The Stacks project

Lemma 67.20.2. 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 65.23.5. Hence the statement of the lemma makes sense. Moreover, computing $R^ if_*\mathcal{F}$ commutes with ├ętale localization on $Y$ (Properties of Spaces, Lemma 64.26.2) and checking whether $R^ if_*\mathcal{F}$ coherent can be done ├ętale locally on $Y$ (Lemma 67.12.2). 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 65.28.7). Thus it is of finite presentation, hence $X$ is Noetherian (Morphisms of Spaces, Lemma 65.28.6). Thus Lemma 67.14.6 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 67.14.6 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 67.12.3 and 67.12.4. 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 support 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 67.8.2 and the fact that a closed immersion is affine. (Morphisms of Spaces, Lemma 65.20.6). 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 67.18.1 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 65.40.4). Hence $i$ is a closed immersion (Morphisms of Spaces, Lemmas 65.40.6 and 65.12.3). 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 65.4.6). Set

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

We may apply Lemma 67.20.1 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 $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}$. 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 64.15.1.) 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 64.29.5. 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 65.15.2). Since $|Z|$ is irreducible and $Z$ is reduced we conclude that the scheme theoretic support of $\mathcal{G}$ is all of $Z$ as desired. $\square$


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