Let $i : A \hookrightarrow X$ be a closed subspace of a topological space $X$, and $j : Z := X \setminus A \hookrightarrow X$ denote its open complement. Given a sheaf $F$ of abelian groups on $X$, one can define sheaf cohomology with support in $A$, by taking the derived functors of the functor $\Gamma_A$ "sections with support in $A$": $$\Gamma_A F := \lbrace s \in \Gamma(X,F): s_{|Z} = 0 \rbrace.$$ Whenever the spaces involved are nice enough, and $F$ is a constant sheaf, these derived functors seem to give the relative singular cohomology groups $H^*(X,Z)$.

I am wondering the following:

- How can we rewrite the expression $R^q \Gamma_A F$ in terms of the functors direct and inverse images associated with $i$ (or $j$) ? That is, can we define the sheaf $R^q \Gamma_A F$ as $R^q i_* i^* F$, or something in this spirit ?
- What if I would like to obtain the relative cohomology groups $H^*(X,A)$ ? In this case, could we define relative sheaf cohomology as something like $R^q j_* j^* F$ ?
- What would be the properties of the functors used to define cohomology relative to the closed $A$ (left adjoint, exact...) ?