The Stacks project

Proof. Denote $f_ i : X \to X_ i$ the projections. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$. Then we have $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits f_ i^{-1}f_{i, *}\mathcal{F}$ by Lemma 59.51.9. Thus $H^ q_{\acute{e}tale}(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ q_{\acute{e}tale}(X_ i, f_{i, *}\mathcal{F})$ by Theorem 59.51.3. The lemma follows. $\square$

Proof. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$. Consider the Leray spectral sequence for the morphism $f : X \to \mathop{\mathrm{Spec}}(K)$. We obtain

converging to $H^{p + q}_{\acute{e}tale}(X, \mathcal{F})$. The stalk of $R^ qf_*\mathcal{F}$ at a geometric point $\mathop{\mathrm{Spec}}(\overline{K}) \to \mathop{\mathrm{Spec}}(K)$ is the cohomology of the pullback of $\mathcal{F}$ to $X_{\overline{K}}$. Hence it vanishes in degrees $\geq 2$ by Theorem 59.83.10. $\square$

Proof. If $\text{trdeg}_ K(L) = \infty $, then this is clear. If not then we can find a sequence of extensions $L= L_ r/L_{r - 1}/ \ldots /L_1/L_0 = K$ such that $\text{trdeg}_{L_ i}(L_{i + 1}) = 1$ and $r = \text{trdeg}_ K(L)$. Hence it suffices to prove the lemma in the case that $r = 1$. In this case we can write $L = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of its finite type $K$-subalgebras. By Lemma 59.95.2 it suffices to prove that $\text{cd}(A_ i) \leq 1 + \text{cd}(K)$. This follows from Lemma 59.95.3. $\square$

Proof. We may assume $X$ is affine. By Noether normalization, after possibly shrinking $X$ again, we can choose a finite morphism $\pi : X \to \mathbf{A}^ d_ K$, see Algebra, Lemma 10.115.5. Since $\kappa (x)$ is a finite extension of the residue field of $\pi (x)$, this residue field has transcendence degree $a$ over $K$ as well. Thus we can find a finite morphism $\pi ' : \mathbf{A}^ d_ K \to \mathbf{A}^ d_ K$ such that $\pi '(\pi (x))$ corresponds to the generic point of the linear subspace $\mathbf{A}^ a_ K \subset \mathbf{A}^ d_ K$ given by setting the last $d - a$ coordinates equal to zero. Hence the composition

of $\pi ' \circ \pi $ and the projection $p$ onto the first $a$ coordinates maps $x$ to the generic point $\eta \in \mathbf{A}^ a_ K$. The induced map

on étale local rings satisfies (1) since it is clear that the residue field of $\mathcal{O}_{X, x}^{sh}$ is an algebraic extension of the separably closed field $K(t_1, \ldots , t_ a)^{sep}$. On the other hand, if $X = \mathop{\mathrm{Spec}}(B)$, then $\mathcal{O}_{X, x}^{sh} = \mathop{\mathrm{colim}}\nolimits B_ j$ is a filtered colimit of étale $B$-algebras $B_ j$. Observe that $B_ j$ is quasi-finite over $K[t_1, \ldots , t_ d]$ as $B$ is finite over $K[t_1, \ldots , t_ d]$. We may similarly write $K(t_1, \ldots , t_ a)^{sep} = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of étale $K[t_1, \ldots , t_ a]$-algebras. For every $i$ we can find an $j$ such that $A_ i \to K(t_1, \ldots , t_ a)^{sep} \to \mathcal{O}_{X, x}^{sh}$ factors through a map $\psi _{i, j} : A_ i \to B_ j$. Then $B_ j$ is quasi-finite over $A_ i[t_{a + 1}, \ldots , t_ d]$. Hence

has dimension $\leq d - a$ as it is quasi-finite over $K(t_1, \ldots , t_ a)^{sep}[t_{a + 1}, \ldots , t_ d]$. The proof of (2) is now finished as $\mathcal{O}_{X, x}^{sh}$ is a filtered colimit¹ of the algebras $B_{i, j}$. Some details omitted. $\square$

Proof. We will prove this by induction on $\dim (X)$. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$.

