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Artin-Schreier sequence

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Definition

Let XX be a reduced scheme of characteristic the prime number pp, hence such that for all points xXx \in X

p𝒪 X,x=0. p \cdot \mathcal{O}_{X,x} = 0 \,.

Write

Fid() p():(𝔾 a) X(𝔾 a) X F - id \coloneqq (-)^p - (-) \;\colon\; (\mathbb{G}_a)_X \longrightarrow (\mathbb{G}_a)_X

for the endomorphism of the additive group over the étale site X etX_{et} of XX (the structure sheaf regarded as just a sheaf of abelian groups) which is the Frobenius endomorphism F()() pF(-) \coloneqq (-)^p minus the identity.

Proposition

There is a short exact sequence of abelian sheaves over the étale site

0(/p) X(𝔾 a) XFid(𝔾 a) X0. 0 \to (\mathbb{Z}/p\mathbb{Z})_X \to (\mathbb{G}_a)_X \stackrel{F-id}{\to} (\mathbb{G}_a)_X \to 0 \,.

This is called the Artin-Schreier sequence (e.g. Tamme, section II 4.2, Milne, example 7.9).

Proof

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 U YXU_Y \to X, and that for every element s𝒪 Xs \in \mathcal{O}_X there exists an étale site covering {U iX}\{U_i \to X\} such that () p()(-)^p- (-) restricts on this to a morphism which hits the restriction of that element.

The first statement is clear, since s=s ps = s^p says that ss is a constant section, hence in the image of the constant sheaf /p\mathbb{Z}/p\mathbb{Z} and hence for each connected U YXU_Y \to X the left morphism is the inclusion

/p𝒪 X \mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathcal{O}_{X'}

induced by including the unit section e Xe_{X'} and its multiples re Xr e_{X'} for 0r<p0 \leq r \lt p. (This uses the “freshman's dream”-fact that in characteristic pp we have (a+b) p=a p+b p(a + b)^p = a^p + b^p).

This is injective by assumption that XX is of characteristic pp.

To show that () p()(-)^p - (-) is an epimorphism of sheaves, it is sufficient to find for each element s𝒪 X=As \in \mathcal{O}_X = A an étale cover Spec(B)Spec(A)Spec(B) \to Spec(A) such that its restriction along this cover is in the image of () p():BB(-)^p - (-) \colon B \to B. The choice

BA[t]/(tt ps) B \coloneqq A[t]/(t- t^p - s)

by construction has the desired property concerning ss, the preimage of ss is the equivalence class of tt.

To see that with this choice Spec(B)Spec(A)Spec(B) \to Spec(A) 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 BTB \to T the generator tt cannot go to a nilpotent element since otherwise ss would have to be nilpotent. This implies formal étaleness analogous to the discussion at étale morphism of schemes – Open immersion is Etale.

References

Last revised on May 26, 2014 at 06:44:25. See the history of this page for a list of all contributions to it.