Representable cdh sheaves and seminormalization

PDF p. 5: Proof of Lemma 2.4, HK18 Theorem 1.4

Let $k$ be a perfect field, and let $X,Y$ be separated finite-type $k$-schemes. Write $h_Y(Z)={\mathrm{Hom}}_k(Z,Y)$ and $h_Y^{\mathrm{c}\mathrm{d}\mathrm{h}}$ for its sheafification on the big cdh site of separated finite-type $k$-schemes. Then there is a natural bijection $h_Y^{\mathrm{c}\mathrm{d}\mathrm{h}}(X)\cong {\mathrm{Hom}}_k(X^{\mathrm{s}\mathrm{n}},Y)\mathrm{.}$ If $X$ is seminormal, this identifies $h_Y^{\mathrm{c}\mathrm{d}\mathrm{h}}(X)$ with ${\mathrm{Hom}}_k(X,Y)$.

A perfect field is one for which every algebraic extension is separable. A finite-type $k$-scheme has a finite affine cover by spectra of finitely generated $k$-algebras; separatedness means its diagonal is a closed immersion. The cdh topology and sheafification are those in Completely decomposed proper covers and the cdh topology, and the seminormalization $X^{\mathrm{s}\mathrm{n}}\to X$ is the subintegral map with seminormal source described in Seminormality and seminormalization. A subintegral map is integral, bijective on points and induces residue-field isomorphisms.