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Let $X$ be a topological space, $\mathcal{F}$ be a sheaf of abelian groups on $X$.

If $i: A \hookrightarrow X$ is a closed subspace of $X$ and $j: X \setminus A \hookrightarrow X$ denotes the open inclusion, we can look at the right derived functor

$$R^q j_{!} j^* \mathcal{F}.$$

It is well-known that when $\mathcal{F}$ is a constant sheaf, this provides the relative singular cohomology groups \begin{equation} H^*(X,A). \end{equation}

I'm wondering how "nice" this functor $j_{!} j^*$ is. In particular:

  1. Is it true that $$R^q j_{!} j^* \mathcal{F} = j_{!} j^* R^q \mathcal{F} ?$$
  2. Does $j_{!} j^*$ commutes with coproducts ?
  3. Does $j_{!} j^*$ commutes with direct image functors ?

Thanks a lot

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