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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