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 20.19.1 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 20.20.3 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
Comments (2)
Comment #6303 by Qilin,Yang on
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