Compact-support acyclicity of c-soft sheaves

Let $C_1,C_2\subseteq X$ be compact, let $V_1,V_2$ be open with $C_k\subseteq V_k$, let $s_k\in \mathcal{F}(V_k)$, and suppose that $W\subseteq V_1\cap V_2$ is open with $C_1\cap C_2\subseteq W$ and $s_1{\mathrm{\mid }}_W=s_2{\mathrm{\mid }}_W$. Then there are an open $V\supseteq C_1\cup C_2$ and $s\in \mathcal{F}(V)$ with $s=s_k$ on an open neighbourhood of $C_k$ for $k=1,2$: the sets $A:=C_1\setminus W$ and $B:=C_2\setminus W$ are compact and disjoint ([F3]), hence separated by disjoint open $G_1\supseteq A$, $G_2\supseteq B$ ([F3]); with $V_k^{\mathrm{\prime }}:=V_k\cap (G_k\cup W)$ one has $C_k\subseteq V_k^{\mathrm{\prime }}$, $V_1^{\mathrm{\prime }}\cap V_2^{\mathrm{\prime }}\subseteq W$, so $s_1{\mathrm{\mid }}_{V_1^{\mathrm{\prime }}}$ and $s_2{\mathrm{\mid }}_{V_2^{\mathrm{\prime }}}$ glue to the required $s$ over $V_1^{\mathrm{\prime }}\cup V_2^{\mathrm{\prime }}\supseteq C_1\cup C_2$ ([F4]).

Let $C\subseteq X$ be compact, $B\subseteq X$ closed, $V\subseteq X$ open with $C\cup B\subseteq V$, $s_C\in \mathcal{F}(V)$ and $s_B\in \mathcal{F}(V)$ with $s_C{\mathrm{\mid }}_W=s_B{\mathrm{\mid }}_W$ for an open $W\subseteq V$ containing $C\cap B$. Then there are an open $V_0\subseteq V$ with $C\cup B\subseteq V_0$ and $s_0\in \mathcal{F}(V_0)$ with $s_0=s_C$ on an open neighbourhood of $C$ and $s_0=s_B$ on an open neighbourhood of $B$. Indeed $A:=C\setminus W$ is compact, $B_0:=B\setminus W$ is closed, and $A\cap B_0=\mathrm{\varnothing }$; covering $A$ by finitely many open $V_x$ with $x\in V_x\subseteq \overline{V_x}\subseteq X\setminus B_0$ and $\overline{V_x}$ compact ([F3]) and setting $G_1:=\bigcup V_x\supseteq A$, $G_2:=X\setminus \bigcup \overline{V_x}\supseteq B\setminus W$ one has $G_1\cap G_2=\mathrm{\varnothing }$; now $V_0^{\mathrm{\prime }}:=V\cap (G_1\cup W)$ contains $C$, $V_0^{\mathrm{\prime }\mathrm{\prime }}:=V\cap (G_2\cup W)$ contains $B$, and $V_0^{\mathrm{\prime }}\cap V_0^{\mathrm{\prime }\mathrm{\prime }}\subseteq W$, so the restrictions of $s_C$ and $s_B$ glue ([F3, F4]).

Clause (1). Let $\mathcal{F}$ be c-soft, let $K\subseteq X$ be compact and let $W\subseteq X$ be an open neighbourhood of $K$; let $\sigma \in \mathrm{\Gamma }(K,i^{-1}\mathcal{F})$ be represented by $s\in \mathcal{F}(U)$ for an open $U\supseteq K$ ([F2]), and replace $U$ by $U\cap W$. Choose open $W_0,W_1$ with $K\subseteq W_0\subseteq \overline{W_0}\subseteq W_1\subseteq \overline{W_1}\subseteq W$ and $\overline{W_1}$ compact (finitely many applications of the shrinking of [F3]); put $C:=\overline{W_1}\setminus W_0$, a compact set disjoint from $K$. Since $\mathcal{F}$ is c-soft, $[s]\in \mathrm{\Gamma }(K,i^{-1}\mathcal{F})$ has a preimage $t_1\in \mathrm{\Gamma }(X,\mathcal{F})$ ([F1]); by [F2] $t_1=s$ on an open neighbourhood of $K$. Apply the gluing of [F2] to the disjoint compact sets $K$ and $C$ with the sections $t_1$ (over $X$) and $0$ (over $X$): there are an open $V\supseteq K\cup C$ and $u\in \mathcal{F}(V)$ with $u=t_1$ on an open neighbourhood of $K$ and $u=0$ on an open neighbourhood of $C$. Choose a compact $K^{\mathrm{\prime }}$ with $K\cup C\subseteq K^{\mathrm{\prime }}\subseteq V$ ([F3]) and extend the class of $u{\mathrm{\mid }}_{K^{\mathrm{\prime }}}$ to a global section $t_2\in \mathrm{\Gamma }(X,\mathcal{F})$ by c-softness ([F1]); then $t_2=u$ on an open neighbourhood of $K^{\mathrm{\prime }}$ by [F2]. Finally $t_2=0$ on the open set $W_1\setminus \overline{W_0}$: it is contained in $C\subseteq K^{\mathrm{\prime }}$ and $u=0$ near $C$; hence $t_2{\mathrm{\mid }}_{W_1}$ and $0$ agree on $W_1\setminus \overline{W_0}$ and glue across the open cover $\{W_1,X\setminus \overline{W_0}\}$ of $X$ to a section $t\in \mathrm{\Gamma }(X,\mathcal{F})$ with $t=t_2$ on $W_1$ and $t=0$ on $X\setminus \overline{W_0}$ ([F4]). Then $\mathrm{Supp}(t)\subseteq \overline{W_0}\subseteq W$, so $t$ is compactly supported with support inside the prescribed $W$, and $t=t_2=t_1=s$ near $K$, so $t$ represents $\sigma$ over $K$.

