Lemma 69.10.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume $\dim (X) \leq d$ for some integer $d$. Let $\mathcal{F}$ be a quasi-coherent sheaf $\mathcal{F}$ on $X$.
$H^ q(X, \mathcal{F}) = 0$ for $q > d$,
$H^ d(X, \mathcal{F}) \to H^ d(U, \mathcal{F})$ is surjective for any quasi-compact open $U \subset X$,
$H^ q_ Z(X, \mathcal{F}) = 0$ for $q > d$ for any closed subspace $Z \subset X$ whose complement is quasi-compact.
Proof. By Properties of Spaces, Lemma 66.22.5 every algebraic space $Y$ étale over $X$ has dimension $\leq d$. If $Y$ is quasi-separated, the dimension of $Y$ is equal to the Krull dimension of $|Y|$ by Decent Spaces, Lemma 68.12.5. Also, if $Y$ is a scheme, then étale cohomology of $\mathcal{F}$ over $Y$, resp. étale cohomology of $\mathcal{F}$ with support in a closed subscheme, agrees with usual cohomology of $\mathcal{F}$, resp. usual cohomology with support in the closed subscheme. See Descent, Proposition 35.9.3 and Étale Cohomology, Lemma 59.79.5. We will use these facts without further mention.
By Decent Spaces, Lemma 68.8.6 there exist an integer $n$ and open subspaces
with the following property: setting $T_ p = U_ p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a quasi-compact separated scheme $V_ p$ and a surjective étale morphism $f_ p : V_ p \to U_ p$ such that $f_ p^{-1}(T_ p) \to T_ p$ is an isomorphism.
As $U_ n = V_ n$ is a scheme, our initial remarks imply the cohomology of $\mathcal{F}$ over $U_ n$ vanishes in degrees $> d$ by Cohomology, Proposition 20.22.4. Suppose we have shown, by induction, that $H^ q(U_{p + 1}, \mathcal{F}|_{U_{p + 1}}) = 0$ for $q > d$. It suffices to show $H_{T_ p}^ q(U_ p, \mathcal{F})$ for $q > d$ is zero in order to conclude the vanishing of cohomology of $\mathcal{F}$ over $U_ p$ in degrees $> d$. However, we have
by Lemma 69.9.3 and as $V_ p$ is a scheme we obtain the desired vanishing from Cohomology, Proposition 20.22.4. In this way we conclude that (1) is true.
To prove (2) let $U \subset X$ be a quasi-compact open subspace. Consider the open subspace $U' = U \cup U_ n$. Let $Z = U' \setminus U$. Then $g : U_ n \to U'$ is an étale morphism such that $g^{-1}(Z) \to Z$ is an isomorphism. Hence by Lemma 69.9.3 we have $H^ q_ Z(U', \mathcal{F}) = H^ q_ Z(U_ n, \mathcal{F})$ which vanishes in degree $> d$ because $U_ n$ is a scheme and we can apply Cohomology, Proposition 20.22.4. We conclude that $H^ d(U', \mathcal{F}) \to H^ d(U, \mathcal{F})$ is surjective. Assume, by induction, that we have reduced our problem to the case where $U$ contains $U_{p + 1}$. Then we set $U' = U \cup U_ p$, set $Z = U' \setminus U$, and we argue using the morphism $f_ p : V_ p \to U'$ which is étale and has the property that $f_ p^{-1}(Z) \to Z$ is an isomorphism. In other words, we again see that
and we again see this vanishes in degrees $> d$. We conclude that $H^ d(U', \mathcal{F}) \to H^ d(U, \mathcal{F})$ is surjective. Eventually we reach the stage where $U_1 = X \subset U$ which finishes the proof.
A formal argument shows that (2) implies (3). $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)