A coherent comparison of the particular open restriction

PDF p. 13: Proposition 6.1 support argument

Assume the Axiom of Choice and the fixed site and universe bounds of the support and comparison constructions. Let $k$ be an algebraically closed field of arbitrary characteristic, let $X$ be an affine finite-type $k$-scheme of Krull dimension three, and let $T\subset X$ be a closed subset contained in the smooth locus $U=X_{\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}}$ of $X\to \mathrm{Spec}k$. Equip $T$ with its reduced induced structure, and put $V=U\setminus T$ and $W=X\setminus T$. For the actual arrow $V\hookrightarrow U$, the weight-three motivic restriction diagram is equivalent to the actual cycle restriction diagram $[\mathbb{Z}(3{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(U)\to \mathbb{Z}(3{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(V)]\simeq [z^3(U,\bullet )[-6]\to z^3(V,\bullet )[-6]]\mathrm{.}$ It follows in particular that its $H^6$ arrow is ordinary Chow restriction. The square is supplied before applying cohomology. In the cycle complexes, homological degree $n$ is reindexed as cochain degree $-n$ before the shift by $-6$.