Finite-stage descent of properness for finitely presented schemes

Finite kernel of a limit quotient. Let $h_i:Z_i\to Y_i$ be a morphism of finitely presented $A_i$-schemes whose limit $h:Z\to Y$ is a closed immersion. On an affine chart $V=\mathrm{Spec}B\subseteq Y$, its inverse image is $\mathrm{Spec}C$ and $B\twoheadrightarrow C$ by [F2]; both $B$ and $C$ are finitely presented $A$-algebras by [F1]. Choose presentations $B=A[x_1,\dots ,x_r]\mathrm{/}(f_1,\dots ,f_s)$ and $C=A[y_1,\dots ,y_m]\mathrm{/}(g_1,\dots ,g_n)$, and polynomials $P_a(y)$ representing the images of the $x_a$. Since $B\to C$ is surjective, choose lifts ${\tilde{y}}_b\in B$ of the $y_b$. Its kernel is generated by the finite list $g_c(\tilde{y})$ and $x_a-P_a(\tilde{y})$: if $F(x)$ is in the kernel, then $F(P(y))$ lies in $(g_c)$ in $A[y]$, so $F(P(\tilde{y}))$ lies in $(g_c(\tilde{y}))$ in $B$, while $F(x)-F(P(\tilde{y}))$ lies in $(x_a-P_a(\tilde{y}))$. Thus every affine limit ideal is finitely generated.

Descend the ideal. Choose a finite affine cover $Y_i={\bigcup }_t{V}_{t,i}$ and write the limit closed subscheme on $V_t=\mathrm{Spec}{B}_t$ as $V(I_t)$; step 1.1 makes $I_t$ finitely generated. Lift its generators to one stage $j$, defining finite ideals $I_{t,j}\subseteq B_{t,j}$. For each pair $t,u$, the stage overlap $V_{t,j}\cap V_{u,j}$ is quasi-compact, so cover it by finitely many principal affine opens inside $V_{t,j}$. On each such chart the two ideals have finitely many generators and become equal after passage to $A$: each generator of either is a finite linear combination of generators of the other in the limit coordinate ring. Lift the finitely many coefficients and equality equations to a later stage. One common stage makes the two ideals equal on every overlap chart, including an overlap whose limit is empty, because its limit coordinate ring is zero and the resulting unit identity is finite. By [F2] these compatible ideals glue to a finite-type quasi-coherent ideal ${\mathcal{I}}_j$ and a closed subscheme $Z_j^{\mathrm{\prime }}\subseteq Y_j$. Its base change is $h(Z)$; the finite ideal generators and [F1] make $Z_j^{\mathrm{\prime }}$ finitely presented over $A_j$.

Descend the isomorphism. Over $A$, the closed immersion identifies $Z$ with $Z_j^{\mathrm{\prime }}{\times }_{A_j}A$ over $Y$. By [F1] descend this isomorphism and its inverse to a common later stage, then descend their two inverse equations and their equality as maps into $Y_k$. At that stage $h_k$ is the composite $Z_k\stackrel{\sim }{\to }{Z}_k^{\mathrm{\prime }}\hookrightarrow Y_k$, hence a closed immersion. This proves finite-stage closed-immersion descent for the stated finite-presentation schemes without using a properness descent theorem.

Separated stage. Apply steps 1.1–3.1 to the diagonal ${\mathrm{\Delta }}_{X_i\mathrm{/}{A}_i}:X_i\to X_i{\times }_{A_i}{X}_i$, whose limit is closed because $f$ is proper. The source and target are finitely presented over $A_i$, so after enlargement the diagonal is closed and $f_j$ is separated. By [F5] each $A_j$ is Noetherian, and $f_j$ is of finite type.

A projective cover. Apply [F4]'s Noetherian Chow lemma to this separated finite-type $f_j$. Obtain a proper surjective ${\pi }_j:X_j^{\mathrm{\prime }}\to X_j$ and an immersion ${\iota }_j:X_j^{\mathrm{\prime }}\to {\mathbb{P}}_{A_j}^N$. Base change to $A$. The composite $f\pi :X^{\mathrm{\prime }}\to \mathrm{Spec}A$ is proper by [F4], and ${\mathbb{P}}_A^N$ is separated over $A$ by [F4]. Hence the $A$-morphism $\iota :X^{\mathrm{\prime }}\to {\mathbb{P}}_A^N$ is proper and has closed image; since it is an immersion, it is a closed immersion.

Close the cover at a stage. The source $X_j^{\mathrm{\prime }}$ is finite type over the Noetherian $A_j$: ${\pi }_j$ is proper, hence finite type, and $X_j$ is finite type over $A_j$. Choose a finite affine cover of $X_j^{\mathrm{\prime }}$; each coordinate ring is a finite-type $A_j$-algebra and therefore finitely presented by [F5] (equivalently, its finite polynomial presentation has a finitely generated kernel by the Hilbert basis theorem). The immersion ${\iota }_j$ is separated by [F4], and ${\mathbb{P}}_{A_j}^N\to \mathrm{Spec}{A}_j$ is separated by [F4], so their composite $X_j^{\mathrm{\prime }}\to \mathrm{Spec}{A}_j$ is separated by [F4]. Thus the finite affine cover has quasi-compact pairwise overlaps, and $X_j^{\mathrm{\prime }}$ is finitely presented over $A_j$. Apply steps 1.1–3.1 to ${\iota }_j$; after enlargement ${\iota }_k$ is closed. Consequently $X_k^{\mathrm{\prime }}\to \mathrm{Spec}{A}_k$ is proper as a closed subscheme of proper projective space.

Descend properness. The base change ${\pi }_k:X_k^{\mathrm{\prime }}\to X_k$ remains proper and surjective. For any $T\to \mathrm{Spec}{A}_k$ and any closed $C\subseteq X_k{\times }_{A_k}T$, the inverse image ${\pi }_{k,T}^{-1}(C)$ is closed. Its image under the proper composite $(f_k{\pi }_k{)}_T$ is closed in $T$. Surjectivity of ${\pi }_{k,T}$ makes this image exactly $f_{k,T}(C)$, so $f_k$ is universally closed. It is separated by step 4.1 and finite type by hypothesis; therefore it is proper by [F3]. If $X_i$ is empty, its empty base changes are proper by the same definition. This proves the Statement without assuming flatness of any transition map.