68.20 Higher direct images of coherent sheaves

In this section we prove the fundamental fact that the higher direct images of a coherent sheaf under a proper morphism are coherent. First we prove a helper lemma.

Lemma 68.20.1. Let $S$ be a scheme. Consider a commutative diagram

$\xymatrix{ X \ar[r]_ i \ar[rd]_ f & \mathbf{P}^ n_ Y \ar[d] \\ & Y }$

of algebraic spaces over $S$. Assume $i$ is a closed immersion and $Y$ Noetherian. Set $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n_ Y}(1)$. Let $\mathcal{F}$ be a coherent module on $X$. Then there exists an integer $d_0$ such that for all $d \geq d_0$ we have $R^ pf_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$.

Proof. Checking whether $R^ pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes d})$ is zero can be done étale locally on $Y$, see Equation (68.3.0.1). Hence we may assume $Y$ is the spectrum of a Noetherian ring. In this case $X$ is a scheme and the result follows from Cohomology of Schemes, Lemma 30.16.2. $\square$

Lemma 68.20.2. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $R^ if_*\mathcal{F}$ is a coherent $\mathcal{O}_ Y$-module for all $i \geq 0$.

Proof. We first remark that $X$ is a locally Noetherian algebraic space by Morphisms of Spaces, Lemma 66.23.5. Hence the statement of the lemma makes sense. Moreover, computing $R^ if_*\mathcal{F}$ commutes with étale localization on $Y$ (Properties of Spaces, Lemma 65.26.2) and checking whether $R^ if_*\mathcal{F}$ coherent can be done étale locally on $Y$ (Lemma 68.12.2). Hence we may assume that $Y = \mathop{\mathrm{Spec}}(A)$ is a Noetherian affine scheme.

Assume $Y = \mathop{\mathrm{Spec}}(A)$ is an affine scheme. Note that $f$ is locally of finite presentation (Morphisms of Spaces, Lemma 66.28.7). Thus it is of finite presentation, hence $X$ is Noetherian (Morphisms of Spaces, Lemma 66.28.6). Thus Lemma 68.14.6 applies to the category of coherent modules of $X$. For a coherent sheaf $\mathcal{F}$ on $X$ we say $\mathcal{P}$ holds if and only if $R^ if_*\mathcal{F}$ is a coherent module on $\mathop{\mathrm{Spec}}(A)$. We will show that conditions (1), (2), and (3) of Lemma 68.14.6 hold for this property thereby finishing the proof of the lemma.

Verification of condition (1). Let

$0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$

be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of higher direct images

$R^{p - 1}f_*\mathcal{F}_3 \to R^ pf_*\mathcal{F}_1 \to R^ pf_*\mathcal{F}_2 \to R^ pf_*\mathcal{F}_3 \to R^{p + 1}f_*\mathcal{F}_1$

Then it is clear that if 2-out-of-3 of the sheaves $\mathcal{F}_ i$ have property $\mathcal{P}$, then the higher direct images of the third are sandwiched in this exact complex between two coherent sheaves. Hence these higher direct images are also coherent by Lemmas 68.12.3 and 68.12.4. Hence property $\mathcal{P}$ holds for the third as well.

Verification of condition (2). This follows immediately from the fact that $R^ if_*(\mathcal{F}_1 \oplus \mathcal{F}_2) = R^ if_*\mathcal{F}_1 \oplus R^ if_*\mathcal{F}_2$ and that a summand of a coherent module is coherent (see lemmas cited above).

Verification of condition (3). Let $i : Z \to X$ be a closed immersion with $Z$ reduced and $|Z|$ irreducible. Set $g = f \circ i : Z \to \mathop{\mathrm{Spec}}(A)$. Let $\mathcal{G}$ be a coherent module on $Z$ whose scheme theoretic support is equal to $Z$ such that $R^ pg_*\mathcal{G}$ is coherent for all $p$. Then $\mathcal{F} = i_*\mathcal{G}$ is a coherent module on $X$ whose scheme theoretic support is $Z$ such that $R^ pf_*\mathcal{F} = R^ pg_*\mathcal{G}$. To see this use the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.7) and the fact that $R^ qi_*\mathcal{G} = 0$ for $q > 0$ by Lemma 68.8.2 and the fact that a closed immersion is affine. (Morphisms of Spaces, Lemma 66.20.6). Thus we reduce to finding a coherent sheaf $\mathcal{G}$ on $Z$ with support equal to $Z$ such that $R^ pg_*\mathcal{G}$ is coherent for all $p$.

