# A Thom isomorphism for sheaves

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?

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