Difference between local cohomology and cohomology with support in a family

Question

Let $X$ be a topological space. A collection of closed subsets of $X$ is called a family of supports (in the sense of Cartan) if: (1) the union of any two elements of $\Phi$ is an element of $\Phi$, and (2) any closed subset of $X$ contained in an element of $\Phi$ is itself an element of $\Phi$.

Given a family of supports $\Phi$, one can define a functor $\Gamma_{\Phi}$ from the category of sheaves of abelian groups on $X$ to the category of abelian groups as follows: $$\Gamma_{\Phi}(\mathcal{F})=\{\sigma\in\Gamma(X,\mathcal{F})\mid\operatorname{Supp}(\sigma)\in\Phi\}.$$ This functor is easily seen to be left exact and one can derive it using resolutions by flasque sheaves. The derived functors of $\Gamma_{\Phi}$ applied to a sheaf of abelian groups $\mathcal{F}$ are denoted by $H^p_{\Phi}(X,\mathcal{F})$ and called the cohomology groups of $X$ with coefficients in $\mathcal{F}$ and support in $\Phi$.

Let $Z$ be a subset of $X$ and let $\Phi_Z$ be the set of all closed subsets of $X$ that are contained in $Z$. Then $\Phi_Z$ is a family of supports and one can consider the cohomology groups $H^p_{\Phi_Z}(X,\mathcal{F})$, for any sheaf of abelian groups $\mathcal{F}$ on $X$. When $Z$ is closed in $X$, Grothendieck defines the functor $\Gamma_Z$ in SGA2 as follows, and notes that the functor $\Gamma_{{\Phi}_Z}$ and $\Gamma_Z$ are the same: $$\Gamma_{Z}(\mathcal{F})=\{\sigma\in\Gamma(X,\mathcal{F})\mid\operatorname{Supp}(\sigma)\subset Z\}.$$ Then he says: "we want to generalize this definition to the case where $Z$ is locally closed in $X$."

Question 1. Which definition exactly does he want to generalize? If he wants tp generalize the definition of $\Gamma_Z$ to the case where $Z$ is locally closed, why doesn't he just consider the functor $\Gamma_{{\Phi}_Z}$ in this case, as well?

In any case, when $Z$ is locally closed in $X$, Grothendieck doesn't define the functor $\Gamma_Z$ as $\Gamma_{{\Phi}_Z}$. Instead, he chooses an open set $V$ that contains $Z$ as a closed subset and defines $$\Gamma_Z(\mathcal{F})=\Gamma_Z(\mathcal{F}|_V).$$ Then he shows this definition is independent of the choice of $V$.

When $Z$ is closed in $X$, if $V$ is any open subset of $X$ that contains $Z$ as a closed subset, and $\sigma\in\Gamma(V,\mathcal{F})$ is any section with support contained in $Z$, then it is easy to see that $\sigma$ is the restriction of a section in $\Gamma(X,\mathcal{F})$. This is no longer true if $Z$ is locally closed in $X$, showing that $\Gamma_{\Phi_Z}(\mathcal{F})$ and $\Gamma_Z(\mathcal{F})$ are not the same in this case.

Question 2. What is the reason that Grothendieck didn't use the functor $\Gamma_{\Phi_Z}$ in the case when $Z$ is locally closed? In other words, what properties of local cohomology did he have in mind, which he couldn't obtain by using the functor $\Gamma_{\Phi_Z}$? Where can we see the difference between local cohomology groups with support in a locally closed subset $Z$, and the cohomology groups $H^p_{\Phi_Z}(X,\mathcal{F})$?

Answer 1

I think here's a simple example that might illustrate the difference. Let's take $X$ to be the real line and $Z$ to be the open interval $(0,1)$. Let $\Phi$ be the closed subsets of $X$. Then $\Phi_Z$ consists of closed subsets of $\mathbb{R}$ contained in $(0,1)$. Since these are bounded, $\Phi_Z$ winds up being the compact subsets of $(0,1)$ and so $H_{\Phi_Z}^i(Z;\mathcal{F})=H^i_c(Z;\mathcal{F}|_Z)$, i.e. you get cohomology with compact supports. By contrast, in this case we have $V=Z$ and so $\Gamma_Z(\mathcal{F}|_V)$ is just the sections of $\mathcal{F}|_Z$. In other words, the resulting cohomology is the ordinary cohomology $H^i(Z;\mathcal{F}|_Z)$.

Either of these is reasonable to study, but I think this suggests that the latter construction is more in line with what one usually considers to be the cohomology of a subspace. I'd suggest having a look at something like Section II.10 of Bredon's Sheaf Theory, which contains a lot of related material.