Coherent higher direct images under proper morphisms

Reduction to an affine base. Coherence of $R^q{f}_{\ast }\mathcal{F}$ is local on $S$ by [F16], so fix an affine open $V=\mathrm{Spec}A\subseteq S$; then $A$ is Noetherian, the inverse image $X_V:=f^{-1}V$ is Noetherian by [F15] since $f$ is of finite type, the base change $f_V:X_V\to V$ of $f$ is proper by [F13], and ${\mathcal{F}}_V:=\mathcal{F}{\mathrm{\mid }}_{X_V}$ is coherent by [F2].

The restriction of a higher direct image. Because $f$ is proper it is quasi-compact and separated, and $\mathcal{F},{\mathcal{F}}_V$ are quasi-coherent, so [F1] applied to $f$ over the affine open $V$ gives $(R^q{f}_{\ast }\mathcal{F}){\mathrm{\mid }}_V\cong \widetilde{H^q(f^{-1}V,\mathcal{F})}$, while [F1] applied to $f_V$ over the affine open $V\subseteq V$ gives $R^q{f}_{V\ast }{\mathcal{F}}_V\cong \widetilde{H^q(f_V^{-1}V,{\mathcal{F}}_V)}=\widetilde{H^q(f^{-1}V,\mathcal{F})}$; hence $(R^q{f}_{\ast }\mathcal{F}){\mathrm{\mid }}_V\cong R^q{f}_{V\ast }{\mathcal{F}}_V$ canonically for every $q\ge 0$.

It therefore suffices to prove the affine statement: for a Noetherian ring $A$, a proper morphism $f:X\to \mathrm{Spec}A$ and a coherent $\mathcal{F}$ on $X$, the sheaf $R^q{f}_{\ast }\mathcal{F}$ is coherent for every $q\ge 0$; indeed the general case then follows from 1.2 because coherence is local on $S$ by [F16].

The affine statement as finiteness of cohomology. Assume now $S=\mathrm{Spec}A$ with $A$ Noetherian and let $X,\mathcal{F}$ be as in 1.3; then $X$ is Noetherian by [F15], so the dévissage lemma [F5] is available on $X$, and applying [F1] with the affine open $S$ itself gives $R^q{f}_{\ast }\mathcal{F}\cong \widetilde{H^q(X,\mathcal{F})}$ for every $q\ge 0$.

For a coherent $\mathcal{H}$ on $X$ the sheaf $R^q{f}_{\ast }\mathcal{H}$ is coherent for all $q\ge 0$ if and only if $H^q(X,\mathcal{H})$ is a finitely generated $A$-module for all $q\ge 0$: by 1.4 the sheaf is the associated sheaf of that module, by [F2] a quasi-coherent module on the locally Noetherian affine scheme $\mathrm{Spec}A$ is coherent precisely when it is of finite type, and an associated sheaf is of finite type exactly when its module is finitely generated.

Define the property $\mathcal{P}$ of coherent ${\mathcal{O}}_X$-modules by: $\mathcal{P}(\mathcal{H})$ holds when $H^q(X,\mathcal{H})$ is a finitely generated $A$-module for every $q\ge 0$. By 1.5 the affine statement of 1.3 is equivalent to the assertion that $\mathcal{P}(\mathcal{H})$ holds for every coherent $\mathcal{H}$ on $X$.

Two-out-of-three, first case. Let $0\to {\mathcal{H}}_1\to {\mathcal{H}}_2\to {\mathcal{H}}_3\to 0$ be a short exact sequence of coherent modules on $X$ and let ${\mathcal{H}}_1,{\mathcal{H}}_2$ satisfy $\mathcal{P}$; the long exact sequence of [F3] gives for each $q$ an exact sequence $H^q(X,{\mathcal{H}}_2)\to H^q(X,{\mathcal{H}}_3)\to H^{q+1}(X,{\mathcal{H}}_1)$, so $H^q(X,{\mathcal{H}}_3)$ is an extension of the submodule $\mathrm{im}(H^q(X,{\mathcal{H}}_3)\to H^{q+1}(X,{\mathcal{H}}_1))$ of the finitely generated module $H^{q+1}(X,{\mathcal{H}}_1)$ by the quotient $\mathrm{im}(H^q(X,{\mathcal{H}}_2)\to H^q(X,{\mathcal{H}}_3))$ of $H^q(X,{\mathcal{H}}_2)$; both are finitely generated over the Noetherian ring $A$ by [F4], and so is $H^q(X,{\mathcal{H}}_3)$, for every $q$, that is, ${\mathcal{H}}_3$ satisfies $\mathcal{P}$.

