (in the sense of [Ber2, §1.5]). An intermediate class between smooth and rig-smooth spaces is that of k-analytic spaces locally embeddable in a smooth space (see [Ber7, §9]). For example, it follows from R. Elkik's results (see [Ber7, 9.7]) that any rig-smooth k-affinoid space is locally embeddable in a smooth space. If X is locally embeddable in a smooth space, the sheaf of differential one-forms Ω¹_X is locally free in the usual topology of X. (If X rig-smooth, the sheaf of differential one-forms is locally free in the more strong G-topology X_G, the Grothendieck topology formed by strictly analytic subdomains of X, see [Ber2, §1.3].) Recall that for any strictly k-analytic space X the set X₀ = {x [member of] X | [H(x) : k] < ∞} is dense in X ([Ber1, 2.1.15]). For a point x [member of] X₀, the field H(x) coincides with the residue field κ(x) = O_X,x/m_x of the local ring O_X,x.

Lemma 1.1.1 Let X be a connected rig-smooth k-analytic space. Then every nonempty Zariski open subset X' [subset] X is dense and connected, and one has c(X) [??] c(X').

Recall that c(X) is the space of global sections of the sheaf of constant analytic functions c_X defined in [Ber9, §8]. If the characteristic of k is zero, then c_X = Ker(O_X [??] Ω¹_X).

Proof. We may assume that the space X = M(A) is strictly k-affinoid, and we may replace X' by a smaller subset of the form X_f = {x [member of] X| f(x) ≠ 0} with f a nonzero element of A. Such a subset is evidently dense in X. We now notice that X_f is the analytification of the affine scheme X_f = Spec(A_f) over X = Spec(A). Since X_f is connected, from [Ber2, Corollary 2.6.6] it follows that X_f is connected. To prove the last property, we can replace k by c(X), and so we may assume that c(X)=k. By [Ber9, Lemma 8.1.4], the strictly k'-affinoid space X [??] k' is connected for any finite extension k' of k and, therefore, the same is true for the space X_f [??] k' = (X [??] k')_f. The latter implies that c(X_f) = k.

Recall that in [Ber1, §9.1] we introduced the following invariants of a point x of a k-analytic space X. The first is the number s(x) = s_k(x) equal to the transcendence degree of [??] over [??], and the second is the number t(x) = t_k(x) equal to the dimension of the Q-vector space √|H(x)*|/√|k|*. One has s(x) + t(x) ≤ dim_x(X), and if x' is a point of X [??] k^a over k then s(x') = s(x) and t (x') = t (x). Moreover, the...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2006
• Sprache: Englisch
• ISBN 10: 0691128626
• ISBN 13: 9780691128627
• Einband: Taschenbuch
• Anzahl der Seiten: 164
• Kontakt zum Hersteller: Princeton University Press; https://press.princeton.edu/books; ; 7th Floor, New South Building; PRINCETON; NJ; 8544; USA
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