Amine Koubaa

Purity in the tame cohomology

Let $(X,Z)$ be a regular pair of pure codimension $r$ over a base scheme $S$ of characteristic $p$. Under the assumption of resolution of singularities, we prove purity for the tame cohomology with coefficient in the logarithmic de Rham-Witt sheaves $ν_m(n)$ for the regular pair $(X,Z)$, i.e. the existence of a natural isomorphism \[ Ri^!ν_{X,m}(n)\cong ν_{Z,m}(n-r)[-r] \] for all $m,n\geq 0$, where the logarithmic de Rham-Witt sheaves $ν_{X,m}(n)$ are the Frobenius-fixed elements in the de Rham-Witt sheaves $W_mΩ^{n}_{X/\mathbb{F}_p}$.