nLab
Kummer sequence

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Idea

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

0μ n𝔾 m() n𝔾 m0, 0 \to \mu_n \to \mathbb{G}_m \stackrel{(-)^n}{\to} \mathbb{G}_m \to 0 \,,

where μ n\mu_n is the group of units of order nn, the group of nnth 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.