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The scheme of connected components of the Néron model of an algebraic torus. (English) Zbl 0764.14009
Let \(T\) be a torus over a local field \(K\). Let \(\varphi\) be the scheme of connected components of the Néron model of \(T\); it is an étale group scheme of finite type. Hence \(\varphi\) is totally determined by a finitely generated \(G_ k\)-module, where \(G_ k\) is the absolute Galois group of \(k\). In this paper we give an explicit description of \(\varphi\) as \(G_ k\)-module in terms of the complex of \(G_ k\)-modules \(R\Gamma(I,X)\), where \(I\) is the inertia subgroup of the absolute Galois group of \(K\), and \(X\) is the character group of \(T\). — The main result is that there exists an isomorphism respecting the \(G_ k\)-action: \(R\text{Hom}_ \mathbb{Z}(\varphi,\mathbb{Z})\cong\tau_{\leq 1} R\Gamma(I,X)\). In particular, we can compute \(\varphi\) as the zero cohomology of the complex \(R\text{Hom}_ \mathbb{Z}(\tau_{\leq 1} R\Gamma(I,X),\mathbb{Z})\) using any \(I\)-acyclic resolution of \(X\). The functor \(\tau_{\leq 1}\) truncates the complex by degree 1, but maintains the 0 and 1 cohomology.

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14L30 Group actions on varieties or schemes (quotients)
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