Elmanto–Morrow rational comparison cofiber

PDF p. 6: EM23 Theorem 4.10(4)

For a quasi-compact quasi-separated Q-scheme X and integer j ≥ 0, Elmanto–Morrow supply a natural map in D(Z)

Z(j)_mot(X) → Z(j)_cdh(X)

whose cofiber is isomorphic to the underlying complex of a complex of Q-vector spaces. The first complex is the Elmanto–Morrow complex of Motivic complexes and cohomology, restricted along Q → k when X is a k-scheme. The second is the cdh-local motivic complex of the cited paper on its category of qcqs Q-schemes. The map is its canonical motivic-to-cdh map. No identification with evaluation on the separately chosen small cdh site is asserted.

This is the rational-cofiber consequence of the fundamental fiber sequence in Elmanto–Morrow Theorem 4.10(3), equivalently of its pullback square in part (2): the derived de Rham terms on the right are rational. The authors explicitly use that rationality to deduce part (4). A quasi-compact scheme has every open cover admitting a finite subcover; quasi-separated means its diagonal is quasi-compact. The assertion is imported with the hypotheses and reading limits in the reference; the underlying construction and proof are not reconstructed here.