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 Post subject: Exercise On Cohomology of Complex SpacesPosted: Sun Mar 05, 2017 12:49 am
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Assuming the following result

THEOREM: Let $X$ be a complex space of dimension $n$ and let $\mathcal{S}$ be any sheaf on $X$. Then $\mathrm{H}^{q}(X, \mathcal{S}) = 0 \, , \, q > 2n$

prove the following results

• LEMMA: Let $X$ be a complex space of dimension $n$ such that $\mathrm{H}^{q}(X, \mathcal{O}) = 0 \, , \, q > 0$
and let $\mathcal{S}$ be any soluble analytic sheaf on $X$, that is, it admits a left resolution
$\mathcal{O}^{p_{r}} \to \mathcal{O}^{p_{r-1}} \to \dots \to \mathcal{O}^{p_{1}} \to \mathcal{O}^{p_{0}} \to \mathcal{S} \to 0$over $X$. Then $\mathrm{H}^{q}(X, \mathcal{S}) = 0 \, , \, q > 0$
• COROLLARY: Let $X$ be a complex space and let $\mathcal{S}$ be a coherent analytic sheaf on $X$. Then for every $x \in X$ there exists an open neighbourhood $U$ such that $\mathrm{H}^{q}(U, \mathcal{S} |_{U}) = 0 \, , \, q>0$

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