Two-out-of-three, second case. If instead ${\mathcal{H}}_2,{\mathcal{H}}_3$ satisfy $\mathcal{P}$, the exact sequence $H^{q-1}(X,{\mathcal{H}}_3)\to H^q(X,{\mathcal{H}}_1)\to H^q(X,{\mathcal{H}}_2)$ of [F3] exhibits $H^q(X,{\mathcal{H}}_1)$ as an extension of $\mathrm{im}(H^q(X,{\mathcal{H}}_1)\to H^q(X,{\mathcal{H}}_2))=\ker (H^q(X,{\mathcal{H}}_2)\to H^q(X,{\mathcal{H}}_3))$, a submodule of the finitely generated module $H^q(X,{\mathcal{H}}_2)$, by the image of $H^{q-1}(X,{\mathcal{H}}_3)$, a quotient of the finitely generated module $H^{q-1}(X,{\mathcal{H}}_3)$; by [F4] these are finitely generated, so ${\mathcal{H}}_1$ satisfies $\mathcal{P}$.

Two-out-of-three, third case. If ${\mathcal{H}}_1,{\mathcal{H}}_3$ satisfy $\mathcal{P}$, the exact sequence $H^q(X,{\mathcal{H}}_1)\to H^q(X,{\mathcal{H}}_2)\to H^q(X,{\mathcal{H}}_3)\to H^{q+1}(X,{\mathcal{H}}_1)$ of [F3] exhibits $H^q(X,{\mathcal{H}}_2)$ as an extension of $\mathrm{im}(H^q(X,{\mathcal{H}}_2)\to H^q(X,{\mathcal{H}}_3))=\ker (H^q(X,{\mathcal{H}}_3)\to H^{q+1}(X,{\mathcal{H}}_1))$, a submodule of the finitely generated module $H^q(X,{\mathcal{H}}_3)$, by the image of $H^q(X,{\mathcal{H}}_1)$, a quotient of the finitely generated module $H^q(X,{\mathcal{H}}_1)$; again [F4] gives finite generation, so ${\mathcal{H}}_2$ satisfies $\mathcal{P}$. Thus $\mathcal{P}$ is stable under two-out-of-three.

The generators: fixing the subscheme. Fix an integral closed subscheme $Z\subseteq X$ with generic point $\xi$; then $Z$ is nonempty, reduced and irreducible by [F18], it is Noetherian by [F15] as a closed subscheme of the Noetherian scheme $X$, and the restriction $g:=f{\mathrm{\mid }}_Z:Z\to \mathrm{Spec}A$ is proper by [F13] as the composite of the closed immersion $Z\to X$ with the proper $f$.

Chow's lemma for the subscheme. By [F6] applied to the proper morphism $g$ over the Noetherian base $\mathrm{Spec}A$ there are an integer $N\ge 0$, a scheme $Z^{\mathrm{\prime }}$, a proper surjective morphism $\pi :Z^{\mathrm{\prime }}\to Z$, a closed immersion $\iota :Z^{\mathrm{\prime }}\hookrightarrow {\mathbb{P}}_A^N$ over $A$, and a dense open $U\subseteq Z$ such that ${\pi }^{-1}U\to U$ is an isomorphism; the generic point $\xi$ of the irreducible $Z$ lies in the dense open $U$ by [F18]. Put $\mathcal{L}:={\iota }^{\ast }{\mathcal{O}}_{{\mathbb{P}}_A^N}(1)$.

