Non-equivalences of motivic codimension filtration quotients

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.