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 φ-torsion spectrum with φ ∈ 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 η-periodic spectra and spectra related to Witt groups.Research is supported by the Russian Science Foundation grant 14-21-00035. Preliminaries on framed correspondencesAll schemes are supposed to be separated and noetherian.Definition 2.1. Let X be a scheme and Z be a closed subscheme of X. Anétale neighborhood of Z in X is a pair of morphisms (p : U → X, r : Z → U ) where p isétale and p • r = i for the closed

Topological models for stable motivic invariants of regular number rings

“…Definition 1. 𝑖 (𝑋) ∈ C ♥ are strongly R-nilpotent. This is clear for free R-modules, and the property is preserved by taking summands and shifts and cofibres by (1). The result follows.…”

“…, 𝑥 𝑛 )) ∈ CAlg(C ♥ ) is idempotent. For varying m, the 1/𝑥 𝑚 𝑖 s form an inverse system indexed on N in an evident way; by taking tensor products, the objects 𝑋/(𝑥 𝑚 1 1 , . .…”

“…By an induction argument, using the octahedral axiom, 𝑋/𝑥 𝑚 is a finite extension of copies of 𝑋/𝑥. Hence, each 𝑋/(𝑥 𝑒 1 1 , . .…”

“…Such results are known as 'rigidity'. Many instances have been proved before, mainly if X is essentially smooth over a field; see, for example, [35,1]. Our main novelty is that we replace the base by a Dedekind domain, at the cost of imposing much stronger assumptions on E. Given a presentably symmetric monoidal stable ∞-category C and a morphism 𝑎 : 𝐿 → 1 with L strongly dualisable, we denote by C ∧ 𝑎 the a-completion; that is, the localisation at maps which become an equivalence after ⊗cof (𝑎).…”

“…If 𝑏 ∈ 𝜋 * * (1), then in our compactly generated situations the b-periodisation E[𝑏 −1 ] is given by the colimit…”