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Let $i : A \hookrightarrow X$ be a closed subspace of a topological space $X$, and $j : Z := X \setminus A \hookrightarrow X$ denote its open complement. Given a sheaf $F$ of abelian groups on $X$, one can define sheaf cohomology with support in $A$, by taking the derived functors of the functor $\Gamma_A$ "sections with support in $A$": $$\Gamma_A F := \lbrace s \in \Gamma(X,F): s_{|Z} = 0 \rbrace.$$ Whenever the spaces involved are nice enough, and $F$ is a constant sheaf, these derived functors seem to give the relative singular cohomology groups $H^*(X,Z)$.

I am wondering the following:

  1. How can we rewrite the expression $R^q \Gamma_A F$ in terms of the functors direct and inverse images associated with $i$ (or $j$) ? That is, can we define the sheaf $R^q \Gamma_A F$ as $R^q i_* i^* F$, or something in this spirit ?
  2. What if I would like to obtain the relative cohomology groups $H^*(X,A)$ ? In this case, could we define relative sheaf cohomology as something like $R^q j_* j^* F$ ?
  3. What would be the properties of the functors used to define cohomology relative to the closed $A$ (left adjoint, exact...) ?
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Let $j:X\backslash A\to X$ and $i:A\to X$ denote the inclusions. Let $j_!$ denote the functor of extension by zero along $j$. Then the relative cohomology is exactly $H^*(X,j_!R)$ where $R$ is the constant sheaf of coefficients. To see this, just consider the exact sequence

$$j_!j^*R\to R \to i_*i^*R$$ and observe that the second map is identified with restriction along $i$ after passing to global sections.

The general scheme to have in mind is that we have 6-functors formalism here:

$i^*$ and $j^*$ are restrictions, $i_*$ and $j_*$ are the usual push maps, right adjoint to $i^*$ and $j_*$. Additinally we have $i_!,i^!,j_!,j^!$ where:

$i_! = i_*$.

$j^!=j^*$.

$j_!$ is extension by zero.

$i^!$ is "sections with support on $A$".

And we have the basic exact triangles of functors $i_!i^!\to Id\to j_*j^*$ and $j_!j^! \to Id \to i_*i^*$.

From those facts you can more or less deduce anything related to the way sheaves and their sections on $X$ decompose into the open and closed part.

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  • $\begingroup$ Thanks a lot @S. Carmeli. What is the difference between $j_!$ and $j_*$ ? If $F$ is a sheaf on $X$, then should I define $R^* j_! j^* F$, or $R^* j_! j^! F$, or something else ? What properties does this functor have ? $\endgroup$ – BrianT Aug 28 '18 at 14:31
  • $\begingroup$ I edited the answer to contain more information. and in general open-closed decomposition is a place where its much easier to understand things on the level of the derived category rather then on the cohomology ($R^*(\bullet)$), so I strongly suggest to try to learn some derived categories in order to understand what's going on here on the cohomological level. $\endgroup$ – S. carmeli Aug 28 '18 at 14:36
  • $\begingroup$ Thanks a lot for your help. I think you made a small writing mistake: "right adjoint to $i^*$ and $j^*$" instead of $j_*$. Last question, does the functor $j_!$ have nice properties, for instance, does it commute with direct sums, derived functors, restrictions etc ? $\endgroup$ – BrianT Aug 28 '18 at 14:51
  • $\begingroup$ If I understand you well, given a sheaf $F$ on $X$, I will define $R^* j_! j^* F$ for relative sheaf cohomology with respect to $A$, and $R^* i_* i^! F$ for relative cohomology with respect to $Z$ ? $\endgroup$ – BrianT Aug 28 '18 at 15:01
  • $\begingroup$ The notation $R^*$ in your comment is non-standard at least. It should be $R^*\Gamma(X,\bullet)$ or so. But again, its a good idea to shift to derived categories and don't take cohomologies. In this context its very helpful. $\endgroup$ – S. carmeli Aug 28 '18 at 15:03

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