Étale cohomology of rigid analytic spaces.

*(English)*Zbl 0922.14012From the introduction: The origin of this paper lies in the question on étale cohomology for rigid analytic spaces posed in a paper by P. Schneider and K. Stuhler [Invent. Math. 105, No. 1, 47-122 (1991; Zbl 0751.14016)]. In that paper an étale site and a corresponding cohomology theory for analytic varieties are defined. We prove here that the axioms for an ‘abstract cohomology’ (as stated in the cited paper) hold for this cohomology theory. In addition, we prove a (quasi-compact) base change theorem for rigid étale cohomology and a comparison theorem comparing rigid and algebraic étale cohomology of algebraic varieties. The main tools in this paper are analytic (resp. étale) points and rigid (resp. étale) overconvergent sheaves. In section 2 we (re)introduce some basic notations concerning analytic points and rigid overconvergent sheaves, which are needed later on. We (re)prove a number of folklore results most importantly: (1) Rigid cohomology agrees with Čech cohomology on quasi-compact spaces. (2) The cohomological dimension of a paracompact space is at most its dimension. (3) A base change theorem.

The rest of the paper deals with étale sites and étale cohomology. Étale points and étale overconvergent sheaves are introduced. A key point is the introduction of special étale morphisms of affinoids \(U\to X\), analogues to rational subdomains in the rigid case. Included in the paper is the proof by R. Huber that any étale morphism of affinoids is special étale. This simplifies the original exposition somewhat. A structure theorem for étale morphisms (3.1.2) allows us to give a proof of the étale base change theorem following closely the proof in the rigid case. We calculate the cohomology groups of one dimensional spaces in section 4. This allows us to prove the basic results mentioned at the beginning of this introduction (sections 5, 6 and 7).

This paper may serve as an introduction to rigid and étale cohomology of rigid analytic spaces.

The rest of the paper deals with étale sites and étale cohomology. Étale points and étale overconvergent sheaves are introduced. A key point is the introduction of special étale morphisms of affinoids \(U\to X\), analogues to rational subdomains in the rigid case. Included in the paper is the proof by R. Huber that any étale morphism of affinoids is special étale. This simplifies the original exposition somewhat. A structure theorem for étale morphisms (3.1.2) allows us to give a proof of the étale base change theorem following closely the proof in the rigid case. We calculate the cohomology groups of one dimensional spaces in section 4. This allows us to prove the basic results mentioned at the beginning of this introduction (sections 5, 6 and 7).

This paper may serve as an introduction to rigid and étale cohomology of rigid analytic spaces.

##### MSC:

14G20 | Local ground fields in algebraic geometry |

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

26E30 | Non-Archimedean analysis |