Actual raw simplicial cycles satisfy descent without separatedness

Let $W=U\cup V$ be a binary open cover and put $Z=W\setminus U=V\setminus (U\cap V)$, with its reduced induced structure. On a pure-dimensional component of dimension $d$, put $q=d-r$. The restriction kernels for $C(W)\to C(U)$ and $C(V)\to C(U\cap V)$ are identified by restriction: an integral cycle supported on the same closed scheme $Z\times {\mathrm{\Delta }}^n$ is closed in both ambients, and its face condition and dimension are identical. With this common kernel denoted $D=z_q(Z,\bullet )$ there are compatible exact rows $0\to D\to C(W)\to C(U)\to Q_1\to 0,$ $0\to D\to C(V)\to C(U\cap V)\to Q_2\to 0.$ The map between their first terms is the identity under the stated identification. By [F1] both cokernels $Q_i$ are acyclic. This uses homological dimension indexing on the possibly singular $Z$, with no smoothness hypothesis on $Z$.

By the kernel-to-fiber lemma in [F2], $D$ maps quasi-isomorphically to both actual restriction fibers, compatibly with the rows of step 1.1. Thus the map between those fibers is a quasi-isomorphism, proving that the raw restriction square is homotopy cartesian. A smooth finite-type scheme has finitely many integral clopen components; cycle complexes and the two rows split over them. Apply the argument with the respective $d-r$ on each component when $W$ is not pure-dimensional.

The canonical unit $\eta :C(-)\to R\mathrm{\Gamma }(-,aC)$ is natural for open restriction. Its target has binary open Mayer–Vietoris by derived sheaf gluing: sections on the union are the homotopy fiber of the difference of the two restriction maps to the intersection. The source has that same property by step 2.1. A morphism between these two fiber diagrams is a quasi-isomorphism on the union whenever it is so on the two opens and their intersection.

Choose a finite affine open cover $X=A_1\cup \cdots \cup A_m$. The comparison $\eta$ is a quasi-isomorphism on each $A_i$ by [F3]. Induct on the number of opens in the union. For $W=A_1\cup \cdots \cup A_{m-1}$, the intersection $W\cap A_m$ is a quasi-compact open in the affine finite-type scheme $A_m$, hence smooth quasi-affine and quasi-projective over the field. Its restriction of $aC$ is the sheafification of the same cycle presheaf there. Thus [F3] supplies the actual unit equivalence on this intersection, and step 3.1 gives it on $W\cup A_m$. Induction proves the claim on $X$, including the empty case. Because the maps throughout are the canonical units, their conclusion is independent of the cover and natural for open restriction.