nLab Kummer sequence

Idea

For $(-)^n \colon \mathbb{G}_m \longrightarrow \mathbb{G}_m$ the endomorphism of powering by $n$ on the multiplicative group over an étale site, then if $n$ is invertible over the site then there is a short exact sequence

$$0 \to \mu_n \to \mathbb{G}_m \stackrel{(-)^n}{\to} \mathbb{G}_m \to 0 \,,$$

where $\mu_n$ is the group of units of order $n$, the group of $n$th roots of unity.

This is called the Kummer sequence.

The analog for the additive group is the Artin-Schreier sequence. Both are unified in the Kummer-Artin-Schreier-Witt exact sequence.

Related concepts

• Kummer theory

• exponential exact sequence, Artin-Schreier sequence

• Kummer-Artin-Schreier-Witt exact sequence

• Tate twist

References

Named after Ernst Kummer.

• Günter Tamme, section II 4.4 of Introduction to Étale Cohomology

• Jordan Watts, The Kummer sequence (pdf)

• Max Lieblich, around prop. 2.3.4.4 Moduli of twisted sheaves (arXiv:math/0411337)