We apply Lemma 68.18.1 to the morphism $Z \to \mathop{\mathrm{Spec}}(A)$. Thus we get a diagram

$\xymatrix{ Z \ar[rd]_ g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_ i & \mathbf{P}^ n_ A \ar[dl] \\ & \mathop{\mathrm{Spec}}(A) & }$

with $\pi : Z' \to Z$ proper surjective and $i$ an immersion. Since $Z \to \mathop{\mathrm{Spec}}(A)$ is proper we conclude that $g'$ is proper (Morphisms of Spaces, Lemma 66.40.4). Hence $i$ is a closed immersion (Morphisms of Spaces, Lemmas 66.40.6 and 66.12.3). It follows that the morphism $i' = (i, \pi ) : \mathbf{P}^ n_ A \times _{\mathop{\mathrm{Spec}}(A)} Z' = \mathbf{P}^ n_ Z$ is a closed immersion (Morphisms of Spaces, Lemma 66.4.6). Set

$\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n_ A}(1) = (i')^*\mathcal{O}_{\mathbf{P}^ n_ Z}(1)$

We may apply Lemma 68.20.1 to $\mathcal{L}$ and $\pi$ as well as $\mathcal{L}$ and $g'$. Hence for all $d \gg 0$ we have $R^ p\pi _*\mathcal{L}^{\otimes d} = 0$ for all $p > 0$ and $R^ p(g')_*\mathcal{L}^{\otimes d} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi _*\mathcal{L}^{\otimes d}$. By the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.7) we have

$E_2^{p, q} = R^ pg_* R^ q\pi _*\mathcal{L}^{\otimes d} \Rightarrow R^{p + q}(g')_*\mathcal{L}^{\otimes d}$

and by choice of $d$ the only nonzero terms in $E_2^{p, q}$ are those with $q = 0$ and the only nonzero terms of $R^{p + q}(g')_*\mathcal{L}^{\otimes d}$ are those with $p = q = 0$. This implies that $R^ pg_*\mathcal{G} = 0$ for $p > 0$ and that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes d}$. Applying Cohomology of Schemes, Lemma 30.16.3 we see that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes d}$ is coherent.

We still have to check that the support of $\mathcal{G}$ is $Z$. This follows from the fact that $\mathcal{L}^{\otimes d}$ has lots of global sections. We spell it out here. Note that $\mathcal{L}^{\otimes d}$ is globally generated for all $d \geq 0$ because the same is true for $\mathcal{O}_{\mathbf{P}^ n}(d)$. Pick a point $z \in Z'$ mapping to the generic point $\xi$ of $Z$ which we can do as $\pi$ is surjective. (Observe that $Z$ does indeed have a generic point as $|Z|$ is irreducible and $Z$ is Noetherian, hence quasi-separated, hence $|Z|$ is a sober topological space by Properties of Spaces, Lemma 65.15.1.) Pick $s \in \Gamma (Z', \mathcal{L}^{\otimes d})$ which does not vanish at $z$. Since $\Gamma (Z, \mathcal{G}) = \Gamma (Z', \mathcal{L}^{\otimes d})$ we may think of $s$ as a global section of $\mathcal{G}$. Choose a geometric point $\overline{z}$ of $Z'$ lying over $z$ and denote $\overline{\xi } = g' \circ \overline{z}$ the corresponding geometric point of $Z$. The adjunction map

