I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f : X \to Y$ be a continuous map between topological space, and $\mathbb{C}$ be the constant sheaf.

Suppose that we are given a closed subset $i : A \hookrightarrow X$, such that $j : f(A) \hookrightarrow Y$ is closed as well. Do we have:

- $j^* R^q f_* \mathbb{C} \simeq R^q f_{|A*} i^* \mathbb{C}$,

where $R^qf_*$ is the higher direct image functor, and $f_{|A}$ is the restriction of $f$ to $A$.

Suppose moreover that we are given a sequence of closed subsets $i_l : A_l \hookrightarrow X$. Do we have:

- $i^* \underset{l}{\oplus} i_{l*} \mathbb{C} \simeq \underset{l}{\oplus} i_{l|A_l \cap A*} \mathbb{C}.$

In other words, **does the restriction functor to a a closed subspace commute with higher direct images, an direct sums of extension functors from closed subspaces?**

Suppose now that all the previous subsets $i : A \hookrightarrow X$, $j : f(A) \hookrightarrow Y$, $i_l : A_l \hookrightarrow X$ are open. I have seen somewhere that the right derived functor applied to $i_! i^*$ gives relative singular cohomology

$$R^q i_! i^* \mathcal{F} \simeq H^q(X, X \setminus A; \mathbb{C}).$$

- Is there a reason why we use $i_!$ instead of $i_*$.

Do we have:

- $j_! j^* R^q f_* \mathbb{C} \simeq R^q f_* i_! i^* \mathbb{C}$;
- $i_! i^* \underset{l}{\oplus} i_{l*} \mathbb{C} \simeq \underset{l}{\oplus} i_{l| A_l \cap A !} i_{l|A_l \cap A}^* \mathbb{C}$.

In other words, **does the functor $i_! i^*$ commute with direct images, higher direct images, and direct sums of extension functors from closed subspaces?**

Let $\Gamma$ denote the global sections functor, and $\mathcal{F}_l$ be a sequence of sheaves of abelian groups. Do we have:

- $R^q \Gamma (\underset{l}{\oplus} \mathcal{F}_l) \simeq \underset{l}{\oplus} R^q \Gamma (\mathcal{F}_l)$.

**In other words, does sheaf cohomology commutes with direct sums?**