# Rigidity for linear framed presheaves and generalized motivic cohomology theories

## TL;DR

This paper proves a rigidity property for certain motivic cohomology theories, showing their values are stable under specific algebraic conditions, extending known rigidity results to a broader class of spectra.

## Contribution

It establishes a rigidity property for homotopy invariant stable linear framed presheaves and derives a variant of Gabber's rigidity theorem for a wide class of motivic cohomology theories.

## Key findings

- Rigidity property for homotopy invariant stable linear framed presheaves.
- A variant of Gabber rigidity theorem for $ ext{GW}(k)$-torsion spectra.
- Values of certain cohomology theories coincide at Henselian rings and residue fields.

## 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 $\phi$-torsion spectrum with $\phi\in\mathrm{GW}(k)$ 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.