Clause (2). Let $\mathcal{I}$ be an injective abelian sheaf and let $U\subseteq X$ be open. By [F5] the canonical map $j_{!}{\mathbb{Z}}_U\to {\mathbb{Z}}_X$ is a monomorphism, so injectivity of $\mathcal{I}$ makes $\mathrm{Hom}({\mathbb{Z}}_X,\mathcal{I})\to \mathrm{Hom}(j_{!}{\mathbb{Z}}_U,\mathcal{I})$ surjective; under the adjunction and the identifications of [F5], $\mathrm{Hom}({\mathbb{Z}}_X,\mathcal{I})\cong \mathrm{\Gamma }(X,\mathcal{I})$ and $\mathrm{Hom}(j_{!}{\mathbb{Z}}_U,\mathcal{I})\cong {\mathrm{Hom}}_U({\mathbb{Z}}_U,\mathcal{I}{\mathrm{\mid }}_U)\cong \mathrm{\Gamma }(U,\mathcal{I})$, and the composite is the restriction map. Hence every section of $\mathcal{I}$ over an open set extends to $X$, that is, $\mathcal{I}$ is flasque, and a flasque sheaf is c-soft by [F1].

Lifting over a compact space. Let $Y$ be a compact Hausdorff space, let $0\to \mathcal{A}\to \mathcal{B}\stackrel{\gamma }{\to }\mathcal{C}\to 0$ be a short exact sequence of abelian sheaves on $Y$ with $\mathcal{A}$ c-soft on $Y$, and let $c\in \mathrm{\Gamma }(Y,\mathcal{C})$. Then $c$ has a preimage in $\mathrm{\Gamma }(Y,\mathcal{B})$. Indeed, the family of all pairs $(U,b)$ with $U\subseteq Y$ open, $b\in \mathcal{B}(U)$ and ${\gamma }_U(b)=c{\mathrm{\mid }}_U$ is cut out by a formula and covers $Y$ (the stalk maps of $\gamma$ are onto, [F4]); by compactness and normality of $Y$ ([F3]) there are finitely many compact $K_1,\dots ,K_m$ covering $Y$ with lifts $b_i\in \mathcal{B}(V_i)$ over open $V_i\supseteq K_i$. We show by induction on $m$ that the pieces can be corrected, without changing their images under $\gamma$, so that they agree near all pairwise intersections and glue ([F2]); for $m=1$ there is nothing to do. For $m\ge 2$, the induction hypothesis gives a section $b^{\mathrm{\prime }}$ over an open neighbourhood of $D:=K_1\cup \cdots \cup K_{m-1}$ with $\gamma (b^{\mathrm{\prime }})=c$ there; over an open neighbourhood of the compact set $D\cap K_m$ the difference $b^{\mathrm{\prime }}-b_m$ has vanishing $\gamma$, so it lies in the image of $\mathcal{A}$ ([F4]); the corresponding section of $\mathcal{A}$ over that neighbourhood extends to $Y$ by c-softness of $\mathcal{A}$, and adding its image to $b_m$ keeps $\gamma (b_m)=c$ while making $b^{\mathrm{\prime }}-b_m$ vanish near $D\cap K_m$; the gluing of step 1.1 applied to the compact pieces $D$ and $K_m$ then produces a section over an open neighbourhood of $D\cup K_m$ mapping to $c$, completing the induction. Since $K_1\cup \cdots \cup K_m=Y$ and a neighbourhood of $Y$ in $Y$ is $Y$, this is a preimage of $c$ in $\mathrm{\Gamma }(Y,\mathcal{B})$.

