External classical coefficient, derived Yoneda and effective-localization interface

PDF p. 7: Lemma2.9: smooth-point supported purity

Over a perfect field, let $\mathcal{A}$ be the category of Nisnevich sheaves with transfers on smooth separated finite-type schemes, with integral coefficients. For $j\ge 0$, the classical complex $\mathbb{Z}(j)$ is bounded above and A¹-local. Its actual restriction functor is naturally modeled by $R{\mathrm{\Gamma }}_{\mathrm{Z}\mathrm{a}\mathrm{r}}(Y,\mathbb{Z}(j))\simeq R{\mathrm{\Gamma }}_{\mathrm{N}\mathrm{i}\mathrm{s}}(Y,\mathbb{Z}(j))\simeq R{\mathrm{Hom}}_{D(\mathcal{A})}({\mathbb{Z}}_{\mathrm{t}\mathrm{r}}(Y),\mathbb{Z}(j))\mathrm{.}$ The first comparison is the canonical topology-change map, not a chosen isomorphism of groups; the second is derived Yoneda evaluation. Against an A¹-local target, localization identifies Hom in $D^{-}(\mathcal{A})$ with Hom in $D{M}_{\mathrm{N}\mathrm{i}\mathrm{s}}^{\mathrm{e}\mathrm{f}\mathrm{f},-}$. These comparisons are natural for actual open maps, and derived Hom takes a cone in the source to the corresponding restriction fiber.

The Tate convention is $\mathbb{Z}(r)=C_{\ast }{\mathbb{Z}}_{\mathrm{t}\mathrm{r}}({\mathbb{G}}_m^{\wedge r})[-r]$ for $r\ge 0$. In the motive category this is the $r$-fold tensor power of $\mathbb{Z}(1)$, so $\mathbb{Z}(r)(d)\cong \mathbb{Z}(r+d)$ for $r,d\ge 0$. Classical negative weights are zero by MVW Definition3.1. The current EM construction itself has weights $j\ge 0$; extending its notation by zero for $j<0$ is an explicit convention, not another EM construction.

Given: Perfect field, integral coefficients, the source's bounded-above effective category, smooth separated finite-type $Y$, AC and fixed small sites/universes.

Goal: Import the concrete classical coefficient model needed for the exact restriction/Hom calculation.

[F1] MVW13.1/13.4 give the sheaf-with-transfers category, enough injectives and actual Yoneda evaluation;13.5 describes its complex version. MVW13.10–13.11 give the canonical Zariski/Nisnevich comparison;14.6.1,14.8–14.12 and14.16 give A¹-local models and representability.

[F2] MVW Definition3.1, Example8.10, Definition14.2 and Lemma14.4 give the stated coefficient and Tate tensor conventions: singular-complex replacement is an A¹-equivalence, while pointed representables tensor as the smash product. This identifies $\mathbb{Z}(r)(d)$ with $\mathbb{Z}(r+d)$ in the motive category.

Import [F1]–[F2] as the concrete derived coefficient package. At the chain level, the derived Yoneda assertion is implemented by resolving the coefficient and using $\mathrm{Hom}({\mathbb{Z}}_{\mathrm{t}\mathrm{r}}(Y),I)=I(Y)$ termwise. Injective sheaves with transfers are Nisnevich-acyclic by13.4, and the complex derived-evaluation interpretation is the standard hyperext construction recorded in13.5; no isomorphism of unrelated group functors is used to choose a restriction arrow. The local relative-Hom lemma applies the actual cone to these Hom complexes. Foundational derived resolutions, A¹-locality and the comparison foundations are imports; this does not claim a new reconstruction of all motivic categories.