Artin-Schreier-Witt extensions and normal bases (Q1940035)

Let \(\Gamma\) be a finite group and \(R\) a ring. Let \(G\) be an affine group schme over \(R\) and \(i: \Gamma \to G\) be a homomorphism. The group scheme \(U(\Gamma)\) represents the functor that sends \(R\) to \(R[\Gamma]^{\times}\), the multiplicative group of invertible elements of the group ring \(R[\Gamma]\). The sculpture problem is to find an isomorphism \(\alpha: \Gamma \to \Gamma\), and homomorphisms \(f: \Gamma \to U(\Gamma)_R\) and \(h: U(\Gamma)_R \to G\) such that \(hf=i\alpha\). As an opposite, the embedding problem is to find an isomorphism \(\alpha: \Gamma \to \Gamma\), and homomorphisms \(f:\Gamma \to U(\Gamma)_R\) and \(g: G \to U(\Gamma)_R\) such that \(gi = f\alpha\). The main result of this paper is an affirmative answer to the two aforementioned problems when \(R\) is a ring of characteristic \(p\), \(\Gamma\) a cyclic group of order \(p^n\) and \(G\) be the group schemes of Witt vectors of length \(n\).