Clause (4), the topological preservations. Let $\mathcal{F}$ be c-soft. (i) For an open subspace $U\subseteq X$: a compact subset of $U$ is compact in $X$ ([F3]) and a section of $\mathcal{F}{\mathrm{\mid }}_U$ over it is represented by an ambient section over a neighbourhood, which clause (1) extends with support control into any prescribed neighbourhood; restricting the result to $U$ extends the given section, so $\mathcal{F}{\mathrm{\mid }}_U$ is c-soft. (ii) For a closed subspace $Z\subseteq X$: compact subsets of $Z$ are compact in $X$ ([F3]) and a section of $\mathcal{F}{\mathrm{\mid }}_Z$ over one is a class of ambient sections over neighbourhoods of it in $X$ ([F2]); clause (1) extends such a class, and restricting to $Z$ gives the required extension, so $\mathcal{F}{\mathrm{\mid }}_Z$ is c-soft. (iii) Let $\mathcal{A}$ be any c-soft sheaf on the open subspace $U$, which is locally compact Hausdorff by [F3], and put $\mathcal{G}:=j_{!}\mathcal{A}$. Its stalks are ${\mathcal{A}}_x$ on $U$ and zero outside $U$, by the local support condition of [F5]. Let $K\subseteq X$ be compact and represent a section over it by $s\in \mathcal{G}(V)$, $V\supseteq K$ open [F2]. The set $L:=K\cap \mathrm{Supp}(s)$ is closed in $K$, hence compact, and lies in $V\cap U$. If $L=\mathrm{\varnothing }$, the zero global section has the required germs on $K$. Otherwise choose opens $W_0,W_1$ with $L\subseteq W_0\subseteq \overline{W_0}\subseteq W_1\subseteq \overline{W_1}\subseteq V\cap U$ and compact closures [F3]. In $U$ the sets $T:=K\cap \overline{W_1}$ and $C:=\overline{W_1}\setminus W_0$ are compact. The class of $s$ on $T$ and the zero class on $C$ agree on $T\cap C$, since every such point belongs to $K\setminus L$ and has zero germ of $s$. By the fusion of step 1.1 and the neighbourhood representation of [F2], they give a section near $T\cup C$ in $U$. C-softness of $\mathcal{A}$ extends this class to $g\in \mathrm{\Gamma }(U,\mathcal{A})$, agreeing with $s$ near $T$ and with zero near $C$. Glue $g{\mathrm{\mid }}_{W_1}$ with zero on $U\setminus \overline{W_0}$; the overlap lies in $C$. The result $e\in \mathrm{\Gamma }(U,\mathcal{A})$ has support closed and contained in the compact $\overline{W_0}\subseteq U$, hence lies in ${\mathrm{\Gamma }}_c(X,j_{!}\mathcal{A})$. On $K\cap W_1$ it has the germs of $s$, and on $K\setminus \overline{W_0}$ both germs vanish because $L\subseteq W_0$. Thus $e$ represents the original section over $K$, proving c-softness of $j_{!}\mathcal{A}$.

Clause (4), finite sums and constant-vector tensors. For a finite family of c-soft abelian sheaves, sections of the direct sum are finite tuples of sections: componentwise locality and gluing show that the finite product presheaf is a sheaf, and the finite product is also the direct sum. A class over a compact $K$ is represented by a tuple on a neighbourhood of $K$ [F2]; extend each component by step 1.3 and form the tuple of global sections. This proves c-softness of finite direct sums. Now assume $\mathcal{F}$ is a sheaf of $k$-vector spaces, let $M$ be any $k$-vector space and $\mathcal{H}:=\mathcal{F}{\otimes }_{k_X}{M}_X$. Choose a basis $B$ of $M$ under AC [F6, F7]. Each coordinate functional ${\lambda }_b:M\to k$ induces a sheaf morphism $\mathcal{H}\to \mathcal{F}$ via $\mathrm{i}\mathrm{d}\otimes ({\lambda }_b{)}_X$ and the canonical $\mathcal{F}{\otimes }_{k_X}{k}_X\cong \mathcal{F}$; the latter map is multiplication and is an isomorphism on stalks [F6]. A germ of $\mathcal{H}$ is a finite sum of $\mathcal{F}$-germs times basis vectors, by [F6]. Represent its coefficients by local sections; equality of germs gives the same finite-sum expression on a smaller neighbourhood. Thus a section $s$ representing a class over compact $K$ has finite coordinate expressions locally near $K$. Compactness gives finitely many such neighbourhoods, whose union $O$ contains $K$ and whose expressions use one finite subset $B_0\subseteq B$. Put $s_b:=(\mathrm{i}\mathrm{d}\otimes {\lambda }_b)(s{\mathrm{\mid }}_O)$ for $b\in B_0$; uniqueness of basis coordinates and locality give $s{\mathrm{\mid }}_O={\sum }_{b\in B_0}{s}_b\otimes b$. Extend the finitely many $s_b$ to global sections $t_b$ agreeing near $K$ by step 1.3. Then ${\sum }_{b\in B_0}{t}_b\otimes b$ is a global section of $\mathcal{H}$ agreeing with $s$ near $K$. The zero-coefficient case $M=0$ is immediate. This proves the tensor clause for arbitrary constant $M$.