The graph map is a closed immersion. The map $\varphi :=(\iota ,\pi ):Z^{\mathrm{\prime }}\to {\mathbb{P}}_A^N{\times }_{A}Z={\mathbb{P}}_Z^N$ factors, via the isomorphism $\iota :Z^{\mathrm{\prime }}\to Z_0^{\mathrm{\prime }}$ onto the image of $\iota$, as the composite of the graph morphism ${\mathrm{\Gamma }}_{{\pi }_0}:Z_0^{\mathrm{\prime }}\to Z_0^{\mathrm{\prime }}{\times }_{A}Z$ of ${\pi }_0:=\pi \circ {\iota }^{-1}$ and the base change $Z_0^{\mathrm{\prime }}{\times }_{A}Z\to {\mathbb{P}}_A^N{\times }_{A}Z$ of the closed immersion $Z_0^{\mathrm{\prime }}\subseteq {\mathbb{P}}_A^N$; the graph is a closed immersion because $Z$ is separated over $A$ by [F18] and a graph into a separated scheme is the base change of the diagonal by [F14], and the base change of a closed immersion is a closed immersion by [F14], so $\varphi$ is a closed immersion.

Ampleness of $\mathcal{L}$ and its restrictions. Since the first projection ${\mathbb{P}}_Z^N\to {\mathbb{P}}_A^N$ composed with $\varphi$ is $\iota$, the defining pullback property of the relative twisting sheaf gives ${\varphi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}_Z^N}(1)\cong {\iota }^{\ast }{\mathcal{O}}_{{\mathbb{P}}_A^N}(1)=\mathcal{L}$, and by [F12] the charts of projective space and their overlaps commute with base change; hence for every affine open $V\subseteq Z$ the base change ${\pi }^{-1}V\to {\mathbb{P}}_V^N$ of the closed immersion $\varphi$ is a closed immersion by [F14] and $\mathcal{L}{\mathrm{\mid }}_{{\pi }^{-1}V}$ is the pullback of ${\mathcal{O}}_{{\mathbb{P}}_V^N}(1)$. Consequently $\mathcal{L}$ is H-very ample relative to the affine base $A$, so it is ample on $Z^{\mathrm{\prime }}$ by [F12], and $\mathcal{L}{\mathrm{\mid }}_{{\pi }^{-1}V}$ is H-very ample relative to the affine base $V$, so it is ample on ${\pi }^{-1}V$ by [F12].

Finiteness and vanishing on $Z^{\mathrm{\prime }}$. The scheme $Z^{\mathrm{\prime }}$ is a closed subscheme of the locally Noetherian scheme ${\mathbb{P}}_A^N$ over the Noetherian ring $A$, hence locally Noetherian by [F15], and each ${\mathcal{L}}^{\otimes n}$ is invertible, hence coherent, by [F2]; so [F8] gives that $H^q(Z^{\mathrm{\prime }},{\mathcal{L}}^{\otimes n})$ is a finitely generated $A$-module for all $q,n\ge 0$, while [F7] applied to the H-projective $A$-scheme $Z^{\mathrm{\prime }}$, the ample $\mathcal{L}$ and the coherent module ${\mathcal{O}}_{Z^{\mathrm{\prime }}}$ gives an integer $d_A$ with $H^q(Z^{\mathrm{\prime }},{\mathcal{L}}^{\otimes n})=0$ for all $q>0$ and all $n\ge d_A$.

Vanishing on the affine pieces of $Z$. For every affine open $V\subseteq Z$ the ring $B={\mathcal{O}}_Z(V)$ is Noetherian by [F15], the morphism ${\pi }^{-1}V\to V$ is the base change of the proper $\pi$, hence proper and quasi-compact by [F13], and it factors as the closed immersion ${\pi }^{-1}V\to {\mathbb{P}}_V^N$ followed by the projection, so ${\pi }^{-1}V$ is projective over the Noetherian ring $B$ in the H-projective convention with ample $\mathcal{L}{\mathrm{\mid }}_{{\pi }^{-1}V}$ by 1.13; hence [F7] applied over the affine base $V$ gives an integer $n_V$ with $H^q({\pi }^{-1}V,{\mathcal{L}}^{\otimes n})=0$ for all $q>0$ and all $n\ge n_V$.

