# Difference between local cohomology and cohomology with support in a family

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 $$H^p_{\Phi_Z}(X,\mathcal{F})$$?

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 $$H^i(Z;\mathcal{F}|_Z)$$.