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? 



Leray spectral sequence of the inclusion of an open subvariety
Here are a few more details:
Suppose that $F=\mathbb{Q}\_X$. 
In this case, 
$$
R^qj\_\*F = 
\begin{cases}
\mathbb{Q}\_X & \text{if $q=0$}\\
\mathbb{Q}\_Z & \text{if $q= 2 c-1$, $c=codim(Z)$}\\
0 & \text{otherwise}
\end{cases} 
$$
To see this, first note that the computation is local, so we may replace the pair $(X,Z)$ by $(\mathbb{C}^n, \mathbb{C}^{n-c})$ using the implicit function theorem. After this, it should be easy. With this calculation the $E\_2$ term of Leray is concentrated along two lines, and it is not hard to see that it reduces to the above "localization" sequence given in [excision in algebraic de Rham cohomology](https://mathoverflow.net/questions/121910/)


Notes: Sorry, I had a typo in the formula yesterday. If $Z$ is smooth but not pure dimensional, then $R^qj\_\*F|\_{Z\_i}= \mathbb{Q}\_{Z\_i}$ where $q=2c\_i-1$ for each component. The final result will be a bit more cumbersome to state. Some related calculations occur
in Deligne's Theorie de Hodge I, II.


