## 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].

### MSC:

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

Zbl 0885.19002