Completely decomposed proper maps are abstract-blowup covers

For every $X$, the reduction $X_{\mathrm{r}\mathrm{e}\mathrm{d}}\to X$ is a $J$-cover: take the abstract blowup family with closed center $X_{\mathrm{r}\mathrm{e}\mathrm{d}}$ and proper member $\mathrm{\varnothing }$. Both open complements are empty, so the required actual isomorphism holds. Properness and complete decomposition survive base change. For the latter, a residue-field lift over $x$ base changes along $x^{\mathrm{\prime }}\to x$ to a $\kappa (x^{\mathrm{\prime }})$-valued lift. Thus, once the assertion is proved for reduced targets, apply it to $Y{\times }_X{X}_{\mathrm{r}\mathrm{e}\mathrm{d}}\to X_{\mathrm{r}\mathrm{e}\mathrm{d}}$ and then use local character along the reduction cover. It suffices to treat reduced targets.

Prove the assertion for reduced targets by induction on $d=\dim X$. The empty case is immediate. If $d=0$, each of the finitely many field components of $X$ has a residue-field-preserving lift in $Y$. These lifts define a section of $p$ over each component and hence over $X$. A map with a section generates the maximal sieve and covers. This proves the base case.

For the induction step, for each generic point ${\eta }_i$ of the reduced $X$ choose a lift $y_i\in Y$ with $\kappa (y_i)=\kappa ({\eta }_i)$. Let $X_i$ be the corresponding reduced irreducible component and $Y_i$ the reduced closure of $y_i$ in $Y$. The map $Y_i\to X$ factors through $X_i$, since $Y_i$ is reduced and its image lies in that closed component. The induced map $Y_i\to X_i$ is proper and birational: the closure is integral with generic residue field $\kappa (y_i)$, and it is closed in the proper $X$-scheme $Y$. Apply [F2] to each such map. After deleting component intersections and the finitely many closed complements of the resulting dense opens, there is an open $U\subseteq X$ meeting every component such that $Y^{\mathrm{\prime }}={\coprod }_i{Y}_i\to X$ is an actual isomorphism over $U$. Let $Z$ be the reduced closed complement of $U$. Then $\dim Z<d$ and $\{Y^{\mathrm{\prime }}\to X,Z\to X\}$ is an actual abstract blowup family.

The base change $Y{\times }_{X}Z\to Z$ is completely decomposed proper and therefore covers by the induction hypothesis, since $Z$ is reduced and has smaller dimension. For the sieve $R$ generated by $p$, its pullback to $Y^{\mathrm{\prime }}$ is maximal because $Y^{\mathrm{\prime }}\to X$ factors through $Y$. Its pullback to $Z$ is covering because it contains the covering sieve generated by $Y{\times }_{X}Z\to Z$. Local character along the abstract blowup family in step 1.3 makes $R$ covering. This proves the induction, and step 1.1 then supplies all nonreduced targets.

Every generator of $J$ covers for the Nisnevich-plus-cdp topology by [F1]; conversely every cdp generator covers for $J$ by step 2.1, and the Nisnevich generators are common. Minimality of the two generated topologies proves equality. Only finitely many lifts and dense opens were selected in each finite-dimensional induction, so the local argument adds no arbitrary-index choice assumption.