Compactly supported global sections and their right derived functors

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a locally compact Hausdorff space, let $\mathrm{A}\mathrm{b}(X)$ be the category of sheaves of abelian groups on $X$ (A sheaf on a topological space) and let $\mathrm{\Gamma }(X,-):\mathrm{A}\mathrm{b}(X)⟶\mathbf{A}\mathbf{b}$ be the additive left exact global-sections functor (Global sections of an abelian sheaf, Sections, restrictions, and global sections of a presheaf). For a sheaf $\mathcal{F}$ on $X$ define the compactly supported global sections of $\mathcal{F}$ to be the subgroup ${\mathrm{\Gamma }}_c(X,\mathcal{F}):=\{s\in \mathrm{\Gamma }(X,\mathcal{F}):\mathrm{Supp}(s)\text{ is a compact subset of }X\}\subseteq \mathrm{\Gamma }(X,\mathcal{F}),$ where the support of a section is the set of points at which its germ is nonzero (Support of a section of a sheaf, Germs of sections, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right). This is a subgroup: the zero section has empty support, negation preserves support, and the support of a sum is closed in $X$ and contained in the finite union of the two compact supports, hence compact (Support of a section of a sheaf, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact). A morphism $\phi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves induces a map ${\mathrm{\Gamma }}_c(X,\phi )$ by restriction of $\mathrm{\Gamma }(X,\phi )$, and this makes ${\mathrm{\Gamma }}_c(X,-)$ an additive functor $\mathrm{A}\mathrm{b}(X)\to \mathbf{A}\mathbf{b}$ (Additive functor, Morphisms of presheaves). Six consistency facts hold; all are immediate from the definitions, and they record the interface used by the compact-support calculus.

