I have a topological space $X$ and two disjoint, closed subspaces $Y$ and $Z$ of $X$. I believe that in this situation, for any abelian sheaf $\mathcal{F}$ on $X$ and any $p \in \mathbb{N}$, there is a natural isomorphism

$$ H^p_Y(X,\mathcal{F}) \oplus H^p_Z(X, \mathcal{F}) \to H^p_{Y \cup Z}(X, \mathcal{F})$$

between local cohomology groups. I can obtain this by taking an injective resolution $0 \to \mathcal{F} \to \mathcal{I}^\bullet$ of $\mathcal{F}$, and constructing by hand a split exact sequence of complexes of sheaves

$$ 0 \to \Gamma_Y(X,\mathcal{I}^\bullet) \to \Gamma_{Y\cup Z}(X,\mathcal{I}^\bullet) \to \Gamma_Z(X,\mathcal{I}^\bullet) \to 0$$

and then passing to cohomology. I am certain that this is completely standard and well known; could someone please point me to a (preferably modern) reference for this result?

  • $\begingroup$ Proposition 2.3.9(iv) in Kashiwara-Schapira's 'Sheaves on Manifolds' suggests that you might need extra conditions on your space $X$ for this to be true. Perhaps $X$ compact Hausdorff will suffice. $\endgroup$ – user91132 Nov 28 '17 at 13:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.