Tame and curve-tame cohomology of ${\mathbb{A}}^1$-invariant étale sheaves

Abstract

We extend the definition of the unramified curve-tame cohomology groups to ${\mathbb{A}}^1$-invariant étale 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.

arXiv comment21 pages; v4: major revision, title changed, material on reciprocity sheaves removed, Section 5 of v3 expanded by adding results about tame and curve-tame cohomology, comments are welcome