Results of “purity” for smooth varieties over a finite field. Appendix to the article of J.-L. Colliot-Thélène. (Résultats de “pureté” pour les variétés lisses sur un corps fini. Appendice à l’article de J.-L. Colliot-Thélène.) (French) Zbl 0885.19003

Goerss, P. G. (ed.) et al., Algebraic \(K\)-theory and algebraic topology. Proceedings of the NATO Advanced Study Institute, Lake Louise, Alberta, Canada, December 12–16, 1991. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 407, 57-62 (1993).
Summary: In this note, we extend the main results of J.-L. Colliot-Thélène [ibid., 35-55 (1993; Zbl 0885.19002)] to more general coefficients than \(\mu_n^{\otimes d}\). For a constant-twisted sheaf \(A\), with geometric fibre \(\mathbb{Z}/ \ell^n\), coming from the ground field (e.g. \(A= \mu_n^{\otimes i})\), we still prove that, with the notation of [loc. cit.], \(H^i(X_{Zar}, {\mathcal H}_X^{d+1} (A))=0\) for \(i=d-1\) and \(d-2\). (If \(\ell=2\), a technical hypothesis on \(A\) is necessary; it holds for \(A= \mu_n^{\otimes i}.)\) For an ind-constant-twisted sheaf \(B\), with geometric fibre \(\mathbb{Q}_\ell/ \mathbb{Z}_\ell\), not isomorphic to \(\mathbb{Q}_\ell/ \mathbb{Z}_\ell (d)\), we prove (under a small technical hypothesis when \(\ell=2)\) that the sheaf \({\mathcal H}_X^{d+1} (B)\) is itself 0, as well as all the terms of its Gersten resolution. The latter result in fact holds for a smooth variety defined over an arbitrary (not necessarily finite) finitely generated field; its proof is much easier than the one for the former result and does not rely on the results of [loc. cit.], while the proof of the first result does.
For the entire collection see [Zbl 0880.00040].


19F05 Generalized class field theory (\(K\)-theoretic aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry


Zbl 0885.19002