Rigidity for linear framed presheaves and generalized motivic cohomology theories

Abstract

A rigidity property for the homotopy invariant stable linear framed presheaves is established. As a consequence a variant of Gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a $\varphi$-torsion spectrum with $\varphi \in \mathrm{GW}(k)$ of rank coprime to the (exponential) characteristic of the base field $k$. It is shown that the values of such cohomology theories at an essentially smooth Henselian ring and its residue field coincide. The result is applicable to cohomology theories representable by $n$-torsion spectra as well as to the ones representable by $\eta$-periodic spectra and spectra related to Witt groups.

arXiv comment27 pages, minor changes. Added Corollary 7.11 (and Corollary 1.3 in the Introduction)