Relative vs. local sheaf cohomology?

Given a sheaf of abelian groups $\mathscr{F}$ on a space $X$, $i: Z \hookrightarrow X$ the inclusion of a closed subspace, and $j: U := X \setminus Z \hookrightarrow X$ the inclusion of its complement, Iversen defines the relative sheaf cohomology groups to be the sheaf cohomology groups $$H^p(X, Z; \mathscr{F}) := H^p(X, j_!j^{-1} \mathscr{F}) := \mathbf{R}^p\Gamma(X,j_!j^{-1} \mathscr{F}).$$

These groups fit into a long exact sequence by taking the derived global sections functor of the short exact sequence $$0 \to j_!j^{-1} \mathscr{F} \to \mathscr{F} \to i_* i^{-1} \mathscr{F} \to 0.$$

On the other hand, earlier in the text, Iversen defines the local sheaf cohomology groups with support in $Z$ to be the derived functors $$H^p_Z( \mathscr{F}) := \mathbf{R}^p\Gamma_Z(X,\mathscr{F}),$$ where $$\Gamma_Z(X, \mathscr{F}) :=\{ s \in \Gamma(X,\mathscr{F}) \mid \mathrm{supp}(s) \subseteq Z\}$$ is the set of sections with support in $Z$. These also fit into a long exact sequence induced by the exact sequence

$$0 \to \Gamma_Z(X, \mathscr{F}) \to \Gamma(X, \mathscr{F}) \to \Gamma(U, \mathscr{F})$$

I have two questions:

1. What is the relationship between relative cohomology and local cohomology?

2. If we take $\mathscr{F}$ to be a constant sheaf of abelian groups, then how do we interpret these two notions of local and relative sheaf cohomology in terms of the usual cohomology theories in algebraic topology (such as singular cohomology on nice spaces)?

These notions are "dual". More precisely, there are two distinguished triangle in the derived category of sheaves on $X$ : $$j_!j^{-1}\rightarrow 1\rightarrow i_*i^{-1}\overset{+1}\rightarrow$$ and $$i_*Ri^!\rightarrow 1\rightarrow Rj_*j^{-1}\overset{+1}\rightarrow$$

Verdier duality exchange both of them. Here, we need to restrict ourselves to the subcategory of constructible sheaves.

Applying $Rf_*$ with $f:X\rightarrow\{*\}$ to the first triangle gives the long exact sequence $$...\rightarrow H^i(X,Z,\mathcal{F})\rightarrow H^i(X,\mathcal{F})\rightarrow H^i(Z,\mathcal{F}_{|Z})\rightarrow ...$$ which has a nice particular case : when $X$ is compact (and so is $Z$), this is the localization exact sequence in cohomology with compact support : $$...\rightarrow H^i_c(U,\mathcal{F}_{|U})\rightarrow H^i(X,\mathcal{F})\rightarrow H^i(Z,\mathcal{F}_{|Z})\rightarrow ...$$ More generally, applying $Rf_!$ to this triangle gives the long exact sequence $$...\rightarrow H^i_c(U,\mathcal{F}_{|U})\rightarrow H^i_c(X,\mathcal{F}) \rightarrow H^i_c(Z,\mathcal{F}_{|Z})\rightarrow ...$$

The second triangle gives the long exact sequence $$...\rightarrow H^i_Z(X,\mathcal{F})\rightarrow H^i(X,\mathcal{F})\rightarrow H^i(U,\mathcal{F}_{|U})\rightarrow ...$$

When everything is smooth, Verdier duality is just Poincaré duality. With some finiteness assumptions and if $X$ and $Z$ are orientables, this last exact sequence is the dual of the third one. We can remove the orientation hypothesis if we use the orientation sheaf instead.

More precisely, $H^i_c(X,\mathcal{F})^\vee=H^{n-i}(X,\mathcal{F}^\vee\otimes\mathfrak{or})$, and similarly for $U$ and $Z$.

If $Z$ is smooth of codimension $c$ and has a tubular neighborhood in $X$, then $H^i_Z(X,\mathbb{Z})=H^{i-c}(Z,\mathbb{Z})$. In fact, one has $Ri^!\mathbb{Z}_X=\mathbb{Z}_Z[-c]$ in the derived category of $Z$. In algebraic geometry, these kind of isomorphism are called purity isomorphism.

Also, when $\mathcal{F}$ is a constant abelian group, on nice spaces, we recover the usual cohomology groups (relative wrt $Z$, or with compact support). In particular $H^i(X,Z,\underline{\mathbb{Z}}_X)$ is canonically isomorphic to $H^i_{sing}(X,Z,\mathbb{Z})$ and $H^i_Z(X,\underline{\mathbb{Z}}_X)$ is canonically isomorphic to $H^i_{sing}(X,U,\mathbb{Z})$.

• Thanks for taking the time to write up such a detailed answer! This is very helpful. Do you happen to know of a textbook reference for this? I tried browsing Iversen and Kashiwara & Schapira, but maybe I missed it. – ಠ_ಠ Apr 6 '17 at 22:32
• Unfortunately I don't know any reference other than Iversen, Kashiwara-Shapira. If it is not in these books, I can't recall where I have read these things. Note that the most difficult part is the comparison theorem, then you "only" need to construct a natural map between the complexes computing the relative cohomology groups (this will be an quasi-isomorphism by the five lemma). – Roland Apr 7 '17 at 23:35
• Take a look at Massey's note (arxiv.org/pdf/math/9908107.pdf) page 10,16,17.. You can also check Dimca's Sheaves in Topology around section 2.6. – Student Oct 7 '18 at 11:33