The case $\dim (X) = 0$. In this case the structure morphism $f : X \to \mathop{\mathrm{Spec}}(K)$ is finite. Hence we see that $R^ if_*\mathcal{F} = 0$ for $i > 0$, see Proposition 59.55.2. Thus $H^ i_{\acute{e}tale}(X, \mathcal{F}) = H^ i_{\acute{e}tale}(\mathop{\mathrm{Spec}}(K), f_*\mathcal{F})$ by the Leray spectral sequence for $f$ (Cohomology on Sites, Lemma 21.14.5) and the result is clear.

The case $\dim (X) = 1$. This is Lemma 59.95.3.

Assume $d = \dim (X) > 1$ and the proposition holds for finite type affine schemes of dimension $< d$ over fields. By Noether normalization, see for example Varieties, Lemma 33.18.2, there exists a finite morphism $f : X \to \mathbf{A}^ d_ K$. Recall that $R^ if_*\mathcal{F} = 0$ for $i > 0$ by Proposition 59.55.2. By the Leray spectral sequence for $f$ (Cohomology on Sites, Lemma 21.14.5) we conclude that it suffices to prove the result for $\pi _*\mathcal{F}$ on $\mathbf{A}^ d_ K$.

Interlude I. Let $j : X \to Y$ be an open immersion of smooth $d$-dimensional varieties over $K$ (not necessarily affine) whose complement is the support of an effective Cartier divisor $D$. The sheaves $R^ qj_*\mathcal{F}$ for $q > 0$ are supported on $D$. We claim that $(R^ qj_*\mathcal{F})_{\overline{y}} = 0$ for $a = \text{trdeg}_ K(\kappa (y)) > d - q$. Namely, by Theorem 59.53.1 we have

Choose a local equation $f \in \mathfrak m_ y \subset \mathcal{O}_{Y, y}$ for $D$. Then we have

Using Lemma 59.95.5 we get an embedding

Since the transcendence degree over $K$ of the fraction field of $\mathcal{O}_{Y, y}^{sh}$ is $d$, we see that $\mathcal{O}_{Y, y}^{sh}[1/f]$ is a filtered colimit of $(d - a - 1)$-dimensional finite type algebras over the field $K(t_1, \ldots , t_ a)^{sep}(x)$ which itself has cohomological dimension $1$ by Lemma 59.95.4. Thus by induction hypothesis and Lemma 59.95.2 we obtain the desired vanishing.

Interlude II. Let $Z$ be a smooth variety over $K$ of dimension $d - 1$. Let $E_ a \subset Z$ be the set of points $z \in Z$ with $\text{trdeg}_ K(\kappa (z)) \leq a$. Observe that $E_ a$ is closed under specialization, see Varieties, Lemma 33.20.3. Suppose that $\mathcal{G}$ is a torsion abelian sheaf on $Z$ whose support is contained in $E_ a$. Then we claim that $H^ b_{\acute{e}tale}(Z, \mathcal{G}) = 0$ for $b > a + \text{cd}(K)$. Namely, we can write $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ with $\mathcal{G}_ i$ a torsion abelian sheaf supported on a closed subscheme $Z_ i$ contained in $E_ a$, see Lemma 59.74.5. Then the induction hypothesis kicks in to imply the desired vanishing for $\mathcal{G}_ i$². Finally, we conclude by Theorem 59.51.3.

Consider the commutative diagram

Observe that $j$ is an open immersion of smooth $d$-dimensional varieties whose complement is an effective Cartier divisor $D$. Thus we may use the results obtained in interlude I. We are going to study the relative Leray spectral sequence

Since $R^ qj_*\mathcal{F}$ for $q > 0$ is supported on $D$ and since $g|_ D : D \to \mathbf{A}^{d - 1}_ K$ is an isomorphism, we find $R^ pg_*R^ qj_*\mathcal{F} = 0$ for $p > 0$ and $q > 0$. Moreover, we have $R^ qj_*\mathcal{F} = 0$ for $q > d$. On the other hand, $g$ is a proper morphism of relative dimension $1$. Hence by Lemma 59.92.2 we see that $R^ pg_*j_*\mathcal{F} = 0$ for $p > 2$. Thus the $E_2$-page of the spectral sequence looks like this

