Noetherian devissage for coherent proper pushforward

Finite-power support calculation. Let $V=\mathrm{Spec}A$ be a Noetherian affine chart, $M$ a finite $A$-module, and $I=(a_1,\dots ,a_t)$ an ideal with $\mathrm{Supp}(\tilde{M})\subseteq V(I)$. By [F1], $V(\mathrm{Ann}M)\subseteq V(I)$, so every $a_j$ lies in $\sqrt{\mathrm{Ann}M}$. Choose $e_j\ge 1$ with $a_j^{e_j}M=0$; then every product of $N=1+{\sum }_j(e_j-1)$ generators of $I$ contains some $a_j^{e_j}$, and $I^{N}M=0$. The same argument applies on any Noetherian open subspace. A finite affine cover gives one common exponent by taking the maximum of finitely many local exponents.

Quasi-coherence of an open pushforward. Let $Z$ be an integral Noetherian scheme, $U\subseteq Z$ a quasi-compact open, $j:U\hookrightarrow Z$, and $\mathcal{H}$ quasi-coherent on $Z$. For an affine $V=\mathrm{Spec}A\subseteq Z$, write $\mathcal{H}{\mathrm{\mid }}_V=\tilde{M}$. The open $U\cap V$ has a finite cover by distinguished opens $D(f_1),\dots ,D(f_t)$ of $V$ by [F1]. The sheaf equalizer [F4] gives $\mathrm{\Gamma }(U\cap V,\mathcal{H})$ as the kernel of ${\prod }_i{M}_{f_i}\to {\prod }_{i,j}{M}_{f_i{f}_j}$. For $g\in A$, the restricted cover of $U\cap D(g)$ is $D(g{f}_i)$; exact localization and its commutation with finite products identify the same equalizer after localizing at $g$ with $\mathrm{\Gamma }(U\cap D(g),\mathcal{H})$. Thus $(j_{\ast }\mathcal{H}{\mathrm{\mid }}_U)(D(g))=(j_{\ast }\mathcal{H}{\mathrm{\mid }}_U)(V{)}_g$ for every $g$, compatibly with further restrictions. The affine criterion in [F1] shows $j_{\ast }\mathcal{H}{\mathrm{\mid }}_U$ is quasi-coherent on $Z$. This includes $U\cap V=\mathrm{\varnothing }$, when both equalizers are zero.

Coherent comparison on an integral scheme. Let $Z$ be integral Noetherian with generic point $\xi$, and let $\mathcal{H},\mathcal{E}$ be coherent ${\mathcal{O}}_Z$-modules whose generic stalks have the same finite dimension $r>0$ over $\kappa (\xi )$. Choose an affine neighbourhood $W=\mathrm{Spec}A$ of $\xi$; $A$ is a domain and the two modules on $W$ are finite. Choose bases of their generic fibres and lift them after clearing denominators to maps $A^r\to \mathrm{\Gamma }(W,\mathcal{H})$ and $A^r\to \mathrm{\Gamma }(W,\mathcal{E})$. Their finite kernels and cokernels vanish after tensoring with the fraction field, so finitely many nonzero denominators annihilate them; on a common nonempty principal open $U=D(f)\subseteq W$, both maps are isomorphisms. Fix the resulting isomorphism $\mathcal{E}{\mathrm{\mid }}_U\cong \mathcal{H}{\mathrm{\mid }}_U$, and put $Q=j_{\ast }(\mathcal{H}{\mathrm{\mid }}_U)$ for $j:U\hookrightarrow Z$. By step 1.2, $Q$ is quasi-coherent. The natural maps $\mathcal{H}\to Q$ and $\mathcal{E}\to Q$ (the latter through the chosen isomorphism) have coherent kernels $K_H,K_E$ and coherent images $H^{\mathrm{\prime }},E^{\mathrm{\prime }}$: their kernels are quasi-coherent submodules of finite modules on Noetherian charts, and their images are coherent quotients by [F2]. They restrict to isomorphisms on $U$, so $K_H,K_E$ have support in the proper closed subset $Z\setminus U$. Form $T=H^{\mathrm{\prime }}\cap E^{\mathrm{\prime }}$ inside $Q$. The kernel of $H^{\mathrm{\prime }}\oplus E^{\mathrm{\prime }}\to Q$, $(h,e)\mapsto h-e$, is canonically isomorphic to $T$ by either projection, so $T$ is a quasi-coherent submodule of the coherent $H^{\mathrm{\prime }}\oplus E^{\mathrm{\prime }}$, hence coherent by [F2]. Both $H^{\mathrm{\prime }}\mathrm{/}T$ and $E^{\mathrm{\prime }}\mathrm{/}T$ are coherent and vanish on $U$, so their supports are proper closed subsets of $Z$.

