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

Berkovich, Vladimir G.

Softcover

ISBN 10: 0691128626 ISBN 13: 9780691128627

Publisher: Princeton University Press, 2006

Synopsis

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.

About this title

About the Author

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.

Excerpt. © Reprinted by permission. All rights reserved.

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 functions s(x) and t (x) are additive in the sense that, given a morphism of k-analytic spaces φ : Y -> X, one has s(y) = s(x)+s_H(x)(y) and t (y) = t (x)+t_H(x)(y), where x = φ(y). Let X_st denote the set of points x [member of] X with s(x) = t (x) = 0. This set contains X₀ and, in particular, if X is strictly k-analytic, X_st is dense in X. By [Ber1, §9], the topology on X_st induced from that on X is totally disconnected. If k' is a closed subfield of the completion [??]^a of an algebraic closure of k and X' = X [??] k', then the image of X'_st under the canonical map X' -> X is contained in X_st. (Notice that if k' is not finite over k the latter fact is not true for the sets X'₀ and X₀.)

For an étale sheaf F on a k-analytic space X, we denote by F_x the stalk at a point x [member of] X of the restriction of F to the usual topology of X and, for a section f [member of] F(X), we denote by f_x its image in F_x. Furthermore, a geometric point of X is a morphism (in the category of analytic spaces over k) of the form [bar.x]: p_H([bar.x]) -> X, where pH([bar.x]) is the spectrum of an algebraically closed non-Archimedean field H([bar.x]) over k. The stalk F[bar.x] of an étale sheaf F at [bar.x] is the stalk of its pullback with respect to the morphism [bar.x], i.e., the inductive limit of F(Y) taken over all pairs (φ, α) consisting of an étale morphism φ : Y -> X and a morphism α : pH([bar.x]) -> Y over [bar.x]. Notice that, if G_{[bar.x]/x is} the Galois group of the separable closure of H(x) in H([bar.x]) over H(x), there is a discrete action of G_{[bar.x]/x is} on F_[bar.x] and, by [Ber2, Proposition 4.2.2], one has [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (Recall that in [Ber2] we denoted by F_x the pullback of F under the canonical morphism p_H(x) -> X, which can be also identified with a discrete G_{[bar.x]/x is} -set.)

Given a finite extension k' of k, the ground field extension functor from the category of strictly k-analytic spaces to that of strictly k'-analytic ones X [??] X [??} k' has a left adjoint functor X' [??] X which associates with a strictly k'-analytic space X' the same space considered as a strictly k-analytic one (see [Ber9, §7.1]). The essential image of the latter functor consists of the strictly k-analytic spaces X for which there exists an embedding of k' to the ring of analytic functions O(X). The canonical morphism X' -> X in the category of analytic spaces over k (see [Ber2, §1.4]) gives rise to an isomorphism of locally ringed spaces, to bijections X'₀ [??] X₀ and X'_st [??} X_st and to isomorphisms of étale sites X'_ét [??} X_ét and of étale topoi X'^~_ét [??] X^~_ét. For an étale sheaf F on X, we denote by F' the corresponding étale sheaf on X'.

1.2 THE SHEAF OF NAIVE ANALYTIC FUNCTIONS

Let X be a strictly k-analytic space. For an étale sheaf F on X, we define an étale presheaf F as follows. Given an étale morphism Y -> X, we set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the direct limit is taken over open neighborhoods V of Y_st in Y. The proof of the following lemma is trivial.

Lemma 1.2.1 Assume that an étale sheaf F on X possesses the following property: for any étale morphism Y -> X and any open neighborhood V of Y_st in Y, the canonical map F(Y) -> F(V) is injective. Then the following are true:

(i) the presheaf F is a sheaf;

(ii) the canonical morphism of sheaves F -> F is injective and gives rise to a bijection of stalks F_[bar.x] [??] F[bar.x] for any geometric point x of X over a point x [member of] X_st;

(iii) the sheaf F possesses the stronger property that, given an étale morphism Y -> X and an open neighborhood V of Y_st in Y, the canonical map F(Y) -> F(V) is bijective;

(iv) one has [??] = [??].

Remarks 1.2.2 (i) The property of the sheaf F implies that for any element f [member of] F(X) there exists a unique maximal open subset X_st [subset] U [subset] X such that f comes from F(U).