Clause (5), local lifting. Let ${\mathcal{F}}^{\mathrm{\prime }}$ be c-soft, let $K\subseteq X$ be compact and let ${\sigma }^{\mathrm{\prime }\mathrm{\prime }}\in \mathrm{\Gamma }(K,i^{-1}{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ be represented by a section $s^{\mathrm{\prime }\mathrm{\prime }}\in {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}(U)$ over an open $U\supseteq K$ ([F2]). Choose open $U_0$ with $K\subseteq U_0\subseteq \overline{U_0}\subseteq U$ and $\overline{U_0}$ compact ([F3]). The restricted sheaves on the compact space $\overline{U_0}$ form a short exact sequence whose kernel is c-soft on $\overline{U_0}$: a compact subset of $\overline{U_0}$ is compact in $X$ ([F3]) and a section of the restricted kernel over it is represented by an ambient section of ${\mathcal{F}}^{\mathrm{\prime }}$ over an open neighbourhood ([F2]), which c-softness of ${\mathcal{F}}^{\mathrm{\prime }}$ extends to $X$. Hence the lifting lemma of step 2.1 applies to the compact space $\overline{U_0}$ and the class of $s^{\mathrm{\prime }\mathrm{\prime }}$ over it: there is $t\in \mathrm{\Gamma }(\overline{U_0},i^{-1}\mathcal{F})$ with $\beta (t)$ equal to that class. Represent $t$ by a section of $\mathcal{F}$ over an open $V\supseteq \overline{U_0}$ ([F2]); shrinking $V$ we may assume that the image of that section under $\beta$ equals $s^{\mathrm{\prime }\mathrm{\prime }}$ on $V$ (they agree as classes over $\overline{U_0}$, hence near it, [F2]). This section of $\mathcal{F}$ over the open neighbourhood $V$ of $K$ lifts the given class ${\sigma }^{\mathrm{\prime }\mathrm{\prime }}$; since $K$ and ${\sigma }^{\mathrm{\prime }\mathrm{\prime }}$ were arbitrary, clause (5) holds locally.

