A unified Kummer-Artin-Schreier sequence
For \(b\) in a commutative ring \(R_0\), let \(G\{b\}\) be the affine group scheme defined by \(\{(x,y)\mid x(1-by)=1\}\). This is smooth with connected fibers of dimension one, and it turns out that \(H^1(R,G\{b\})\) is a relative Picard group familiar from \(K\)-theory for any \(R_0\)-algebra \(R\). Now let \(p\) be a prime, and let \(\zeta\) be a \(p\)-th root of unity. Then over \(R_0 = \mathbb{Z}[\zeta]\) there is an exact sequence \[ 1\longrightarrow \mathbb{Z}/p\mathbb{Z}\longrightarrow G\{1-\zeta \}\longrightarrow G\{(1-\zeta)^p\}\longrightarrow 1. \] The cohomology for this sequence reduces to the Kummer sequence when \(1/p\in R\), and it reduces to the Artin-Schreier sequence when \(p=0\) in \(R\). For \(p=2\) (and arbitrary \(R)\) it yields the description of étale quadratic algebras previously derived by adhoc methods.
scientific article; zbMATH DE number 3983328