1. Well-definedness of ${\mathrm{\Gamma }}_c(X,\phi )$. If $x\mathrm{\notin }\mathrm{Supp}(s)$ then $s_x=0$ in the stalk ${\mathcal{F}}_x$ (The stalk of a presheaf at a point), and since the stalk maps of a morphism commute with the germ maps, $\phi (s{)}_x={\phi }_x(s_x)=0$, so $\mathrm{Supp}(\phi (s))\subseteq \mathrm{Supp}(s)$. A support $\mathrm{Supp}(t)$ is closed in $X$: if $x\mathrm{\notin }\mathrm{Supp}(t)$ then $t$ agrees with the zero section on some open neighbourhood of $x$, by the description of equality of germs (Support of a section of a sheaf). Hence $\mathrm{Supp}(\phi (s))$ is a closed subset of $X$ contained in the compact set $\mathrm{Supp}(s)$; being closed in $X$ it is closed in the subspace $\mathrm{Supp}(s)$, so it is a compact subset of that compact space (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact) and therefore a compact subset of $X$ (A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it), and ${\mathrm{\Gamma }}_c(X,\phi )$ maps ${\mathrm{\Gamma }}_c(X,\mathcal{F})$ into ${\mathrm{\Gamma }}_c(X,\mathcal{G})$. Composites and identities are computed componentwise, so ${\mathrm{\Gamma }}_c(X,-)$ is a functor; it is additive because $\mathrm{\Gamma }(X,-)$ is.
2. Sheaves of $k$-vector spaces. If $\mathcal{F}$ is a sheaf of $k$-vector spaces, then ${\mathrm{\Gamma }}_c(X,\mathcal{F})$ is a $k$-vector subspace of $\mathrm{\Gamma }(X,\mathcal{F})$: the support of $c\cdot s$ is $\mathrm{Supp}(s)$ for $c\mathrm{\ne }0$ and $\mathrm{\varnothing }$ for $c=0$, and the zero section has empty support. For a morphism $\phi$ of sheaves of $k$-vector spaces the map ${\mathrm{\Gamma }}_c(X,\phi )$ is $k$-linear, since $\mathrm{\Gamma }(X,\phi )$ is. The higher derived functors below are computed from an injective resolution datum on abelian sheaves, so they are abelian groups; the degree-zero part is the $k$-vector space ${\mathrm{\Gamma }}_c(X,\mathcal{F})$.
3. Left exactness. For every short exact sequence $0\to {\mathcal{F}}^{\mathrm{\prime }}\stackrel{{\phi }^{\mathrm{\prime }}}{\to }\mathcal{F}\stackrel{\phi }{\to }{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of abelian sheaves the sequence $0\to {\mathrm{\Gamma }}_c(X,{\mathcal{F}}^{\mathrm{\prime }})\stackrel{{\mathrm{\Gamma }}_c({\phi }^{\mathrm{\prime }})}{\to }{\mathrm{\Gamma }}_c(X,\mathcal{F})\stackrel{{\mathrm{\Gamma }}_c(\phi )}{\to }{\mathrm{\Gamma }}_c(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ is exact. Injectivity: a monomorphism of sheaves has injective stalk maps (A sequence of abelian sheaves is exact exactly when it is exact on every stalk), so $\mathrm{Supp}({\phi }^{\mathrm{\prime }}(s^{\mathrm{\prime }}))=\mathrm{Supp}(s^{\mathrm{\prime }})$ and a section killed by ${\phi }^{\mathrm{\prime }}$ has all germs zero, hence is zero. Exactness at the middle: the kernel of $\mathrm{\Gamma }(X,\phi )$ is the image of $\mathrm{\Gamma }(X,{\phi }^{\mathrm{\prime }})$ by left exactness of global sections (Global sections are left exact but need not preserve epimorphisms); if $s={\phi }^{\mathrm{\prime }}(s^{\mathrm{\prime }})$ has compact support, then $s^{\mathrm{\prime }}$ has the same support, $\mathrm{Supp}(s^{\mathrm{\prime }})=\mathrm{Supp}({\phi }^{\mathrm{\prime }}(s^{\mathrm{\prime }}))=\mathrm{Supp}(s)$, because ${\phi }^{\mathrm{\prime }}$ is stalk-injective, so $s^{\mathrm{\prime }}$ lies in ${\mathrm{\Gamma }}_c(X,{\mathcal{F}}^{\mathrm{\prime }})$, and the reverse inclusion is immediate from $\phi \circ {\phi }^{\mathrm{\prime }}=0$. Thus ${\mathrm{\Gamma }}_c(X,-)$ is an additive left exact functor on $\mathrm{A}\mathrm{b}(X)$ (Left exact and right exact functors).
4. The derived functors. Abelian sheaves on $X$ have enough injectives, and the embeddings of Enough injective abelian sheaves supply one functorial injective resolution datum $\mathcal{F}\mapsto I^{\bullet }(\mathcal{F})$ on the whole category $\mathrm{A}\mathrm{b}(X)$, under the Axiom of Choice (Right derived objects relative to supplied injective resolution data). The compactly supported cohomology of $\mathcal{F}$ is $H_c^q(X,\mathcal{F}):=R^q{\mathrm{\Gamma }}_c(X,\mathcal{F}):=H^q({\mathrm{\Gamma }}_c(X,I^{\bullet }(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}))(q\ge 0),H_c^q(X,\mathcal{F}):=0(q<0),$ the cohomology of the deleted resolution, as in the pattern of Sheaf cohomology as right derived global sections; for a morphism $\phi$ we write $H_c^q(X,\phi ):=R^q{\mathrm{\Gamma }}_c(X,\phi )$ for the induced map.
5. Independence of the datum. If $J$ is another supplied injective resolution datum on $\mathrm{A}\mathrm{b}(X)$, then $R^q{\mathrm{\Gamma }}_c(X,-)$ computed from $J$ is naturally isomorphic to $R^q{\mathrm{\Gamma }}_c(X,-)$ computed from $I$ (Two supplied injective resolution data define naturally isomorphic right derived functors); the Axiom of Dependent Choice needed for that comparison follows from AC (AC implies DC implies countable choice). The notation $H_c^q(X,\mathcal{F})$ therefore does not depend on the datum up to this canonical natural isomorphism.
6. Delta functor and the compact case. Since ${\mathrm{\Gamma }}_c(X,-)$ is additive and left exact and the datum is defined on all of $\mathrm{A}\mathrm{b}(X)$, the functors $R^q{\mathrm{\Gamma }}_c(X,-)$, $q\ge 0$, carry connecting maps that make them a cohomological delta functor with $R^0{\mathrm{\Gamma }}_c(X,-)\cong {\mathrm{\Gamma }}_c(X,-)$, and every short exact sequence of abelian sheaves has a long exact compact-support cohomology sequence, natural in the sequence (Right derived functors form a cohomological delta functor). If $X$ is compact, then for every global section $s$ its support is closed in $X$ by fact 1, hence a closed subset of the compact space $X$ and therefore compact, so every global section is compactly supported; hence ${\mathrm{\Gamma }}_c(X,-)=\mathrm{\Gamma }(X,-)$ and $H_c^q(X,\mathcal{F})=H^q(X,\mathcal{F})$ for every $q$, where $H^q(X,-)$ is the sheaf cohomology of Sheaf cohomology as right derived global sections computed from the same datum.

• Terminology and provenance. For the terminal map $a_X:X\to \mathrm{p}\mathrm{t}$ the functor $a_{X!}$ is the compactly supported global-sections functor, left exact, and its right derived functors are $H_c^q(X,-)$; this is Verdier's starting point for the duality theory of locally compact spaces, and Schapira writes the same functor ${\mathrm{\Gamma }}_c(X;-)$. The construction above is the special case of that pattern with $Y$ a point.

• Choice accounting. The Axiom of Choice is used exactly once in the definition, through the injective resolution datum for abelian sheaves (Enough injective abelian sheaves); the independence of the datum and the delta-functor statement use the Axiom of Dependent Choice, which follows from AC (AC implies DC implies countable choice). Nothing else in the definition selects from infinitely many sets.