We conclude that $R^ qf_*\mathcal{F} = g_*R^ qj_*\mathcal{F}$ for $q > 2$. By interlude I we see that the support of $R^ qf_*\mathcal{F}$ for $q > 2$ is contained in the set of points of $\mathbf{A}^{d - 1}_ K$ whose residue field has transcendence degree $\leq d - q$. By interlude II

On the other hand, by Theorem 59.53.1 we have $R^2f_*\mathcal{F}_{\overline{\eta }} = H^2(\mathbf{A}^1_{\overline{\eta }}, \mathcal{F}) = 0$ (vanishing by the case of dimension $1$) where $\eta $ is the generic point of $\mathbf{A}^{d - 1}_ K$. Hence by interlude II again we see

Finally, we have

by induction hypothesis. Combining everything we just said with the Leray spectral sequence $H^ p(\mathbf{A}^{d - 1}_ K, R^ qf_*\mathcal{F}) \Rightarrow H^{p + q}(\mathbf{A}^ d_ K, \mathcal{F})$ we conclude. $\square$

Proof. We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ with $\mathcal{F}_ i$ a torsion abelian sheaf supported on a closed subscheme $Z_ i$ contained in $E_ a$, see Lemma 59.74.5. Then Proposition 59.95.6 gives the desired vanishing for $\mathcal{F}_ i$. Details omitted; hints: first use Proposition 59.46.4 to write $\mathcal{F}_ i$ as the pushforward of a sheaf on $Z_ i$, use the vanishing for this sheaf on $Z_ i$, and use the Leray spectral sequence for $Z_ i \to Z$ to get the vanishing for $\mathcal{F}_ i$. Finally, we conclude by Theorem 59.51.3. $\square$

Proof. The question is local on $Y$ hence we can assume $Y$ is affine. Then $X$ is affine too and we can choose a diagram

where the horizontal arrows are closed immersions and the vertical arrow on the right is the projection (details omitted). Then $j_*R^ qf_*\mathcal{F} = R^ q\text{pr}_*i_*\mathcal{F}$ by the vanishing of the higher direct images of $i$ and $j$, see Proposition 59.55.2. Moreover, the description of the stalks of $j_*$ in the proposition shows that it suffices to prove the vanishing for $j_*R^ qf_*\mathcal{F}$. Thus we may assume $f$ is the projection morphism $\text{pr} : \mathbf{A}^{n + m}_ K \to \mathbf{A}^ n_ K$ and an abelian torsion sheaf $\mathcal{F}$ on $\mathbf{A}^{n + m}_ K$ satisfying the assumption in the statement of the lemma.

Let $y$ be a point in $\mathbf{A}^ n_ K$. By Theorem 59.53.1 we have

Say $b = \text{trdeg}_ K(\kappa (y))$. From Lemma 59.95.5 we get an embedding

Write $\mathcal{O}_{Y, y}^{sh} = \mathop{\mathrm{colim}}\nolimits B_ i$ as the filtered colimit of finite type $L$-subalgebras $B_ i \subset \mathcal{O}_{Y, y}^{sh}$ containing the ring $K[T_1, \ldots , T_ n]$ of regular functions on $\mathbf{A}^ n_ K$. Then we get

If $z \in \mathbf{A}^ m_{B_ i}$ is a point in the support of $\mathcal{F}$, then the image $x$ of $z$ in $\mathbf{A}^{m + n}_ K$ satisfies $\text{trdeg}_ K(\kappa (x)) \leq a$ by our assumption on $\mathcal{F}$ in the lemma. Since $\mathcal{O}_{Y, y}^{sh}$ is a filtered colimit of étale algebras over $K[T_1, \ldots , T_ n]$ and since $B_ i \subset \mathcal{O}_{Y, y}^{sh}$ we see that $\kappa (z)/\kappa (x)$ is algebraic (some details omitted). Then $\text{trdeg}_ K(\kappa (z)) \leq a$ and hence $\text{trdeg}_ L(\kappa (z)) \leq a - b$. By Lemma 59.95.7 we see that

Thus by Theorem 59.51.3 we get $(Rf_*\mathcal{F})_{\overline{y}} = 0$ for $q > a - b$ as desired. $\square$