Étale morphism

Definition

The following conditions are equivalent for a morphism of schemes $f:X\to Y$:

1. $f$ is flat and unramified.
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is flat and the sheaf of Kähler differentials is zero; $\Omega _{X/Y}=0$.
3. $f$ is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle smooth} of relative dimension 0.

The small étale site

The category of étale $S$-schemes becomes a Grothendieck topology, if one defines the sets of coverings to be jointly-surjective collections of $S$-morphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{f_i:U_i\to U\}} ; i.e., such that the union of images $\bigcup f_{i}(U_{i})$ covers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} . That this forms a grothendieck essentially follows from the following three facts:

1. Open immersions are étale.
2. The étale property lifts by base change: that is, if $f:X\to Y$ is an étale morphism, and $g:Y'\to Y$ is any morphism, then the canonical fibered projection $f'X\times _{Y}Y'\to Y'$ is again étale.
3. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\to Y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g:Y\to Z} are such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\circ f} is étale, then $f$ is étale as well.

Étale cohomology

One begins by defining a presheaf to be a contravariant functor from the underlying category of a small étale site Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} into an abelian category $A$. A sheaf on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is then

Applications

Deligne proved the Weil-Riemann hypothesis using étale cohomology.