Non-equivalences of motivic codimension filtration quotients

TL;DR

This paper investigates motivic equivalences of certain quotient objects in motivic homotopy categories, establishing conditions for isomorphisms and equivalences of categories related to residue fields and cycle modules.

Contribution

It proves that motivic equivalence of specific quotient objects implies residue field isomorphisms and establishes categorical equivalences involving these objects and Rost cycle modules.

Findings

Motivic equivalence implies residue field isomorphism.

Isomorphism of Hom groups corresponds to correspondences between points.

Equivalence of Rost cycle modules and the homotopy heart of DM(k).

Abstract

We prove that a motivic equivalence of objects of the form \begin{equation*} X/(X-x)\simeq X^\prime/(X^\prime-x^\prime) \end{equation*} in $\mathbf{H}^\bullet(B)$ or $\mathbf{DM}(B)$ over a scheme $B$, where $x$ and $x^\prime$ are closed points of smooth $B$-schemes $X$ and $X^\prime$, implies an isomorphism of residue fields, i.e. \[x\cong x^\prime.\] For a given $d\geq 0$, $X,X^\prime\in\mathrm{Sm}_B$, $\operatorname{dim}_B X=d=\operatorname{dim}_B X^\prime$, and closed points $x$ and $x^\prime$ that residue fields are simple extensions of the ones of $B$, we show an isomorphism of groups \[\mathrm{Hom}_{\mathbf{DM}(B)}(X/(X-x),X^\prime/(X^\prime-x^\prime)))\cong\mathrm{Cor}(x,x^\prime),\] and prove that it leads to an equivalence of subcategories. Additionally, using the result on perverse homotopy heart by F.~D\'eglise and N.~Feld and F.~Jin and the strict homotopy invariance theorem for presheaves with transfers over fields by the first author, we prove an equivalence of the Rost cycle modules category and the homotopy heart of $\mathbf{DM}(k)$ over a field $k$ with integral coefficients.