Huber–Kelly representable cdh sheafification input

PDF p. 5: HK18 Theorem 1.4

Let $S$ be noetherian and Nagata, and let $X,Y$ be separated finite-type $S$-schemes. The canonical map identifies the cdh sheafification of the presheaf $h_Y(Z)={\mathrm{Hom}}_S(Z,Y)$, evaluated at $X$, with ${\mathrm{Hom}}_S(X^{\mathrm{s}\mathrm{n}},Y)$, naturally in $X$ and $Y$. Here noetherian means admitting a finite affine cover by spectra of noetherian rings; Nagata means that the normalization of every finite-type $S$-scheme is finite over that scheme. The notation $X^{\mathrm{s}\mathrm{n}}\to X$ denotes the seminormalization of the reduction, with its canonical map to $X$.

For this statement, use a suitably small category equivalent to the category of separated finite-type $S$-schemes, with all $S$-morphisms. Its cdh topology is the least Grothendieck topology in which Nisnevich covering families and singleton proper completely decomposed morphisms generate covering sieves. A Nisnevich family consists of étale morphisms such that every target point has a lift inducing an isomorphism of residue fields. A morphism is completely decomposed if it has such a lift at every target point, and proper means separated, of finite type and universally closed. On a field base this is the supplied field-site topology.

Seminormality and seminormalization

Completely decomposed proper covers and the cdh topology

The formula is the cdh case of Huber–Kelly Theorem 1.4: its rh formula (2) and rh-to-cdh comparison (3) give the stated identification. The source includes the spectrum of a perfect field as an example of a noetherian Nagata base. Its general-base site convention is Section 2A and Remark 2.2(1), using the cdp-morphisms of Definition 2.1; the statement above supplies that convention explicitly rather than extending the field-only interface by reference. The source's Proposition 2.8 also places $X^{\mathrm{s}\mathrm{n}}$ in this finite-type site under the Nagata hypothesis. These are precise external inputs; their proofs are not reconstructed here. The local finite-subintegral descent lemmas do not by themselves prove this representable-sheaf formula.