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...) ?

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:

$i_! = i_*$.


$j_!$ 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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.