Introduction to Étale Cohomology

Frontmatter

Chapter 0. Preliminaries

Abstract

Let C be a category and let u : A → B be a morphism in C. Then u is called a monomorphism (or injective) if the map Hom(C, A) → Hom(C,B) which sends v to uv is injective for all objects C in C. Analogously, u is called an epimorphism (or surjective) if the map Hom(B, C) → Hom(A, C) which sends w to wu is injective for all C ∈ C. The morphism u is bijective if u is both injective and surjeetive. An isomorphism, i.e. a morphism having an inverse, is always bijective. The converse is not true in general.

Güter Tamme

Chapter I. Topologies and Sheaves

Abstract

Let X be a topological space, and let T denote the family of all open subsets of X. T becomes a category if we define

$$ Hom(U,V) = \left\{ {\begin{array}{*{20}{c}} 0&{if U \not\subset V} \\ {inclusion U \to V}&{if U \subset V} \end{array}} \right.$$

for U,V ∈ T. X is a final object in the category T. The intersection ∩U _i of finitely many U₁,…, U _n in T is equal to the product of the U₁,…, U _n in the category T. The union ∪ U _i of arbitrarily many U _i in T is equal to the direct sum of the U _i in the category T (cp. 0.1.1).

Güter Tamme

Chapter II. Étale Cohomology

Abstract

A morphism f : X → Y of schemes is called locally finitely presented, if for each point x ∈ X there are affine open neighbourhoods V of y = f(x) and U of x with f(U) ⊂ V, such that the Γ (V, O_Y)-algebra Γ (U,O _x ) is finitely presented (cp. [17], 6.2.).

Güter Tamme

Backmatter