Motivic complexes and cohomology

PDF p. 4: Definition 2.2

Let $k$ and ${\mathrm{S}\mathrm{c}\mathrm{h}}_k$ have the conventions of Article scheme conventions: objects of ${\mathrm{S}\mathrm{c}\mathrm{h}}_k$ are quasi-compact quasi-separated schemes over $k$. For an integer $j\ge 0$, the motivic complex of weight $j$ is the restriction to ${\mathrm{S}\mathrm{c}\mathrm{h}}_k$ of the Elmanto–Morrow Nisnevich sheaf $\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}$ supplied by Elmanto–Morrow motivic complex input. It is a contravariant functor with values in $D(\mathbb{Z})$, the derived category of complexes of abelian groups.

For $X\in {\mathrm{S}\mathrm{c}\mathrm{h}}_k$ and $i\in \mathbb{Z}$, its motivic cohomology group of degree $i$ and weight $j$ is $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i,j}(X,\mathbb{Z}):=H^i(\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X))\mathrm{.}$ For a positive integer $n$, define finite coefficients objectwise by $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i,j}(X,\mathbb{Z}\mathrm{/}n):=H^i(\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X){\otimes }_{\mathbb{Z}}^{\mathbf{L}}\mathbb{Z}\mathrm{/}n)\mathrm{.}$ The derived tensor product here is equivalently the mapping cone of multiplication by $n$. A morphism $X\to Y$ of $k$-schemes induces pullback from the group of $Y$ to that of $X$.

The external input Elmanto–Morrow motivic complex input supplies the specified complex-valued functor and its Nisnevich descent on the stated category. By Cohomology descends to the derived category, $H^i$ is a functor on $D(\mathbb{Z})$, so the integral group and pullback are independent of representatives and respect identities and composition. Functorial finite coefficients for derived complexes supplies the derived quotient, its equivalence with the cone, and its functoriality; applying cohomology gives the finite-coefficient group and its pullback. Evaluation in this definition is evaluation of the supplied complex-valued sheaf; no identification with derived sections on a separately chosen site is assumed.

The article's Definition 2.2 and the introduction and opening statement of Elmanto–Morrow Theorem 1.1 were read. The construction is used through the explicitly marked external input; its proof is not reconstructed here.