Elmanto–Morrow motivic complex input

PDF p. 4: Definition 2.2, citing EM23 Theorem 1.1

Let $F$ be a prime field, namely $\mathbb{Q}$ or ${\mathbb{F}}_p$ for a prime $p$. For each integer $j\ge 0$, Elmanto and Morrow construct a finitary Nisnevich sheaf $\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}$ on the category of quasi-compact quasi-separated $F$-schemes, valued in the derived category $D(\mathbb{Z})$ of complexes of abelian groups. Finitary means compatibility with the filtered inverse limits to which the cited theorem applies. The sheaf is understood in the derived, homotopy-coherent sense of the cited source.

For a field $k$ with prime subfield $F$, restriction along the forgetful functor from qcqs $k$-schemes to qcqs $F$-schemes supplies the corresponding contravariant complex-valued functor on ${\mathrm{S}\mathrm{c}\mathrm{h}}_k$ of Article scheme conventions, with Nisnevich descent for covers in that category. Nisnevich covers are as in Nisnevich covers and distinguished squares.

The existence and Nisnevich descent of the prime-field sheaves are imported from the opening assertion of Elmanto–Morrow, Theorem 1.1, under its stated hypotheses. This item does not reconstruct their construction or import the filtration, spectral sequence, or comparison theorems.