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Let say $\mathcal{F}$ is a locally free sheaf of abelian groups over $X$, where $X$ is an algebraic variety over $\mathbb{C}$ (or a field $k$) with analytic (or étale) topology and $Z$ is a closed subvariety of $X$ of codimension $d$. I want to ask if there exists, or under which condition there exists, an isomorphism $$H^{2d}_{Z}(X, \mathcal{F}) \xrightarrow{\sim} H^0(Z,\mathcal{F}_{|Z}).$$ More specifically, can one deduce such an isomorphism from a suitable spectral sequence?

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    $\begingroup$ A sheaf of what? Over what kind of field, for what topology? $\endgroup$ – abx May 20 at 8:34
  • $\begingroup$ I have just edited $\endgroup$ – Matvey Tizovsky May 20 at 9:29

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