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

## 1 Answer

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.

• Thank you for your response. One of the goals of my question was to try to understand what should be "called" local cohomology. For example, suppose we want to generalize the definition of local cohomology to the case where the support is not locally closed. What should we be looking for? In other words, what characterizes local cohomology? Do you have any comments on this question? – Mahdi Majidi-Zolbanin May 26 at 16:48
• Again, I'd suggest looking at Bredon's book. It also has a lot of material about sheaf cohomology on subspaces. – Greg Friedman May 27 at 4:36