$(g')^*\mathcal{G} = (g')^*g'_*\mathcal{L}^{\otimes d} \longrightarrow \mathcal{L}^{\otimes d}$

induces a map of stalks $\mathcal{G}_{\overline{\xi }} \to \mathcal{L}_{\overline{z}}$, see Properties of Spaces, Lemma 65.29.5. Moreover the adjunction map sends the pullback of $s$ (viewed as a section of $\mathcal{G}$) to $s$ (viewed as a section of $\mathcal{L}^{\otimes d}$). Thus the image of $s$ in the vector space which is the source of the arrow

$\mathcal{G}_{\overline{\xi }} \otimes \kappa (\overline{\xi }) \longrightarrow \mathcal{L}^{\otimes d}_{\overline{z}} \otimes \kappa (\overline{z})$

isn't zero since by choice of $s$ the image in the target of the arrow is nonzero. Hence $\xi$ is in the support of $\mathcal{G}$ (Morphisms of Spaces, Lemma 66.15.2). Since $|Z|$ is irreducible and $Z$ is reduced we conclude that the scheme theoretic support of $\mathcal{G}$ is all of $Z$ as desired. $\square$

Lemma 68.20.3. Let $A$ be a Noetherian ring. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of algebraic spaces. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $H^ i(X, \mathcal{F})$ is finite $A$-module for all $i \geq 0$.

Proof. This is just the affine case of Lemma 68.20.2. Namely, by Lemma 68.3.1 we know that $R^ if_*\mathcal{F}$ is a quasi-coherent sheaf. Hence it is the quasi-coherent sheaf associated to the $A$-module $\Gamma (\mathop{\mathrm{Spec}}(A), R^ if_*\mathcal{F}) = H^ i(X, \mathcal{F})$. The equality holds by Cohomology on Sites, Lemma 21.14.6 and vanishing of higher cohomology groups of quasi-coherent modules on affine schemes (Cohomology of Schemes, Lemma 30.2.2). By Lemma 68.12.2 we see $R^ if_*\mathcal{F}$ is a coherent sheaf if and only if $H^ i(X, \mathcal{F})$ is an $A$-module of finite type. Hence Lemma 68.20.2 gives us the conclusion. $\square$

Lemma 68.20.4. Let $A$ be a Noetherian ring. Let $B$ be a finitely generated graded $A$-algebra. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of algebraic spaces. Set $\mathcal{B} = f^*\widetilde B$. Let $\mathcal{F}$ be a quasi-coherent graded $\mathcal{B}$-module of finite type. For every $p \geq 0$ the graded $B$-module $H^ p(X, \mathcal{F})$ is a finite $B$-module.

Proof. To prove this we consider the fibre product diagram

$\xymatrix{ X' = \mathop{\mathrm{Spec}}(B) \times _{\mathop{\mathrm{Spec}}(A)} X \ar[r]_-\pi \ar[d]_{f'} & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(B) \ar[r] & \mathop{\mathrm{Spec}}(A) }$

Note that $f'$ is a proper morphism, see Morphisms of Spaces, Lemma 66.40.3. Also, $B$ is a finitely generated $A$-algebra, and hence Noetherian (Algebra, Lemma 10.31.1). This implies that $X'$ is a Noetherian algebraic space (Morphisms of Spaces, Lemma 66.28.6). Note that $X'$ is the relative spectrum of the quasi-coherent $\mathcal{O}_ X$-algebra $\mathcal{B}$ by Morphisms of Spaces, Lemma 66.20.7. Since $\mathcal{F}$ is a quasi-coherent $\mathcal{B}$-module we see that there is a unique quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$ such that $\pi _*\mathcal{F}' = \mathcal{F}$, see Morphisms of Spaces, Lemma 66.20.10. Since $\mathcal{F}$ is finite type as a $\mathcal{B}$-module we conclude that $\mathcal{F}'$ is a finite type $\mathcal{O}_{X'}$-module (details omitted). In other words, $\mathcal{F}'$ is a coherent $\mathcal{O}_{X'}$-module (Lemma 68.12.2). Since the morphism $\pi : X' \to X$ is affine we have

$H^ p(X, \mathcal{F}) = H^ p(X', \mathcal{F}')$

by Lemma 68.8.2 and Cohomology on Sites, Lemma 21.14.6. Thus the lemma follows from Lemma 68.20.3. $\square$

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