c-soft sheaves

Let $X$ be a locally compact Hausdorff space (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and let $\mathcal{F}$ be a sheaf of abelian groups on $X$ (A sheaf on a topological space); this covers in particular every sheaf of $k$-vector spaces over a field $k$, which is the case used by the compact-support calculus, and for such a sheaf the groups below are $k$-vector spaces and the maps $k$-linear. Global sections are written $\mathrm{\Gamma }(X,-)$ (Global sections of an abelian sheaf). For a compact subset $K\subseteq X$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right) let $i:K\hookrightarrow X$ be the inclusion and consider the canonical restriction map $\mathrm{\Gamma }(X,\mathcal{F})⟶\mathrm{\Gamma }(K,i^{-1}\mathcal{F})=(i^{-1}\mathcal{F})(K),s⟼s{\mathrm{\mid }}_K,$ which sends a global section (Sections, restrictions, and global sections of a presheaf) to the section of the inverse image sheaf (Inverse image presheaf and inverse image sheaf) that it induces on $K$. Then $\mathcal{F}$ is c-soft, also soft for compact supports, when this map is surjective for every compact subset $K\subseteq X$, that is: every section of $\mathcal{F}$ over a compact subset of $X$ extends to a global section of $\mathcal{F}$. For sheaves of abelian groups ${\mathrm{\Gamma }}_c$ and $H_c^q$ of Compactly supported global sections and their right derived functors are the compactly supported analogues of $\mathrm{\Gamma }$ and $H^q$.

Two consistency facts hold; both are immediate from the definitions and from the extension lemma for sections on a compact set, and they are recorded here because the compact-support calculus uses the first constantly and the second for injective sheaves.

1. Neighbourhood formulation. By Sections on a compact set extend to a neighbourhood, the target of the restriction map is the defining colimit of the inverse-image presheaf, , so that $\mathcal{F}$ is c-soft if and only if for every compact $K$ and every open neighbourhood $U$ of $K$, every section $s\in \mathcal{F}(U)$ agrees with a global section of $\mathcal{F}$ on some open neighbourhood of $K$. It therefore suffices, for c-softness, to test the extended sections only locally near $K$.
2. Flasque implies c-soft. If $\mathcal{F}$ is flasque (Flasque sheaf) then $\mathcal{F}$ is c-soft: a class in $(i^{-1}\mathcal{F})(K)$ is represented by a section $s$ over an open neighbourhood $U$ of $K$, and flasqueness extends $s$ to a global section $t\in \mathcal{F}(X)$ with $t{\mathrm{\mid }}_U=s$, so $t$ is a preimage of that class.

C-softness is a property of the pair $(X,\mathcal{F})$ and not of the sheaf alone: the compact subsets of the base space are part of the test. Nothing is assumed about the support of the extended global sections, and c-softness is not the same condition as flasqueness, only implied by it.

• Terminology and provenance. The term is Schapira's c-soft ("mou pour les compacts"), Definition 11.2.1 of his lecture notes; Verdier's expose uses c-mou for the same condition on a locally compact space. The definition needs only the compact subsets of $X$, and local compactness together with Hausdorffness supplies the ambient facts the later arguments use: compact subsets are closed, and disjoint compact sets are separated by disjoint open sets.

• The neighbourhood formulation is the form in which c-softness is verified in the compact-support calculus of this page: there one is given sections over a compact set as representatives on an open neighbourhood, and must produce, or compare with, global sections agreeing near the compact set.