The Stacks project

Proof. Looking at stalks we see that this is a special case of Theorem 59.91.11. $\square$

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

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

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

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

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

Proof. It is enough to prove this when $Y$ and $Y'$ are quasi-compact. By Morphisms, Lemma 29.28.5 we see that the dimension of the fibres of $f : X \to Y$ and $f' : X' \to Y'$ are bounded. Thus Lemma 59.92.2 implies that

and

have finite cohomological dimension in the sense of Derived Categories, Lemma 13.32.2. Choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathbf{Z}/n\mathbf{Z}$-modules each of whose terms $\mathcal{I}^ n$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules representing $E$. See Injectives, Theorem 19.12.6. By the usual proper base change theorem we find that $R^ qf'_*(g')^{-1}\mathcal{I}^ n = 0$ for $q > 0$, see Theorem 59.91.11. Hence we conclude by Derived Categories, Lemma 13.32.2 that we may compute $Rf'_*(g')^{-1}E$ by the complex $f'_*(g')^{-1}\mathcal{I}^\bullet $. Another application of the usual proper base change theorem shows that this is equal to $g^{-1}f_*\mathcal{I}^\bullet $ as desired. $\square$

Proof. Say $H^ i(E) = 0$ for $i \geq a$ and $H^ j(K) = 0$ for $j \geq b$. We may represent $K$ by a bounded below complex $K^\bullet $ of torsion free $\mathbf{Z}$-modules. (Choose a K-flat complex $L^\bullet $ representing $K$ and then take $K^\bullet = \tau _{\geq b - 1}L^\bullet $. This works because $\mathbf{Z}$ has global dimension $1$. See More on Algebra, Lemma 15.66.2.) We may represent $E$ by a bounded below complex $\mathcal{E}^\bullet $. Then $E \otimes _\mathbf {Z}^\mathbf {L} \underline{K}$ is represented by

Using distinguished triangles

and the trivial vanishing

and

we reduce to the case where $K^\bullet $ is a bounded complex of flat $\mathbf{Z}$-modules. Repeating the argument we reduce to the case where $K^\bullet $ is equal to a single flat $\mathbf{Z}$-module sitting in some degree. Next, using the stupid trunctions for $\mathcal{E}^\bullet $ we reduce in exactly the same manner to the case where $\mathcal{E}^\bullet $ is a single abelian sheaf sitting in some degree. Thus it suffices to show that

when $M$ is a flat $\mathbf{Z}$-module and $\mathcal{E}$ is an abelian sheaf on $X$. In this case we write $M$ is a filtered colimit of finite free $\mathbf{Z}$-modules (Lazard's theorem, see Algebra, Theorem 10.81.4). By Theorem 59.51.3 this reduces us to the case of finite free $\mathbf{Z}$-module $M$ in which case the result is trivially true. $\square$

Proof. There is a canonical map from left to right by Cohomology on Sites, Section 21.50. We will check the equality on stalks. Recall that computing derived tensor products commutes with pullbacks. See Cohomology on Sites, Lemma 21.18.4. Thus we have

where $\overline{y}$ is the image of $\overline{x}$ in $Y$. Since $\mathbf{Z}$ has global dimension $1$ we see that this complex has vanishing cohomology in degree $< - 1 + a + b$ if $H^ i(E) = 0$ for $i \geq a$ and $H^ j(K) = 0$ for $j \geq b$. Moreover, since $H^ i(E)$ is a torsion abelian sheaf for each $i$, the same is true for the cohomology sheaves of the complex $E \otimes _\mathbf {Z}^\mathbf {L} K$. Namely, we have

which is zero in the derived category. In this way we see that Lemma 59.91.13 applies to both sides to see that it suffices to show

This is shown in Lemma 59.92.4. $\square$