(ii) The assumption of Lemma 1.2.1 for an étale abelian sheaf F on X is equivalent to the property that, for any étale morphism Y -> X and any element f [member of] F(Y), the intersection Supp(f)[intersection] Y_st is dense in the support Supp(f) of f. For the sheaves we are going to consider even the smaller intersection Supp(f) [intersection] Y₀ is dense in Supp(f).

Let K be a filtered k-algebra K, i.e., a commutative k-algebra with unity provided with an increasing sequence of k-vector subspaces K₀ [subset] K¹ [subset] K² [subset] ··· such that Kⁱ • K^j [subset] K^i+j and K = U^∞_i=0Kⁱ. Given a strictly k-analytic space X, we set O^K,i_X = O_X [cross product]_k Kⁱ. Notice that, if the number of connected components of X is finite, then O^K,i(X) = O(X)[cross product]_k Kⁱ. The sheaf O^K_X = OX [cross product]_k K is an example of a filtered O_X-algebra which is defined as a sheaf of O_X-algebras A provided with an increasing sequence of O_X-modules A⁰ [subset] A¹ [subset] A² [subset] ··· such that Aⁱ • A^j [subset] A^i+j and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If X is reduced, we set C^K,i_X = c_X [cross [product]_k Kⁱ and C^KX = c_X [cross product]_k K.

The sheaf of N^K,i-analytic functions N^K,i_X is the sheaf F associated to F = O^K,i_X (which evidently satisfies the assumption of Lemma 1.2.1). The inductive limit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a sheaf of filtered O_X-algebras. Notice that for every function f [member of] N^K(X) there exists a unique maximal open subset X_st [subset] U [subset] X, called the analyticity set of f, such that f comes from O^K(U). Furthermore, assume we are given a non-Archimedean field k', a strictly k'-analytic space X', a filtered k'-algebra K', a morphism of analytic spaces X' -> X, and a homomorphism of filtered algebras K -> K' over an isometric embedding of fields k [??] k'. If the analyticity set of a function f [member of] N^K(X) contains the image of X'_st in X, then there is a well-defined function φ*(f) [member of] N^K'(X'). For example, if k' [subset] [??}^a, then the image of X'_st is contained in X_st and the latter property is true for all local sections of N^K_X.

If X is reduced, elements of N^K,i(X) can be interpreted as the maps f that take a point x [member of] X_st to an element f(x) [member of] H(x)[cross product]_k Kⁱ and such that every point from X_st admits an open neighborhood X' [subset] X and an analytic function g [member of] O^K,i(X') with f (x) = g(x) for all x [member of] X'_st. In particular, if K = k, N^K_X is the sheaf n_X introduced in the introduction.

More generally, the sheaf of N^K,i-differential q-form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is the sheaf F associated to F = Ω^q_X [cross product]k Kⁱ (which also satisfies the assumption of Lemma 1.2.1). We also set _[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If X is locally embeddable in a smooth space, the sheaf Ω¹_X is locally free over O_X and, therefore, there is a canonical isomorphism of filtered O_X-modules [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume now that X is reduced, and that c(X) contains a finite extension k' of k. Let X' be the same X considered as a strictly k'-analytic space, and let K' be the filtered k'-algebra K [cross product]k k'. Then the sheaf (O^K_X)' on X' that corresponds to O^K_X coincides with O^K'X'. It follows that (N^KX)' = N^K'_X' and (C^KX)' = C^K'_X'.

1.3 D_X-MODULES AND D_X-MODULES

Till the end of this section, the field k is assumed to be of characteristic zero.

Let X be a smooth k-analytic space. A D_X-modules on X is an étale O_X-module F provided with an integrable connection [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For such a D_X-modules F, the subsheaves of horizontal sections F^[nabla] = Ker([nabla]) and of closed one-forms ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) are étale sheaves of modules over c_X. If F = O_X, the former is c_X and the latter is denoted by Ω^1,cl_X.

(Continues...)

Excerpted from Integration of One-Forms on P-adic Analytic Spacesby Vladimir G. Berkovich Copyright © 2007 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Edition details

• Publisher: Princeton University Press
• Publication date: 2006
• Language: English
• ISBN 10: 0691128626
• ISBN 13: 9780691128627
• Binding: Paperback
• Edition number: 1
• Number of pages: 168