Effective descent and base change of embedded Hilbert families

For a faithfully flat ring map $A\to B$, write a module descent datum as an overlap isomorphism $\theta :N{\otimes }_{A}B\to B{\otimes }_{A}N$ satisfying its cocycle identity. Set $\rho (n)=\theta (n\otimes 1)$. The diagonal identity and cocycle identity give $\mu \rho ={\mathrm{id}}_N$ and $({\mathrm{id}}_B\otimes \rho )\rho (n)=\sum b_j\otimes 1\otimes n_j$ when $\rho (n)=\sum b_j\otimes n_j$; also $\rho (bn)=b\rho (n)$, with $b$ acting in the first factor. Put $M=\{n\in N:\rho (n)=1\otimes n\}$. Since $B$ is flat over $A$, tensoring the equalizer defining $M$ identifies $B{\otimes }_{A}M$ with the equalizer of ${\mathrm{id}}_B\otimes \rho$ and insertion of $1$ in the middle factor. The cocycle formula therefore makes $\rho$ land in $B{\otimes }_{A}M$. Multiplication $\mu :B{\otimes }_{A}M\to N$ is inverse to that map: $\mu \rho =\mathrm{id}$ by the diagonal identity, and $\rho (bm)=b\otimes m$ for $m\in M$. This proves effectivity. The same equalizer identifies compatible module maps with maps on $M$, giving uniqueness and descent of maps.

Apply step 1.1 on affine pieces of an fpqc cover to the ideal of the compatible embedded subschemes, viewed as a submodule of the structure algebra of $X_T$. Descent of its inclusion and multiplication stability yields an ideal in that structure algebra; the descended quotient algebra defines the unique closed subscheme. Equalizers commute with restriction to affine opens, so these ideals glue. Finite presentation descends as well: descend finitely many generators by [F1], obtain a finite free surjection onto the descended module, and descend finite generation of its relation kernel by [F1]. For the ideal defining a closed immersion, finite generation alone gives finite presentation of the quotient algebra. Flatness of the quotient descends by [F1]. The fpqc cover on $X_T$ is the base change of that on $T$, so these conclusions apply to the embedded families in question.

Pulling a quotient structure algebra back gives its scheme theoretic inverse image, still finitely presented and flat, since finite presentations tensor and flatness is stable under base change. On a geometric fibre the new fibre is a field extension of the old one, so [F2] preserves its polynomial. This also shows that fixed-polynomial membership can be checked on a surjective fpqc cover. Finally finitely many values of the polynomial determine it: its degree is bounded locally by an ambient projective-space dimension. Each such value is locally constant by [F2], so near each point the entire polynomial is constant. Its loci are therefore open and, since their complements are unions of the other loci, closed. This proves the sheaf and stratum assertions.