nLab Kummer sequence

Contents

Context

Algebra

higher algebra

universal algebra

topos theory

Contents

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.

References

Named after Ernst Kummer.

Last revised on May 26, 2014 at 06:38:09. See the history of this page for a list of all contributions to it.