Tame and curve-tame cohomology of $\mathbb A^1$-invariant \'etale sheaves

Abstract

We extend the definition of the unramified curve-tame cohomology groups to $\mathbb{A}^1$-invariant \'etale sheaves under some additional hypotheses. We define a pairing of this group with the Suslin homology satisfying desirable properties and using this, we show that the unramified curve-tame cohomology of a smooth geometrically connected variety over a field of positive characteristic agrees with the cohomology of the base field.