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30.19 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.

reference

Proposition 30.19.1. Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a proper morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $R^ if_*\mathcal{F}$ is a coherent $\mathcal{O}_ S$-module for all $i \geq 0$.

Proof. Since the problem is local on $S$ we may assume that $S$ is a Noetherian scheme. Since a proper morphism is of finite type we see that in this case $X$ is a Noetherian scheme also. Consider the property $\mathcal{P}$ of coherent sheaves on $X$ defined by the rule

\[ \mathcal{P}(\mathcal{F}) \Leftrightarrow R^ pf_*\mathcal{F}\text{ is coherent for all }p \geq 0 \]

We are going to use the result of Lemma 30.12.6 to prove that $\mathcal{P}$ holds for every coherent sheaf on $X$.

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 Lemma 30.9.2 and 30.9.3. Hence property $\mathcal{P}$ holds for the third as well.

Let $Z \subset X$ be an integral closed subscheme. We have to find a coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z$, whose stalk at the generic point $\xi $ of $Z$ is a $1$-dimensional vector space over $\kappa (\xi )$ such that $\mathcal{P}$ holds for $\mathcal{F}$. Denote $g = f|_ Z : Z \to S$ the restriction of $f$. Suppose we can find a coherent sheaf $\mathcal{G}$ on $Z$ such that (a) $\mathcal{G}_\xi $ is a $1$-dimensional vector space over $\kappa (\xi )$, (b) $R^ pg_*\mathcal{G} = 0$ for $p > 0$, and (c) $g_*\mathcal{G}$ is coherent. Then we can consider $\mathcal{F} = (Z \to X)_*\mathcal{G}$. As $Z \to X$ is a closed immersion we see that $(Z \to X)_*\mathcal{G}$ is coherent on $X$ and $R^ p(Z \to X)_*\mathcal{G} = 0$ for $p > 0$ (Lemma 30.9.9). Hence by the relative Leray spectral sequence (Cohomology, Lemma 20.13.8) we will have $R^ pf_*\mathcal{F} = R^ pg_*\mathcal{G} = 0$ for $p > 0$ and $f_*\mathcal{F} = g_*\mathcal{G}$ is coherent. Finally $\mathcal{F}_\xi = ((Z \to X)_*\mathcal{G})_\xi = \mathcal{G}_\xi $ which verifies the condition on the stalk at $\xi $. Hence everything depends on finding a coherent sheaf $\mathcal{G}$ on $Z$ which has properties (a), (b), and (c).

We can apply Chow's Lemma 30.18.1 to the morphism $Z \to S$. Thus we get a diagram

\[ \xymatrix{ Z \ar[rd]_ g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_ i & \mathbf{P}^ n_ S \ar[dl] \\ & S & } \]

as in the statement of Chow's lemma. Also, let $U \subset Z$ be the dense open subscheme such that $\pi ^{-1}(U) \to U$ is an isomorphism. By the discussion in Remark 30.18.2 we see that $i' = (i, \pi ) : Z' \to \mathbf{P}^ n_ Z$ is a closed immersion. Hence

\[ \mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1) \cong (i')^*\mathcal{O}_{\mathbf{P}^ n_ Z}(1) \]

is $g'$-relatively ample and $\pi $-relatively ample (for example by Morphisms, Lemma 29.39.7). Hence by Lemma 30.16.2 there exists an $n \geq 0$ such that both $R^ p\pi _*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$ and $R^ p(g')_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi _*\mathcal{L}^{\otimes n}$. Property (a) holds because $\pi _*\mathcal{L}^{\otimes }|_ U$ is an invertible sheaf (as $\pi ^{-1}(U) \to U$ is an isomorphism). Properties (b) and (c) hold because by the relative Leray spectral sequence (Cohomology, Lemma 20.13.8) we have

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

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

Lemma 30.19.2. Let $S = \mathop{\mathrm{Spec}}(A)$ with $A$ a Noetherian ring. Let $f : X \to S$ be a proper morphism. 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 Proposition 30.19.1. Namely, by Lemmas 30.4.5 and 30.4.6 we know that $R^ if_*\mathcal{F}$ is the quasi-coherent sheaf associated to the $A$-module $H^ i(X, \mathcal{F})$ and by Lemma 30.9.1 this is a coherent sheaf if and only if $H^ i(X, \mathcal{F})$ is an $A$-module of finite type. $\square$

Lemma 30.19.3. 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. Set $\mathcal{B} = f^*\widetilde B$. Let $\mathcal{F}$ be a quasi-coherent graded $\mathcal{B}$-module of finite type.

  1. For every $p \geq 0$ the graded $B$-module $H^ p(X, \mathcal{F})$ is a finite $B$-module.

  2. If $\mathcal{L}$ is an ample invertible $\mathcal{O}_ X$-module, then there exists an integer $d_0$ such that $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$ and $d \geq d_0$.

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, Lemma 29.41.5. Also, $B$ is a finitely generated $A$-algebra, and hence Noetherian (Algebra, Lemma 10.31.1). This implies that $X'$ is a Noetherian scheme (Morphisms, Lemma 29.15.6). Note that $X'$ is the relative spectrum of the quasi-coherent $\mathcal{O}_ X$-algebra $\mathcal{B}$ by Constructions, Lemma 27.4.6. 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, Lemma 29.11.6 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 30.9.1). Since the morphism $\pi : X' \to X$ is affine we have

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

by Lemma 30.2.4. Thus (1) follows from Lemma 30.19.2. Given $\mathcal{L}$ as in (2) we set $\mathcal{L}' = \pi ^*\mathcal{L}$. Note that $\mathcal{L}'$ is ample on $X'$ by Morphisms, Lemma 29.37.7. By the projection formula (Cohomology, Lemma 20.51.2) we have $\pi _*(\mathcal{F}' \otimes \mathcal{L}') = \mathcal{F} \otimes \mathcal{L}$. Thus part (2) follows by the same reasoning as above from Lemma 30.16.2. $\square$


Comments (2)

Comment #2712 by Zhang on

Reference: EGA, Chapitre III "Étude cohomologique des faisceaux cohérents", Théorème (3.2.1)


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