The Stacks project

Lemma 59.92.2. Let $f : X \to Y$ be a proper morphism of schemes all of whose fibres have dimension $\leq n$. Then for any abelian torsion sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $R^ qf_*\mathcal{F} = 0$ for $q > 2n$.

Proof. We will prove this by induction on $n$ for all proper morphisms.

If $n = 0$, then $f$ is a finite morphism (More on Morphisms, Lemma 37.44.1) and the result is true by Proposition 59.55.2.

If $n > 0$, then using Lemma 59.91.13 we see that it suffices to prove $H^ i_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $i > 2n$ and $X$ a proper scheme, $\dim (X) \leq n$ over an algebraically closed field $k$ and $\mathcal{F}$ is a torsion abelian sheaf on $X$.

If $n = 1$ this follows from Theorem 59.83.11. Assume $n > 1$. By Proposition 59.45.4 we may replace $X$ by its reduction. Let $\nu : X^\nu \to X$ be the normalization. This is a surjective birational finite morphism (see Varieties, Lemma 33.27.1) and hence an isomorphism over a dense open $U \subset X$ (Morphisms, Lemma 29.50.5). Then we see that $c : \mathcal{F} \to \nu _*\nu ^{-1}\mathcal{F}$ is injective (as $\nu $ is surjective) and an isomorphism over $U$. Denote $i : Z \to X$ the inclusion of the complement of $U$. Since $U$ is dense in $X$ we have $\dim (Z) < \dim (X) = n$. By Proposition 59.46.4 have $\mathop{\mathrm{Coker}}(c) = i_*\mathcal{G}$ for some abelian torsion sheaf $\mathcal{G}$ on $Z_{\acute{e}tale}$. Then $H^ q_{\acute{e}tale}(X, \mathop{\mathrm{Coker}}(c)) = H^ q_{\acute{e}tale}(Z, \mathcal{F})$ (by Proposition 59.55.2 and the Leray spectral sequence) and by induction hypothesis we conclude that the cokernel of $c$ has cohomology in degrees $\leq 2(n - 1)$. Thus it suffices to prove the result for $\nu _*\nu ^{-1}\mathcal{F}$. As $\nu $ is finite this reduces us to showing that $H^ i_{\acute{e}tale}(X^\nu , \nu ^{-1}\mathcal{F})$ is zero for $i > 2n$. This case is treated in the next paragraph.

Assume $X$ is integral normal proper scheme over $k$ of dimension $n$. Choose a nonconstant rational function $f$ on $X$. The graph $X' \subset X \times \mathbf{P}^1_ k$ of $f$ sits into a diagram

\[ X \xleftarrow {b} X' \xrightarrow {f} \mathbf{P}^1_ k \]

Observe that $b$ is an isomorphism over an open subscheme $U \subset X$ whose complement is a closed subscheme $Z \subset X$ of codimension $\geq 2$. Namely, $U$ is the domain of definition of $f$ which contains all codimension $1$ points of $X$, see Morphisms, Lemmas 29.49.9 and 29.42.5 (combined with Serre's criterion for normality, see Properties, Lemma 28.12.5). Moreover the fibres of $b$ have dimension $\leq 1$ (as closed subschemes of $\mathbf{P}^1$). Hence $R^ ib_*b^{-1}\mathcal{F}$ is nonzero only if $i \in \{ 0, 1, 2\} $ by induction. Choose a distinguished triangle

\[ \mathcal{F} \to Rb_*b^{-1}\mathcal{F} \to Q \to \mathcal{F}[1] \]

Using that $\mathcal{F} \to b_*b^{-1}\mathcal{F}$ is injective as before and using what we just said, we see that $Q$ has nonzero cohomology sheaves only in degrees $0, 1, 2$ sitting on $Z$. Moreover, these cohomology sheaves are torsion by Lemma 59.78.2. By induction we see that $H^ i(X, Q)$ is zero for $i > 2 + 2\dim (Z) \leq 2 + 2(n - 2) = 2n - 2$. Thus it suffices to prove that $H^ i(X', b^{-1}\mathcal{F}) = 0$ for $i > 2n$. At this point we use the morphism

\[ f : X' \to \mathbf{P}^1_ k \]

whose fibres have dimension $< n$. Hence by induction we see that $R^ if_*b^{-1}\mathcal{F} = 0$ for $i > 2(n - 1)$. We conclude by the Leray spectral sequence

\[ H^ i(\mathbf{P}^1_ k, R^ jf_*b^{-1}\mathcal{F}) \Rightarrow H^{i + j}(X', b^{-1}\mathcal{F}) \]

and the fact that $\dim (\mathbf{P}^1_ k) = 1$. $\square$

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