Let be a reduced scheme of characteristic the prime number , hence such that for all points
Write
for the endomorphism of the additive group over the étale site of (the structure sheaf regarded as just a sheaf of abelian groups) which is the Frobenius endomorphism minus the identity.
There is a short exact sequence of abelian sheaves over the étale site
This is called the Artin-Schreier sequence (e.g. Tamme, section II 4.2, Milne, example 7.9).
By the discussion at category of sheaves – Epi-/Mono-morphisms we need to show that the left morphism is an injection over any étale morphism , and that for every element there exists an étale site covering such that restricts on this to a morphism which hits the restriction of that element.
The first statement is clear, since says that is a constant section, hence in the image of the constant sheaf and hence for each connected the left morphism is the inclusion
induced by including the unit section and its multiples for . (This uses the “freshman's dream”-fact that in characteristic we have ).
This is injective by assumption that is of characteristic .
To show that is an epimorphism of sheaves, it is sufficient to find for each element an étale cover such that its restriction along this cover is in the image of . The choice
by construction has the desired property concerning , the preimage of is the equivalence class of .
To see that with this choice is indeed an étale morphism of schemes it is sufficient to observe that it is a morphism of finite presentation and a formally étale morphism. The first is true by construction. For the second observe that for a ring homomorphism the generator cannot go to a nilpotent element since otherwise would have to be nilpotent. This implies formal étaleness analogous to the discussion at étale morphism of schemes – Open immersion is Etale.
Günter Tamme, section II 4.2 Introduction to Étale Cohomology
James Milne, example 7.9 of Lectures on Étale Cohomology
Last revised on May 26, 2014 at 06:44:25. See the history of this page for a list of all contributions to it.