Compact Support Sheaf Calculus — Examples

Sheaves on a discrete space are families. Let $\mathcal{V}$ be a sheaf of $k$-vector spaces on $X$ and write $V_x:=\mathcal{V}(\{x\})$, an open set by [F2]. For open $U\subseteq X$ the singletons $\{x\}$, $x\in U$, are pairwise disjoint open sets covering $U$, so the gluing and locality axioms [F1] identify $s\in \mathcal{V}(U)$ with the family of its restrictions $s_x\in V_x$, giving $\mathcal{V}(U)\cong {\prod }_{x\in U}{V}_x$; these identifications are compatible with restriction to smaller opens. The neighbourhood $\{x\}$ is the smallest open neighbourhood of $x$ [F2], so the filtered colimit of [F3] is already $\mathcal{V}(\{x\})$ and the stalk at $x$ is $V_x$. Since a morphism of sheaves is a morphism of presheaves [F1], it is exactly a family of $k$-linear maps ${\phi }_x:V_x\to W_x$ determining ${\phi }_U={\prod }_{x\in U}{\phi }_x$; conversely any such family satisfies the restriction squares.

Every sheaf is c-soft. The space $X$ is Hausdorff because distinct singletons are disjoint open neighbourhoods. Each singleton is compact (one member of any cover covers it) and open, hence a compact neighbourhood; thus $X$ is locally compact. Let $K\subseteq X$ be compact; by [F6] the set $K$ is finite, so a section of $\mathcal{V}$ over $K$ is a family $(v_x{)}_{x\in K}\in {\prod }_{x\in K}{V}_x$ under the identification of [step 1.1]. Extending it by $v_x:=0$ for $x\in X\setminus K$ defines a global section of $\mathcal{V}$ whose restriction to $K$ is the given family, since the restriction maps are the projections of the product [step 1.1]. Hence the restriction map $\mathrm{\Gamma }(X,\mathcal{V})\to \mathrm{\Gamma }(K,\mathcal{V}{\mathrm{\mid }}_K)$ is surjective for every compact $K$, that is, $\mathcal{V}$ is c-soft [F4]; no choice is used, only the formula for the extension.

Compact support in degree zero. The finite set $X$ is compact: for any open cover, choose one member containing each $x\in X$; these finitely many members cover $X$, by finite choice [F9], and for $X=\mathrm{\varnothing }$ take the empty subcover. Hence every global section of $\mathcal{V}$ has compact support and ${\mathrm{\Gamma }}_c(X,\mathcal{V})=\mathrm{\Gamma }(X,\mathcal{V})={\prod }_{x\in X}{V}_x$ by [F5] and [step 1.1]; since $X$ is finite, the product coincides with the direct sum, so ${\mathrm{\Gamma }}_c(X,\mathcal{V})={⨁}_{x\in X}{V}_x$ [F8].

Vanishing in positive degrees and the adjunction. Every sheaf on $X$ is c-soft by [step 2.1], so the acyclicity lemma gives $H_c^q(X,\mathcal{V})=0$ for every $q>0$, and $R{\mathrm{\Gamma }}_c(X,\mathcal{V})$ is concentrated in degree zero with value ${\mathrm{\Gamma }}_c(X,\mathcal{V})$ as computed in [step 2.2]; this proves (2) and, with [step 2.1], also (1). For a $k$-vector space $W$ the constant sheaf $W_X$ has $(W_X{)}_x\cong W$ for every $x$ [F7], so by [step 1.1] a morphism $\phi :\mathcal{V}\to W_X$ is exactly a family of $k$-linear maps ${\phi }_x:V_x\to W$, and such a family defines $v\mapsto {\sum }_{x\in X}{\phi }_x(v_x)$ on the direct sum, and restricting a linear map to the coordinate inclusions recovers the family. Every vector is the finite sum of its coordinate vectors, so these constructions are inverse [F8]; under the identification of [step 2.2] this is ${\mathrm{Hom}}_k({\mathrm{\Gamma }}_c(X,\mathcal{V}),W)$. The bijection is natural in $\mathcal{V}$ and in $W$, because it is obtained by composing with the canonical inclusions $V_x\to {⨁}_x{V}_x$ and is compatible with the action of morphisms on the components.

Conclusion. Clause (1) is [step 2.1], clause (2) is [step 2.2] together with [step 3.1], and clause (3) is [step 3.1]. The Axiom of Choice is inherited from the compact-support acyclicity lemma used for the vanishing; the only selections made here are over the finite set $X$ and its finite subsets, supplied by finite choice in ZF [F9].