Commutativity between functors on sheaves of abelian groups

Question

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.

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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:

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

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:

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

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?

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

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

Do we have:

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

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

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

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

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

Answer 1

1) No. For example, if $A$ is non-empty and $f: X \to \ast$ is the map to the point then $j^*R^qf_*\mathbb{C} \cong H^q(X,\mathbb{C})$ while $R^q(f|_A)_*i^*\mathbb{C} \cong R^q(f|_A)_*\mathbb{C} \cong H^q(A,\mathbb{C})$ (all cohomologies being sheaf cohomologies, which are the same as singluar cohomology if $X$ and $A$ are sufficiently nice). These are of course generally different.

2) Yes, at least if the spaces involved are sufficiently nice: since $i^*$ is a left functor it commutes with direct sums, and hence it suffices to show that for every $l$ we have $i^*(i_l)_*\mathbb{C} \cong ({i_l}|_{A_l \cap A})_*\mathbb{C}$ as sheaves on $A$. But this is an instance of the proper base change theorem (which requires some separation conditions, hence the niceness requirement).

3) The functor $i_!$, also called direct image with compact support, coincides with $i_*$ when $i$ is a closed embedding. Maybe you mean the functor $i^!: Sh(X) \to Sh(A)$, which is right adjoint to $i_!$? Then the total derived functor $Ri^!\mathbb{C}$ is a complex of sheaves on $A$ whose derived global sections compute the relative cohomology groups $H^q(X,X\setminus A)$ when $X$ and $A$ are sufficiently nice.

4) No. Use the same type of counter-example as in (1).

5) I'm not sure I understand the question. On what space are we supposed to consider the sheaves on the left and right hand side?

6) Yes if there are finitely many $A_l$'s or if $X$ is compact Hausdorff. In general no.

Edit

Some additional remarks:

1) The statement does hold when $A = f^{-1}(f(A))$ and $X$ and $A$ are sufficiently nice by the proper base change theorem. It's possible that a suitable version of (4) may also work in this case.

3) If $X$ is a space and $A \subseteq X$ is a closed subspace then the functor $\Gamma_A: Sh(X,Ab) \to Ab$ which takes a sheaf of abelian groups $F$ to the abelian group $$\Gamma_A(F) := \{s \in \Gamma(F) | s|_{X \setminus A}=0\}$$ is a left exact functor. Its derived functors $H^q_A(X,F)$ are then referred to as the cohomology of $F$ with support on $A$. If $X$ and $A$ are sufficiently nice then $H^q_A(X,\mathbb{C}) \cong H^q(X,X\setminus A;\mathbb{C})$. There is also a sheaf on $X$ whose cohomologies are the cohomologies of $F$ with support on $A$: this is the sheaf $U \mapsto \Gamma_{U \cap A}(F|_U)$ which gives to every open subset the subgroup of those local sections which vanish outside $A$. One can then check that $\Gamma_{U \cap A}(F|_U)$ only depends on $U \cap A$, and hence this sheaf is of the form $i_*F^{!}$ for some sheaf $F^{!}$ on $A$. The sheaf $F^{!}$ has the property that $H^q(A,F^{!}) \cong H^q(X,i_*F^{!}) \cong H^q_A(X,F)$. The association $F \mapsto F^{!}$ then determines a left exact functor $i^!:Sh(X,Ab) \to Sh(A,Ab)$ which is sometimes called the exceptional inverse image.