Exercise On Cohomology of Complex Spaces
Posted: Sun Mar 05, 2017 12:49 am
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
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 \[ \mathrm{H}^{q}(U, \mathcal{S} |_{U}) = 0 \, , \, q>0 \]