Comparison of the MVW and cdp cdh topologies

Given: The field and small scheme category of the Statement. A covering morphism means a morphism whose singleton family generates a covering sieve.

The external definition of $J_{\mathrm{M}\mathrm{V}\mathrm{W}}$ uses the same Nisnevich families and the maps $g:Z⨿Y\to X$, where $Z\to X$ is a closed immersion, $Y\to X$ is proper, and the induced map on reduced open complements is an isomorphism (MVW Definitions 12.21 and 12.23). For a scheme $T$, its reduction $T_{\mathrm{r}\mathrm{e}\mathrm{d}}$ means the unique reduced closed subscheme with underlying space $\mathrm{\mid }T\mathrm{\mid }$; this construction is supplied by Reduced induced closed subscheme construction, applied to the whole space. The open complements, fibre products and coproducts are those in Completely decomposed proper covers and the cdh topology.

Import two precise source facts: every $J_{\mathrm{M}\mathrm{V}\mathrm{W}}$ covering morphism has a residue-field-preserving lift of each point (MVW Example 12.24), and every proper morphism with that lifting property is $J_{\mathrm{M}\mathrm{V}\mathrm{W}}$ covering (MVW Lemma 12.26). The pointwise lifting property is exactly complete decomposition, without the étaleness requirement of a Nisnevich family.

Completely decomposed proper covers and the cdh topology defines $J_{\mathrm{c}\mathrm{d}\mathrm{p}}$ as the least topology containing Nisnevich families and singleton cdp families. Its Justification, step 1.1, proves that $Z⨿Y\to X$ is proper whenever $Z\to X$ is a closed immersion and $Y\to X$ is proper, independently of any condition on the complements. It also constructs the finite coproduct in the chosen category. Nisnevich covers and distinguished squares supplies the same residue-field-lifting and étaleness definition of a Nisnevich family as MVW printed p. 91.