The Stacks project

[Theorem 3.6.5, Tohoku].

Proposition 20.20.7 (Grothendieck). Let $X$ be a Noetherian topological space. If $\dim (X) \leq d$, then $H^ p(X, \mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\mathcal{F}$ on $X$.

Proof. We prove this lemma by induction on $d$. So fix $d$ and assume the lemma holds for all Noetherian topological spaces of dimension $< d$.

Let $\mathcal{F}$ be an abelian sheaf on $X$. Suppose $U \subset X$ is an open. Let $Z \subset X$ denote the closed complement. Denote $j : U \to X$ and $i : Z \to X$ the inclusion maps. Then there is a short exact sequence

\[ 0 \to j_{!}j^*\mathcal{F} \to \mathcal{F} \to i_*i^*\mathcal{F} \to 0 \]

see Modules, Lemma 17.7.1. Note that $j_!j^*\mathcal{F}$ is supported on the topological closure $Z'$ of $U$, i.e., it is of the form $i'_*\mathcal{F}'$ for some abelian sheaf $\mathcal{F}'$ on $Z'$, where $i' : Z' \to X$ is the inclusion.

We can use this to reduce to the case where $X$ is irreducible. Namely, according to Topology, Lemma 5.9.2 $X$ has finitely many irreducible components. If $X$ has more than one irreducible component, then let $Z \subset X$ be an irreducible component of $X$ and set $U = X \setminus Z$. By the above, and the long exact sequence of cohomology, it suffices to prove the vanishing of $H^ p(X, i_*i^*\mathcal{F})$ and $H^ p(X, i'_*\mathcal{F}')$ for $p > d$. By Lemma 20.20.1 it suffices to prove $H^ p(Z, i^*\mathcal{F})$ and $H^ p(Z', \mathcal{F}')$ vanish for $p > d$. Since $Z'$ and $Z$ have fewer irreducible components we indeed reduce to the case of an irreducible $X$.

If $d = 0$ and $X$ is irreducible, then $X$ is the only nonempty open subset of $X$. Hence every sheaf is constant and higher cohomology groups vanish (for example by Lemma 20.20.2).

Suppose $X$ is irreducible of dimension $d > 0$. By Lemma 20.20.4 we reduce to the case where $\mathcal{F} = j_!\underline{\mathbf{Z}}_ U$ for some open $U \subset X$. In this case we look at the short exact sequence

\[ 0 \to j_!(\underline{\mathbf{Z}}_ U) \to \underline{\mathbf{Z}}_ X \to i_*\underline{\mathbf{Z}}_ Z \to 0 \]

where $Z = X \setminus U$. By Lemma 20.20.2 we have the vanishing of $H^ p(X, \underline{\mathbf{Z}}_ X)$ for all $p \geq 1$. By induction we have $H^ p(X, i_*\underline{\mathbf{Z}}_ Z) = H^ p(Z, \underline{\mathbf{Z}}_ Z) = 0$ for $p \geq d$. Hence we win by the long exact cohomology sequence. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 20.20: Vanishing on Noetherian topological spaces

Post a comment

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02UZ. Beware of the difference between the letter 'O' and the digit '0'.