Let $X$ be a smooth variety over a field $k \subset \mathbb{C}$ and $Z$ a smooth subvariety. Let U=X-Z$. I'm trying to understand what information do the Leray spectral sequences attached to the inclusions

$j: U(\mathbb{C}) \hookrightarrow X(\mathbb{C})$

and

$i: Z(\mathbb{C}) \hookrightarrow X(\mathbb{C})$

provide. More precisely, if $F$ is a sheaf on $U$ one has a spectral sequence

$ E_2^{p,q}:=H^p(X, j_\ast F) \Longrightarrow H^{p+q}(U, F)$

Is there a long exact sequence associated to it such that, when $F$ is the constant sheaf $\mathbb{Q}$, gives something like excision for usual cohomology?

Let $X$ be a smooth variety over a field $k \subset \mathbb{C}$ and $Z$ a smooth subvariety. Let $U=X-Z$. I'm trying to understand what information do the Leray spectral sequences attached to the inclusions

$j: U(\mathbb{C}) \hookrightarrow X(\mathbb{C})$

and

$i: Z(\mathbb{C}) \hookrightarrow X(\mathbb{C})$

provide. More precisely, if $F$ is a sheaf on $U$ one has a spectral sequence

$ E_2^{p,q}:=H^p(X, R^q j_\ast F) \Longrightarrow H^{p+q}(U, F)$

Is there a long exact sequence associated to it such that, when $F$ is the constant sheaf $\mathbb{Q}$, gives something like excision for usual cohomology?