Étale Morphisms of Schemes

Étale Morphisms of Schemes 024j

Source file: src/etale.tex THE ÉTALE TOPOLOGY ON SCHEMES

In this Chapter, we study étale morphisms of schemes. Our principal goal is to equip the reader with enough (commutative) algebraic tools to approach a treatise on étale cohomology. An auxiliary goal is to provide enough evidence to ensure that the reader stops calling the phrase “the étale topology of schemes” an exercise in general nonsense, if (s)he does indulge in such blasphemy.

Étale Morphisms of Schemes

Contents 1. Introduction 1 2. Conventions 2 3. Unramified morphisms 2 4. Three other characterizations of unramified morphisms 4 5. The functorial characterization of unramified morphisms 5 6. Topological properties of unramified morphisms 6 7. Universally injective, unramified morphisms 7 8. Examples of unramified morphisms 9 9. Flat morphisms 10 10. Topological properties of flat morphisms 11...

On the Space of Morphisms between Étale Groupoids

Given two étale groupoids G and G, we consider the set of pointed morphisms from G to G. Under suitable hypothesis we introduce on this set a structure of Banach manifold which can be considered as the space of objects of an étale groupoid whose space of orbits is the space of morphisms from G to G. This is a drawing up of a talk given at the IHP Paris in february 2007, in the framework of the ...

PRO-ÉTALE COHOMOLOGY Contents

Contents 1. Introduction 1 2. Some topology 2 3. Local isomorphisms 4 4. Ind-Zariski algebra 6 5. Constructing w-local affine schemes 6 6. Identifying local rings versus ind-Zariski 10 7. Ind-´ etale algebra 14 8. Constructing ind-´ etale algebras 15 9. Weaklyétale versus pro-´ etale 18 10. Constructing w-contractible covers 18 11. The pro-´ etale site 21 12. Points of the pro-´ etale site 29 1...