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An extension of Artin-Verdier duality to non-torsion sheaves. (English) Zbl 0599.14017

Let X be the spectrum of the ring of integers of a totally imaginary number \(field\quad K\) or a smooth complete curve over a finite field. Artin-Verdier duality comes from the Yoneda-pairing \(H^ i(X,F)\times Ext^{3-i}(F,{\mathbb{G}}_ m)\to H^ 3(X_ m)\cong {\mathbb{Q}}/{\mathbb{Z}}\) when F is a constructible (étale) sheaf. The author generalizes this to \({\mathbb{Z}}\)-constructible sheaves. Locally these are finitely generated abelian groups, rather than finite. In this case the Yoneda-pairing gives isomorphisms \(H^ i(X,F)^*\cong Ext^{3-i}(F,{\mathbb{G}}_ m)\) for \(i=0, 1\) and \(Ext^{3-i}(F,{\mathbb{G}}_ m)^*\cong H^ i(X,F)\) for \(i=2, 3\). For \(i\geq 4\) the groups are zero. Further the pairings are perfect when the groups are suitably topologized (profinite, discrete). The author also improves some of this for locally constant sheaves of finite type.
In his final section the author extends the result to sheaves that are inductive limits of sequences of \({\mathbb{Z}}\)-constructible sheaves. This leads to appropriate isomorphisms when F is a separated commutative group scheme of finite type, in particular if \({\mathcal A}\) is the Néron model of an abelian variety over K, \(Ext^ 2({\mathcal A},{\mathbb{G}}_ m)=H^ 1(X,{\mathcal A})^*\).
Reviewer: G.Horrocks

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
12G05 Galois cohomology
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