Étale cohomology of rigid analytic varieties and adic spaces.

*(English)*Zbl 0868.14010
Aspects of Mathematics. E30. Wiesbaden: Vieweg. x, 450 p. (1996).

Tate introduced rigid analytic spaces over a non-archimedean field. More general spaces are: relative rigid spaces, defined by Bosch and Lütkebohmert; non-archimedean analytic spaces, defined by Berkovich; and adic spaces, defined by the author. – The relative rigid spaces are obtained from formal schemes by localizing with respect to a certain class of blowing-ups. The affine non-archimedean analytic space, corresponding to a given Tate-algebra, consists of the space of all continuous rank one valuations on this algebra. The affine adic space, corresponding to a given Tate-algebra, consists of all continuous valuations of any rank on this algebra. – R. Huber introduces an affinoid ring, which is a generalization of a Tate-algebra, and associates with this an affinoid adic space. This affinoid adic space has the structure \((X,{\mathcal O}_X,(v_x|x\in X))\), where \(X\) is a topological space, \({\mathcal O}_X\) is a sheaf of topological rings on \(X\) and, for every \(x\in X\), \(v_x\) is an equivalence class of valuations on the stalk \({\mathcal O}_{X,x}\). A general adic space is obtained by gluing affinoid adic spaces. It seems that the adic spaces “include” the other generalizations of rigid analytic spaces. Tate’s rigid analytic spaces are still the ones used in applications, i.e., degenerations of abelian varieties, moduli spaces for Drinfeld modules, cohomology of \(p\)-adic symmetric spaces, Jacquet-Langlands theory for function fields.

Chapter 0 of the book under review gives a summary of his results on étale cohomology in the context of rigid analytic spaces. We give some items: cohomology of constant sheaves, base change theorems, cohomology with compact support, finiteness theorems, Poincaré duality, comparison theorems with étale cohomology of schemes, comparison with Berkovich’s étale cohomology. Thus the connection with Berkovich’s work and with the work of de Jong and van der Put on étale cohomology is made clear. – Chapter 1 introduces the theory of adic spaces. The later chapters develop the étale theory of those spaces and provide proofs (for adic spaces) of the results stated in chapter 0. A large part of the book is concerned with étale cohomology with compact support. An important point is the use of compactifications of morphism between adic spaces and the calculation of the étale cohomology of “adic curves”.

The importance of this book lies in the systematic treatment of étale cohomology for adic spaces, comparable with SGA for étale cohomology of schemes. The subject of the book is highly technical in nature and one should not expect an introduction to the theme for the non-expert.

Chapter 0 of the book under review gives a summary of his results on étale cohomology in the context of rigid analytic spaces. We give some items: cohomology of constant sheaves, base change theorems, cohomology with compact support, finiteness theorems, Poincaré duality, comparison theorems with étale cohomology of schemes, comparison with Berkovich’s étale cohomology. Thus the connection with Berkovich’s work and with the work of de Jong and van der Put on étale cohomology is made clear. – Chapter 1 introduces the theory of adic spaces. The later chapters develop the étale theory of those spaces and provide proofs (for adic spaces) of the results stated in chapter 0. A large part of the book is concerned with étale cohomology with compact support. An important point is the use of compactifications of morphism between adic spaces and the calculation of the étale cohomology of “adic curves”.

The importance of this book lies in the systematic treatment of étale cohomology for adic spaces, comparable with SGA for étale cohomology of schemes. The subject of the book is highly technical in nature and one should not expect an introduction to the theme for the non-expert.

Reviewer: M.van der Put (Groningen)

##### MSC:

14F20 | Étale and other Grothendieck topologies and (co)homologies |

32P05 | Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32-XX describing the type of problem) |