Twisted Kummer and Kummer-Artin-Schreier theories
This paper is devoted to developing a form of Kummer theory extending results of \textit{T. Komatsu} [Manuscr. Math. 114, No. 3, 265--279 (2004; Zbl 1093.11068)]. If \(A\) is a ring, \(B\) a quadratic extension of \(A\), then the author defines group schemes \(U_{B/A}\), \(G_{B/A}\) with exact sequences \[ \begin{aligned} & 0 \rightarrow U_{B/A} \rightarrow \Pi_{B/A} G_{m,B} \rightarrow G_{m,A}\rightarrow 0 \\ & 0 \rightarrow G_{m,A} \rightarrow \Pi_{B/A} G_{m,B} \rightarrow G_{B,A}\rightarrow 0 \end{aligned} \] (with \(\Pi_{B/A}\) denoting Weil restriction) and establishes an analog of Hilbert's theorem 90. He then proves that if \(R\) is a local \({\mathbb Z} [w, 1/n]\) algebra with \(w = e^{2 \pi i/n} + e^{-2\pi i/n}\) then for \(n\) odd \(H^1(R, {\mathbb Z}/n {\mathbb Z})\) is isomorphic to \(U_{B/A}(R)/n\) and that if \(R\) is a local \({\mathbb Z}[w]\)-algebra with \(w = e^{2\pi i/p} + e^{-2\pi i/p}\) then \(H^1(R, {\mathbb Z}/p {\mathbb Z})\) is isomorphic to \(\text{Coker} (G_{B/A} (R) \to (G_{\tilde{B}/A } (R))\). He also uses an equivariant compactification \(G_{B/A} \to {\mathbb P}^1_A\) to establish some results concerning unramified cyclic extension of \(R\).
scientific article; zbMATH DE number 5318513
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