Let $S$ be an affine scheme, $X$ be a projective $S$-scheme, $W,Z\to X$ two reduced, irreducible closed $S$-subschemes, flat over $S$. Let $S'\to S$ be a faithfully flat map, with $S'$ affine.


Assume there exists an $S'$-isomorphism $g: W\_{S'}\xrightarrow{\simeq} Z\_{S'}$, commuting with the closed immersions into $X\_{S'}$.


When does there exist an $S$-morphism $f : W\to Z$ such that $f\_{S'} = g$, and (hence) $f$ is an $S$-isomorphism?



Descent of isomorphisms between irreducible closed subschemes
$W,Z\to X$ are classified by two points $s,s'\in\text{Hilb}\_{X/S}(S)$. Here I mean the Hilbert functor, ie. I am not using its representability.


Since the Hilbert functor is an fpqc sheaf (ie. with no need of a sheafification) and $S'\to S$ is an fpqc cover, the map


$$\text{Hilb}\_{X/S}(S)\to (S'\to S)^{-1}\_{\rm fpqc}\text{Hilb}\_{X/S}(S')$$
is injective.


We have $(S'\to S)^{-1}\_{\rm fpqc}\text{Hilb}\_{X/S} = \text{Hilb}\_{X\_{S'}/S'}$, and so if $s,s'$ agree in $\text{Hilb}\_{X\_{S'}/S'}(S')$ under the above map, we have $s = s'$ in $\text{Hilb}\_{X/S}(S)$.


Am I missing something?


