nLab Artin-Schreier sequence

Definition

Let $X$ be a reduced scheme of characteristic the prime number $p$, hence such that for all points $x \in X$

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

Write

$$F - id \coloneqq (-)^p - (-) \;\colon\; (\mathbb{G}_a)_X \longrightarrow (\mathbb{G}_a)_X$$

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

Proposition

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

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

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

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

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

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

This is injective by assumption that $X$ is of characteristic $p$.

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

$$B \coloneqq A[t]/(t- t^p - s)$$

by construction has the desired property concerning $s$, the preimage of $s$ is the equivalence class of $t$.

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

Related concepts

• étale cohomology

• Kummer sequence

• Kummer-Artin-Schreier-Witt exact sequence

• exponential exact sequence

References

• Günter Tamme, section II 4.2 Introduction to Étale Cohomology

• James Milne, example 7.9 of Lectures on Étale Cohomology