Finite constant cdh sections

Let $C_A$ be the constant presheaf. There is a natural map $C_A\to h_{Y_A}$ sending $a\in A$ to the composite $T\to \mathrm{Spec}k\to Y_A$ through the summand labelled $a$. The same definition covers $A=\mathrm{\varnothing }$, when the map has empty domain at each object.

For any cdh sheaf $F$, restriction along this map gives a function $\mathrm{Hom}(h_{Y_A},F)\to \mathrm{Hom}(C_A,F)$. It is bijective. Indeed, given a presheaf map $u:C_A\to F$ and a morphism $f:T\to Y_A$, write $T_a$ for the inverse image of the $a$-summand. The $T_a$ are pairwise disjoint open-and-closed subschemes covering $T$; because $A$ is finite, they are objects of the site. On $T_a$ take $u_{T_a}(a)$. These sections agree on all overlaps: the overlaps are empty, and the sheaf axiom for the empty open cover gives $F(\mathrm{\varnothing })$ exactly one element. By the sheaf axiom they glue uniquely to an element of $F(T)$. Pullback of this decomposition and uniqueness of gluing show that the construction is natural in $T$, hence defines a presheaf map $h_{Y_A}\to F$. If $A$ is empty, a map $T\to Y_A=\mathrm{\varnothing }$ exists only for $T=\mathrm{\varnothing }$, and the same empty-cover argument applies.

The construction in step 2.1 restricts to $u$: the constant morphism labelled $a$ has $T_a=T$ and all other inverse images empty. Conversely, any presheaf map $h_{Y_A}\to F$ must take $f$ to the glued section in step 2.1, since its restriction to each $T_a$ is determined by naturality and the constant map labelled $a$. Thus the two operations are inverse. By [F2], the induced map $a_{\mathrm{c}\mathrm{d}\mathrm{h}}{C}_A\to a_{\mathrm{c}\mathrm{d}\mathrm{h}}{h}_{Y_A}$ is an isomorphism: both sheaves represent the same natural functor of cdh sheaves. This proves the asserted canonical identification of sheaves.

The finite disjoint union $Y_A$ is separated and of finite type over $k$, including when it is empty. An algebraically closed field is perfect, since every algebraic extension is the field itself. Thus [F3] applies to the given $X$ and $Y_A$, and the seminormality hypothesis gives $h_{Y_A}^{\mathrm{c}\mathrm{d}\mathrm{h}}(X)\cong {\mathrm{Hom}}_k(X,Y_A)$. Together with step 3.1 this yields the first displayed bijection.

A finite-type scheme over a field is Noetherian: its finite affine cover has finitely generated coordinate algebras, which are Noetherian by the Hilbert basis theorem, and a finite union of Noetherian open subspaces is Noetherian. Applying [F4] to $X$ gives the second displayed bijection. The two bijections are natural in $X$ and $A$: the first is induced by natural sheaf maps and the natural comparison in [F3], and the second is natural by [F4].

Steps 3.1, 4.1 and 5.1 prove the sheaf identification and both claimed bijections, including the empty cases covered by the constructions and [F4].