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Artin-Verdier duality for \(n\)-dimensional local fields involving higher algebraic \(K\)-sheaves. (English) Zbl 0608.12016

The authors give a generalization of the cohomological formulation of class field theory for local fields of dimension 1 to higher dimension. Let \(K\) be an \(n\)-dimensional local ring, \(\mathcal O\subset K\) be its first discrete valuation subring, \(X= \operatorname{Spec}\mathcal O\), \(p\) be the characteristic of the last residue field of \(K\). The following result holds modulo \(p\)-torsion. There exists a canonical trace isomorphism \[ H_c^{n+2}(X, K_{2n- 1})\,{\tilde \to}\, {\mathbb{Q}}/ {\mathbb{Z}}, \] where \(K_{2n-1}=K_{2n-1}(\mathcal O_X)\) is the sheaf of Quillen’s \(K\)-functors in the étale topology. For every constructible étale sheaf \(F\) on \(X\) the Yoneda pairing \[ H^i_c(X, F) \times \mathrm{Ext}_X^{n+2-i}(F, K_{2n-1}) \to H_c^{n+2}(X, K_{2n-1}) \] is a nondegenerate pairing of finite groups (all cohomologies are with compact support).

MSC:

11S31 Class field theory; \(p\)-adic formal groups
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
11S70 \(K\)-theory of local fields
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References:

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