A uniform twist. The Noetherian scheme $Z$ has a finite affine open cover $V_1,\dots ,V_r$ by [F15], and every intersection indexed by one or more members of this cover is affine by [F9], so 1.15 applies to each of the finitely many nonempty intersections $V_I$; since $n_0:=\max (d_A,\{n_{V_I}\})$ ranges over a finite set of integers, for every $n\ge n_0$ one has $H^q(Z^{\mathrm{\prime }},{\mathcal{L}}^{\otimes n})=0$ and $H^q({\pi }^{-1}{V}_I,{\mathcal{L}}^{\otimes n})=0$ for all $q>0$.

The Čech–Leray comparison. Fix $n\ge n_0$ and put $\mathcal{G}:={\pi }_{\ast }{\mathcal{L}}^{\otimes n}$; then $\mathcal{G}$ is quasi-coherent because it is $R^0{\pi }_{\ast }$ of the quasi-coherent ${\mathcal{L}}^{\otimes n}$ by [F1], and the ordered Čech complexes agree entrywise with equal differentials, $C^q(\mathcal{V},\mathcal{G})={\prod }_{\mathrm{\mid }I\mathrm{\mid }=q+1}\mathcal{G}(V_I)={\prod }_{\mathrm{\mid }I\mathrm{\mid }=q+1}{\mathcal{L}}^{\otimes n}({\pi }^{-1}{V}_I)=C^q({\pi }^{-1}\mathcal{V},{\mathcal{L}}^{\otimes n})$, by the definition of the direct image [F17]; the scheme $Z$ is quasi-compact and separated by [F18], so [F9] gives ${\check{H}}^q(\mathcal{V},\mathcal{G})\cong H^q(Z,\mathcal{G})$, while the cover ${\pi }^{-1}\mathcal{V}$ is ${\mathcal{L}}^{\otimes n}$-acyclic by 1.16, so [F10] gives ${\check{H}}^q({\pi }^{-1}\mathcal{V},{\mathcal{L}}^{\otimes n})\cong H^q(Z^{\mathrm{\prime }},{\mathcal{L}}^{\otimes n})$; altogether $H^q(Z,\mathcal{G})\cong H^q(Z^{\mathrm{\prime }},{\mathcal{L}}^{\otimes n})$ for every $q\ge 0$.

Coherence of $\mathcal{G}$ and its generic fibre. For every affine open $V\subseteq Z$ the localization [F1] gives $\mathcal{G}{\mathrm{\mid }}_V\cong \widetilde{H^0({\pi }^{-1}V,{\mathcal{L}}^{\otimes n})}$ with $H^0({\pi }^{-1}V,{\mathcal{L}}^{\otimes n})$ a finitely generated module over the Noetherian ring ${\mathcal{O}}_Z(V)$ by [F8], so $\mathcal{G}$ is coherent on the locally Noetherian scheme $Z$ by [F2] and [F16]. Moreover ${\pi }^{-1}U\to U$ is an isomorphism, so the stalk of $\mathcal{G}$ at the point $\xi \in U$ is the stalk of the invertible sheaf ${\mathcal{L}}^{\otimes n}$ at ${\pi }^{-1}\xi$, a free module of rank one over the local ring ${\mathcal{O}}_{Z^{\mathrm{\prime }},{\pi }^{-1}\xi }\cong {\mathcal{O}}_{Z,\xi }$ by [F17], and hence the fibre ${\mathcal{G}}_{\xi }{\otimes }_{{\mathcal{O}}_{Z,\xi }}\kappa (\xi )$ is one-dimensional over the residue field $\kappa (\xi )$.

The pushed module on $X$. Let $j:Z\to X$ be the closed immersion and put ${\mathcal{F}}_Z:=j_{\ast }\mathcal{G}$; then ${\mathcal{F}}_Z$ is coherent by [F11]. Its support is contained in $Z$: for $x\mathrm{\notin }Z$ the complement $X\setminus Z$ is an open neighbourhood of $x$ with $j^{-1}(X\setminus Z)=\mathrm{\varnothing }$, and the stalk is the colimit over a cofinal system of such neighbourhoods, hence zero, by [F17], while for $x\in Z$ the stalk of $j_{\ast }\mathcal{G}$ at $x$ is ${\mathcal{G}}_x$ by [F17]. At the generic point $\xi$ of the integral scheme $Z$, ${\mathcal{O}}_{Z,\xi }=\kappa (\xi )$ and the action of ${\mathcal{O}}_{X,\xi }$ on $(j_{\ast }\mathcal{G}{)}_{\xi }$ factors through ${\mathcal{O}}_{X,\xi }\twoheadrightarrow {\mathcal{O}}_{Z,\xi }=\kappa (\xi )$; hence ${\mathfrak{m}}_{\xi }({\mathcal{F}}_Z{)}_{\xi }=0$, and the stalk itself has dimension one over $\kappa (\xi )$ by 1.18.

