The Stacks project

Lemma 59.95.8. Let $f : X \to Y$ be an affine morphism of schemes of finite type over a field $K$. Let $E_ a(X)$ be the set of points $x \in X$ with $\text{trdeg}_ K(\kappa (x)) \leq a$. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$ whose support is contained in $E_ a$. Then $R^ qf_*\mathcal{F}$ has support contained in $E_{a - q}(Y)$.

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

\[ \xymatrix{ X \ar[d]_ f \ar[r]_ i & \mathbf{A}^{n + m}_ K \ar[d]^{\text{pr}} \\ Y \ar[r]^ j & \mathbf{A}^ n_ K } \]

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

\[ (R^ q\text{pr}_*\mathcal{F})_{\overline{y}} = H^ q(\mathbf{A}^{n + m}_ K \times _{A^ n_ K} \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}^{sh}), \mathcal{F}) = H^ q(\mathbf{A}^ m_{\mathcal{O}_{Y, y}^{sh}}, \mathcal{F}) \]

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

\[ L = K(t_1, \ldots , t_ b)^{sep} \longrightarrow \mathcal{O}_{Y, y}^{sh} \]

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

\[ \mathbf{A}^ m_{\mathcal{O}_{Y, y}^{sh}} = \mathop{\mathrm{lim}}\nolimits \mathbf{A}^ m_{B_ i} \]

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

\[ H^ q(\mathbf{A}^ m_{B_ i}, \mathcal{F}) = 0\text{ for }q > a - b \]

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

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