Cohomology with compact support for sheaves in separated schemes of finite type over a Noetherian scheme: three different definitions

Question

usually there are three notions of cohomology with compact (proper) support. The first one usually done in the étale site. However the second one is used in Verdier duality. The third one is done in algebraic topology.

Let $X$ be a separated scheme of finite type over a Noetherian scheme $S$. Then Nagata compactification guarantees an open immersion $j : X \hookrightarrow \overline{X}$. In this context, the cohomology with compact support can be defined as $\text{H}^q (X, \mathscr{F}) = \text{H}_c^q (\overline{X}, j_{!}\mathscr{F})$. In this case, $R^{p}\text{H}_c^0 \neq \text{H}_c^{p}$.

In the other approach, $\text{H}_c^{q} (X, \mathscr{F})$ is defined as the $q$-th right derived functor of the proper supported global sections.

In the third approach, $\text{H}_c^{q} (X, \mathscr{F})$ is defined as the $q$-th right derived functor of the compact supported global sections.

What's is the relation between these three approaches? Why the first one is used in étale cohomology? Would Verdier duality holds in the first and third approach? Why the first and second approach is called compact supported anyway (I can't see the sections that are compact supported!)?

Thanks in advance.

Answer 1

This is a longer version of my previous comments.

In topology, along with the ordinary cohomology, there is also the so called cohomology with compact support, denoted by $H_c^*(X)$. It can be defined

• using compactly supported differential forms
• using singular cochains that vanish outside a compact set
• using the derived functor of the compactly supported global sections of a sheaf.

all of them give the same result (with corresponding coefficients) for reasonable spaces and have the following fundamental properties :

• they coincide with ordinary cohomology if $X$ is compact.
• they are covariant wrt open immersion and contravariant wrt to proper maps.
• there exists localization long exact sequence : if $U\subset X$ is an open subset and $Z$ the closed complement, the sequence : $$ \dots\rightarrow H^*_c(U)\rightarrow H^*_c(X)\rightarrow H_c^*(Z)\rightarrow H^{*+1}_c(U)\rightarrow\dots$$ is exact.

Lets talk about sheaves now. Along with the compactly supported global sections functor $\Gamma_c(X,\cdot)$, there exists a relative version, namely the functor $f_!$ for $f:X\rightarrow Y$. It is defined as $$f_!\mathcal{F}(U)=\{ s\in\mathcal{F}(f^{-1}(U)), \text{such that } f_{|\mathrm{supp} (s)}:\mathrm{supp}(s)\rightarrow Y \text{ is proper}\} $$ and as the following properties : $(f\circ g)_!=f_!\circ g_!$, for $j$ an open immersion, $j_!$ is the extension by zero functor and is exact, $f_!=f_*$ if $f$ is proper, and for any $y\in Y$ $$(f_!\mathcal{F})_y=H^*_c(f^{-1}(y),\mathcal{F})$$ When $f:X\rightarrow pt$, you get the functor $\Gamma_c$.

In topology, and only in topology, because of the so called soft sheaves, deriving $f_!$ et $\Gamma_c$ makes sense and give the expected cohomology. I believe this justify the name.

In other context, like in algebraic geometry, there are not enough proper subset or there are too much compact ones to have interesting results. Besides, in the context of étale cohomology (which was developed in order to have a cohomology theory like the singular one), the only definition that would make sense is something that can be compared to the topological cohomology with compact support. Hence your second and third definitions cannot be used.

However, the first make perfect sense : the functor $j_!$ exists, this is the left adjoint of $j^*$ for an open immersion, the so called extension by zero functor. So that, for $f:X\rightarrow Y$ we can define $f_!$ to be $\overline{f}_*\circ j_!$ where $j:X\rightarrow \overline{X}$ is any compactification with $f=\overline{f}\circ j$ and $\overline{f}:\overline{X}\rightarrow Y$ proper. Similarly, we can define $\Gamma_c$ to be $\Gamma\circ j_!$.

It turns out, that their derived functors are not interesting, but $R^i\overline{f}_*j_!$ and $R^i\Gamma\circ j_!$ are and are exactly what we wanted :

• they can be compared to the topological ones
• they are covariant wrt to open immersion, contravariant wrt proper morphisms
• there are localization long exact sequences.