The generator has property $\mathcal{P}$. By [F11] and 1.17 there are isomorphisms $H^q(X,{\mathcal{F}}_Z)\cong H^q(Z,\mathcal{G})\cong H^q(Z^{\mathrm{\prime }},{\mathcal{L}}^{\otimes n})$ for every $q\ge 0$, and each $H^q(Z^{\mathrm{\prime }},{\mathcal{L}}^{\otimes n})$ is a finitely generated $A$-module by [F8]; hence $H^q(X,{\mathcal{F}}_Z)$ is finitely generated for every $q$, that is, $\mathcal{P}({\mathcal{F}}_Z)$ holds.

Dévissage concludes the affine case. The zero sheaf has zero cohomology in every degree, so $\mathcal{P}(0)$ holds by 1.6. By 1.7-1.9 the property $\mathcal{P}$ is stable under two-out-of-three, and steps 1.10-1.20 produce for every integral closed subscheme $Z\subseteq X$ with generic point $\xi$ a coherent module ${\mathcal{F}}_Z$ with support contained in $Z$, a one-dimensional $\kappa (\xi )$-stalk annihilated by ${\mathfrak{m}}_{\xi }$ as checked in 1.19, and $\mathcal{P}$; the dévissage lemma [F5] applied to the Noetherian scheme $X$ therefore gives $\mathcal{P}(\mathcal{F})$ for the given coherent $\mathcal{F}$, so $H^q(X,\mathcal{F})$ is a finitely generated $A$-module for every $q\ge 0$ and 1.5 yields the coherence of $R^q{f}_{\ast }\mathcal{F}$ on $\mathrm{Spec}A$ for every $q\ge 0$. This proves the affine statement of 1.3.

Conclusion for general $S$. For every affine open $V\subseteq S$ the restriction $(R^q{f}_{\ast }\mathcal{F}){\mathrm{\mid }}_V$ is, by 1.2, the coherent sheaf $R^q{f}_{V\ast }{\mathcal{F}}_V$ supplied by 1.21, and coherence is local on $S$ by [F16]; hence $R^q{f}_{\ast }\mathcal{F}$ is a coherent ${\mathcal{O}}_S$-module for every $q\ge 0$.

Boundary and choice accounting. If $X=\mathrm{\varnothing }$ then $\mathcal{F}=0$ and $R^q{f}_{\ast }\mathcal{F}=0$ for every $q$, and the zero module is coherent; if $S=\mathrm{\varnothing }$ then also $X=\mathrm{\varnothing }$; if $\mathcal{F}=0$ the same vanishing holds; the degree $q=0$ is not exceptional because the dévissage argument treats all $q\ge 0$ at once and in particular yields coherence of $f_{\ast }\mathcal{F}$. The case $Z=X$ with $X$ integral is one instance of 1.10-1.20, the case of a one-member affine cover of 1.16 and the case $U=Z$ are included in the same argument, and no reducedness or irreducibility of $X$ itself is assumed. The base $S$ is only locally Noetherian, so the reduction 1.1-1.3 covers non-affine and infinite-dimensional bases by locality, and the twist range starts at the endpoint $n=n_0$ of 1.16, which is allowed since all bounds are closed conditions $n\ge n_V$. The Axiom of Choice [F19] is consumed through Chow's lemma [F6], the dévissage lemma [F5] and Serre vanishing [F7], and the Axiom of Dependent Choice through the affine localization [F1] and the Čech comparison [F9]; the finitely many affine pieces and the finitely many intersection indices of 1.16 are indexed by finite sets, and no further selection is made. ∎