La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique. (The Gersten conjecture for the logarithmic Hodge-Witt sheaves)
Let X be a smooth scheme over a perfect field k of characteristic \(p>0,\) \(W_ n\Omega^ i_{X,\log}\) be the logarithmic abelian étale Hodge-Witt sheaf, \(\epsilon: X_{et}\to X_{zar}\) be the projection of the étale topos onto the Zariski topos. For every \(m>0\), the authors constructs a flat resolution of the sheaf \(R^ m\epsilon_*W_ n\Omega^ i_{X,\log}\) in the Zariski topology. From this, in particular, the equality \(R^ m\epsilon_*W_ n\Omega^ i_{X,\log}=0\) follows for all \(m>1\). In view of the notation \({\mathcal H}^ r({\mathbb{Z}}/p^ n{\mathbb{Z}}(i))=R^{r-1}\epsilon_*W_ n\Omega^ i_{X,\log}\), the authors note that the constructed resolution is analogous in the case \(m=0\) to the corresponding Bloch-Ogus resolution for the \(\ell\)-adic sheaves \({\mathcal H}^ r({\mathbb{Z}}/\ell^ n{\mathbb{Z}}(i))\) in the case \(\ell \neq p\) and allows one to obtain the ``Bloch-Quillen formula modulo \(p^ n{}'': CH^ d(X)/p^ n\overset \sim \rightarrow H^ d(X_{zar},{\mathcal H}^ d({\mathbb{Z}}/p^ n{\mathbb{Z}}(d))\).
logarithmic abelian étale Hodge-Witt sheaf
scientific article; zbMATH DE number 4177251
scientific article