Minimal support. Suppose some coherent module lacks $\mathcal{P}$. Its support is closed by [F1], and the empty support gives the zero module, which has $\mathcal{P}$ by hypothesis. The Noetherian descending-chain condition therefore selects a counterexample $\mathcal{F}$ whose nonempty support $Z$ is minimal among counterexample supports. Every coherent module with support strictly contained in $Z$ has $\mathcal{P}$. If $Z$ is reducible, choose a proper irreducible component $Z_1⊊Z$ with reduced ideal sheaf ${\mathcal{I}}_1$ and generic point ${\eta }_1$. The point ${\eta }_1$ is minimal in the support of $\mathcal{F}$; a prime strictly below its prime in an affine neighbourhood would otherwise be a point of the support specializing to ${\eta }_1$, contradicting that $Z_1$ is a component. Thus the finite module ${\mathcal{F}}_{{\eta }_1}$ is supported only at the maximal ideal of the Noetherian local ring ${\mathcal{O}}_{X,{\eta }_1}$, and the local form of step 1.1 gives ${\mathcal{I}}_{1,{\eta }_1}^n{\mathcal{F}}_{{\eta }_1}=0$ for some $n$. The coherent submodule ${\mathcal{I}}_1^n\mathcal{F}$ therefore misses ${\eta }_1$, so its closed support is a proper subset of $Z$. The coherent quotient $\mathcal{F}\mathrm{/}{\mathcal{I}}_1^n\mathcal{F}$ is supported on $Z_1$, because ${\mathcal{I}}_1={\mathcal{O}}_X$ off $Z_1$, and its support is also a proper subset of $Z$. Both have $\mathcal{P}$ by minimality, and the exact sequence between them gives $\mathcal{P}(\mathcal{F})$ by two-out-of-three, a contradiction. Consequently $Z$ is irreducible.

Reduction to modules on the integral closed subscheme. Give the irreducible closed set $Z$ its reduced integral closed subscheme structure $i:Z_{\mathrm{r}\mathrm{e}\mathrm{d}}\hookrightarrow X$ with coherent ideal $\mathcal{J}$. By step 1.1, since $\mathrm{Supp}\mathcal{F}=Z=V(\mathcal{J})$, some $N$ satisfies ${\mathcal{J}}^N\mathcal{F}=0$. The finite filtration by ${\mathcal{J}}^k\mathcal{F}$ has coherent successive quotients ${\mathcal{H}}_k={\mathcal{J}}^k\mathcal{F}\mathrm{/}{\mathcal{J}}^{k+1}\mathcal{F}$ by [F2]. Each ${\mathcal{H}}_k$ is annihilated by $\mathcal{J}$, hence is $i_{\ast }{\bar{\mathcal{H}}}_k$ for a coherent module on $Z_{\mathrm{r}\mathrm{e}\mathrm{d}}$ by [F3]. If a quotient has zero stalk at the generic point $\xi$ of $Z$, its support is a proper closed subset of $Z$ and it has $\mathcal{P}$ by minimality. It remains to treat a quotient of generic rank $r>0$.

The one-generator witness on the reduced subscheme. Take the witness ${\mathcal{G}}_Z$ from the Statement for $Z_{\mathrm{r}\mathrm{e}\mathrm{d}}$. Because ${\mathcal{J}}_{\xi }={\mathfrak{m}}_{\xi }$ and ${\mathfrak{m}}_{\xi }{\mathcal{G}}_{Z,\xi }=0$, the coherent submodule $\mathcal{J}{\mathcal{G}}_Z$ has zero stalk at $\xi$, hence support strictly contained in $Z$ and property $\mathcal{P}$ by minimality. The exact sequence $0\to \mathcal{J}{\mathcal{G}}_Z\to {\mathcal{G}}_Z\to \mathcal{E}:={\mathcal{G}}_Z\mathrm{/}\mathcal{J}{\mathcal{G}}_Z\to 0$ and the given $\mathcal{P}({\mathcal{G}}_Z)$ imply $\mathcal{P}(\mathcal{E})$. The coherent $\mathcal{E}$ is annihilated by $\mathcal{J}$, so $\mathcal{E}=i_{\ast }\bar{\mathcal{E}}$ for a coherent module on $Z_{\mathrm{r}\mathrm{e}\mathrm{d}}$; its generic stalk is a one-dimensional vector space over $\kappa (\xi )$. Since $\mathcal{P}(0)$ holds and $\mathcal{P}$ is stable under extensions, each finite direct sum ${\mathcal{E}}^{\oplus r}$ has $\mathcal{P}$.

Transfer to each generic-rank-$r$ quotient. Apply step 2.1 on the integral scheme $Z_{\mathrm{r}\mathrm{e}\mathrm{d}}$ to ${\bar{\mathcal{H}}}_k$ and ${\bar{\mathcal{E}}}^{\oplus r}$. Push its coherent kernels, images, common intersection $T$, and quotient sequences forward by the closed immersion $i$; [F5] preserves their exactness and [F3] their coherence. Every pushed-forward kernel and quotient from step 2.1 has support in the proper closed subset $Z\setminus U$, so it has $\mathcal{P}$ by step 2.2. Starting from $\mathcal{P}({\mathcal{E}}^{\oplus r})$ of step 3.2, two-out-of-three in succession gives $\mathcal{P}(i_{\ast }{E}^{\mathrm{\prime }})$, then $\mathcal{P}(i_{\ast }T)$, then $\mathcal{P}(i_{\ast }{H}^{\mathrm{\prime }})$, and finally $\mathcal{P}({\mathcal{H}}_k)$. Thus every successive quotient in step 3.1 has $\mathcal{P}$.

Conclusion. Apply the extension direction of two-out-of-three to the finite filtration of step 3.1, beginning with ${\mathcal{J}}^N\mathcal{F}=0$, which has $\mathcal{P}$ by hypothesis. Step 4.1 gives $\mathcal{P}$ for each successive quotient, so induction through the filtration gives $\mathcal{P}(\mathcal{F})$, contradicting step 2.2. There is no counterexample. If $X=\mathrm{\varnothing }$, its sole coherent module is zero and the explicit $\mathcal{P}(0)$ hypothesis gives the same conclusion. The use of AC is confined to the finite-cover and associated-sheaf suppliers in [F1]–[F4] and the stipulated witnesses ${\mathcal{G}}_Z$; all subsequent choices are finite.