The Stacks project

Lemma 57.89.7. Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$. Let $E_ a \subset X$ 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 $H^ b_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $b > a + \text{cd}(K)$.

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 57.73.5. Then Proposition 57.89.6 gives the desired vanishing for $\mathcal{F}_ i$. Details omitted; hints: first use Proposition 57.46.4 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 57.51.3. $\square$

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