The Stacks project

This is a special case of [Proposition 3.6.1, Tohoku].

Lemma 20.21.4. Let $X$ be a topological space. Let $d \geq 0$ be an integer. Assume

  1. $X$ is quasi-compact,

  2. the quasi-compact opens form a basis for $X$, and

  3. the intersection of two quasi-compact opens is quasi-compact.

  4. $H^ p(X, j_!\underline{\mathbf{Z}}_ U) = 0$ for all $p > d$ and any quasi-compact open $j : U \to X$.

Then $H^ p(X, \mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\mathcal{F}$ on $X$.

Proof. Let $S = \coprod _{U \subset X} \mathcal{F}(U)$ where $U$ runs over the quasi-compact opens of $X$. For any finite subset $A = \{ s_1, \ldots , s_ n\} \subset S$, let $\mathcal{F}_ A$ be the subsheaf of $\mathcal{F}$ generated by all $s_ i$ (see Modules, Definition 17.4.5). Note that if $A \subset A'$, then $\mathcal{F}_ A \subset \mathcal{F}_{A'}$. Hence $\{ \mathcal{F}_ A\} $ forms a system over the directed partially ordered set of finite subsets of $S$. By Modules, Lemma 17.4.6 it is clear that

\[ \mathop{\mathrm{colim}}\nolimits _ A \mathcal{F}_ A = \mathcal{F} \]

by looking at stalks. By Lemma 20.20.1 we have

\[ H^ p(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits _ A H^ p(X, \mathcal{F}_ A) \]

Hence it suffices to prove the vanishing for the abelian sheaves $\mathcal{F}_ A$. In other words, it suffices to prove the result when $\mathcal{F}$ is generated by finitely many local sections over quasi-compact opens of $X$.

Suppose that $\mathcal{F}$ is generated by the local sections $s_1, \ldots , s_ n$. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf generated by $s_1, \ldots , s_{n - 1}$. Then we have a short exact sequence

\[ 0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}/\mathcal{F}' \to 0 \]

From the long exact sequence of cohomology we see that it suffices to prove the vanishing for the abelian sheaves $\mathcal{F}'$ and $\mathcal{F}/\mathcal{F}'$ which are generated by fewer than $n$ local sections. Hence it suffices to prove the vanishing for sheaves generated by at most one local section. These sheaves are exactly the quotients of the sheaves $j_!\underline{\mathbf{Z}}_ U$ where $U$ is a quasi-compact open of $X$.

Assume now that we have a short exact sequence

\[ 0 \to \mathcal{K} \to j_!\underline{\mathbf{Z}}_ U \to \mathcal{F} \to 0 \]

with $U$ quasi-compact open in $X$. It suffices to show that $H^ q(X, \mathcal{K})$ is zero for $q \geq d + 1$. As above we can write $\mathcal{K}$ as the filtered colimit of subsheaves $\mathcal{K}'$ generated by finitely many sections over quasi-compact opens. Then $\mathcal{F}$ is the filtered colimit of the sheaves $j_!\underline{\mathbf{Z}}_ U/\mathcal{K}'$. In this way we reduce to the case that $\mathcal{K}$ is generated by finitely many sections over quasi-compact opens. Note that $\mathcal{K}$ is a subsheaf of $\underline{\mathbf{Z}}_ X$. Thus by Lemma 20.21.3 there exists a finite filtration of $\mathcal{K}$ whose successive quotients $\mathcal{Q}$ fit into a short exact sequence

\[ 0 \to j''_!\underline{\mathbf{Z}}_ W \to j'_!\underline{\mathbf{Z}}_ V \to \mathcal{Q} \to 0 \]

with $j'' : W \to X$ and $j' : V \to X$ the inclusions of quasi-compact opens. Hence the vanishing of $H^ p(X, \mathcal{Q})$ for $p > d$ follows from our assumption (in the lemma) on the vanishing of the cohomology groups of $j''_!\underline{\mathbf{Z}}_ W$ and $j'_!\underline{\mathbf{Z}}_ V$. Returning to $\mathcal{K}$ this, via an induction argument using the long exact cohomology sequence, implies the desired vanishing for it as well. $\square$


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