Lemma 20.11.9. (Variant of Lemma 20.11.8.) Let X be a ringed space. Let \mathcal{B} be a basis for the topology on X. Let \mathcal{F} be an \mathcal{O}_ X-module. Assume there exists a set of open coverings \text{Cov} with the following properties:
For every \mathcal{U} \in \text{Cov} with \mathcal{U} : U = \bigcup _{i \in I} U_ i we have U, U_ i \in \mathcal{B} and every U_{i_0 \ldots i_ p} \in \mathcal{B}.
For every U \in \mathcal{B} the open coverings of U occurring in \text{Cov} is a cofinal system of open coverings of U.
For every \mathcal{U} \in \text{Cov} we have \check{H}^ p(\mathcal{U}, \mathcal{F}) = 0 for all p > 0.
Then H^ p(U, \mathcal{F}) = 0 for all p > 0 and any U \in \mathcal{B}.
Proof.
Let \mathcal{F} and \text{Cov} be as in the lemma. We will indicate this by saying “\mathcal{F} has vanishing higher Čech cohomology for any \mathcal{U} \in \text{Cov}”. Choose an embedding \mathcal{F} \to \mathcal{I} into an injective \mathcal{O}_ X-module. By Lemma 20.11.1 \mathcal{I} has vanishing higher Čech cohomology for any \mathcal{U} \in \text{Cov}. Let \mathcal{Q} = \mathcal{I}/\mathcal{F} so that we have a short exact sequence
0 \to \mathcal{F} \to \mathcal{I} \to \mathcal{Q} \to 0.
By Lemma 20.11.7 and our assumption (2) this sequence gives rise to an exact sequence
0 \to \mathcal{F}(U) \to \mathcal{I}(U) \to \mathcal{Q}(U) \to 0.
for every U \in \mathcal{B}. Hence for any \mathcal{U} \in \text{Cov} we get a short exact sequence of Čech complexes
0 \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{Q}) \to 0
since each term in the Čech complex is made up out of a product of values over elements of \mathcal{B} by assumption (1). In particular we have a long exact sequence of Čech cohomology groups for any open covering \mathcal{U} \in \text{Cov}. This implies that \mathcal{Q} is also an \mathcal{O}_ X-module with vanishing higher Čech cohomology for all \mathcal{U} \in \text{Cov}.
Next, we look at the long exact cohomology sequence
\xymatrix{ 0 \ar[r] & H^0(U, \mathcal{F}) \ar[r] & H^0(U, \mathcal{I}) \ar[r] & H^0(U, \mathcal{Q}) \ar[lld] \\ & H^1(U, \mathcal{F}) \ar[r] & H^1(U, \mathcal{I}) \ar[r] & H^1(U, \mathcal{Q}) \ar[lld] \\ & \ldots & \ldots & \ldots \\ }
for any U \in \mathcal{B}. Since \mathcal{I} is injective we have H^ n(U, \mathcal{I}) = 0 for n > 0 (see Derived Categories, Lemma 13.20.4). By the above we see that H^0(U, \mathcal{I}) \to H^0(U, \mathcal{Q}) is surjective and hence H^1(U, \mathcal{F}) = 0. Since \mathcal{F} was an arbitrary \mathcal{O}_ X-module with vanishing higher Čech cohomology for all \mathcal{U} \in \text{Cov} we conclude that also H^1(U, \mathcal{Q}) = 0 since \mathcal{Q} is another of these sheaves (see above). By the long exact sequence this in turn implies that H^2(U, \mathcal{F}) = 0. And so on and so forth.
\square
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