Clause (5), compact-support exactness. Let ${\mathcal{F}}^{\mathrm{\prime }}$ be c-soft; we show that ${\mathrm{\Gamma }}_c(\beta ):{\mathrm{\Gamma }}_c(X,\mathcal{F})\to {\mathrm{\Gamma }}_c(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ is surjective. Let $s^{\mathrm{\prime }\mathrm{\prime }}\in {\mathrm{\Gamma }}_c(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ and put $S:=\mathrm{Supp}(s^{\mathrm{\prime }\mathrm{\prime }})$, compact. Choose open $W_0,W_1$ with $S\subseteq W_0\subseteq \overline{W_0}\subseteq W_1\subseteq \overline{W_1}$ and $\overline{W_1}$ compact (finitely many shrinkings, [F3]); put $C:=\overline{W_1}\setminus W_0$, a compact set disjoint from $S$. Apply the lifting lemma of step 2.1 to the compact space $\overline{W_1}$ and the class of $s^{\mathrm{\prime }\mathrm{\prime }}$ over it: there is $t\in \mathrm{\Gamma }(\overline{W_1},i^{-1}\mathcal{F})$ with $\beta (t)=s^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\mid }}_{\overline{W_1}}$; represent $t$ by $a\in \mathcal{F}(V_0)$ over an open $V_0\supseteq \overline{W_1}$ with $\beta (a)=s^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\mid }}_{V_0}$ ([F2]). On the open set $O:=V_0\setminus S$ the section $a$ has $\beta (a)=s^{\mathrm{\prime }\mathrm{\prime }}=0$, hence $a{\mathrm{\mid }}_O=\alpha (e)$ for a uniquely determined $e\in {\mathcal{F}}^{\mathrm{\prime }}(O)$: the family of all open $O_x\subseteq V_0$ over which $a$ lies in the image of $\alpha$ covers $V_0\setminus S$ ([F4], the kernel of $\beta$ is the image of $\alpha$ as subsheaves) and the local sections glue because $\alpha$ is injective on sections ([F4]). The compact set $C=\overline{W_1}\setminus W_0$ is contained in $O$, so $e$ is a section of the c-soft ${\mathcal{F}}^{\mathrm{\prime }}$ over an open neighbourhood of the compact set $C$; by clause (1) applied to ${\mathcal{F}}^{\mathrm{\prime }}$ there is $e_X\in {\mathrm{\Gamma }}_c(X,{\mathcal{F}}^{\mathrm{\prime }})$ with $e_X=e$ near $C$ ([F2]). Set $a_1:=a-\alpha (e_X){\mathrm{\mid }}_{V_0}\in \mathcal{F}(V_0)$: then $\beta (a_1)=s^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\mid }}_{V_0}$, and $a_1$ vanishes on the open set $O_C:=\{x\in V_0:a_x=\alpha (e_X{)}_x\}$, which contains $C$. The overlap $W_1\setminus \overline{W_0}$ of the open cover $\{W_1,X\setminus \overline{W_0}\}$ of $X$ is contained in $C\subseteq O_C$, where $a_1$ vanishes, so $a_1{\mathrm{\mid }}_{W_1}$ and the zero section on $X\setminus \overline{W_0}$ agree there and glue to a section $t^{\mathrm{\prime }}\in \mathrm{\Gamma }(X,\mathcal{F})$ with $t^{\mathrm{\prime }}=a_1$ on $W_1$ and $t^{\mathrm{\prime }}=0$ on $X\setminus \overline{W_0}$ ([F4]). Then $\mathrm{Supp}(t^{\mathrm{\prime }})\subseteq \overline{W_0}$ is compact, and $\beta (t^{\mathrm{\prime }})=s^{\mathrm{\prime }\mathrm{\prime }}$: on $W_1$ one has $\beta (a_1)=s^{\mathrm{\prime }\mathrm{\prime }}$, while on $X\setminus \overline{W_0}$ both sides vanish since $S\subseteq W_0\subseteq \overline{W_0}$, and $W_1\cup (X\setminus \overline{W_0})=X$. Therefore $t^{\mathrm{\prime }}\in {\mathrm{\Gamma }}_c(X,\mathcal{F})$ with ${\mathrm{\Gamma }}_c(\beta )(t^{\mathrm{\prime }})=s^{\mathrm{\prime }\mathrm{\prime }}$.

Clause (5), quotient. Let ${\mathcal{F}}^{\mathrm{\prime }}$ and $\mathcal{F}$ be c-soft. Let $K\subseteq X$ be compact and let ${\sigma }^{\mathrm{\prime }\mathrm{\prime }}\in \mathrm{\Gamma }(K,i^{-1}{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$: by step 3.1 it is lifted by a section $u$ of $\mathcal{F}$ over an open neighbourhood of $K$, and clause (1) applied to the c-soft $\mathcal{F}$ with the compact set $K$ and the prescribed neighbourhood of $u$ produces $t\in {\mathrm{\Gamma }}_c(X,\mathcal{F})$ with $t=u$ near $K$; then $\beta (t)\in \mathrm{\Gamma }(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ restricts to ${\sigma }^{\mathrm{\prime }\mathrm{\prime }}$ over $K$, so every class over a compact set extends to a global section and ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ is c-soft.

Clause (3). Let $\mathcal{F}$ be c-soft and resolve it by injectives, $0\to \mathcal{F}\to I^0\to I^1\to \cdots$, using the functorial datum of [F1]; each $I^j$ is c-soft by clause (2) (Enough injective abelian sheaves supplies the resolution terms as injective objects). The syzygies $Z^0:=\mathcal{F}$, $Z^{j+1}:=\mathrm{im}(I^j\to I^{j+1})$ are c-soft: $Z^1=\mathrm{coker}(\mathcal{F}\to I^0)$ is a quotient of c-soft sheaves in $0\to \mathcal{F}\to I^0\to Z^1\to 0$, and inductively $Z^{j+1}$ is a quotient of c-soft sheaves in $0\to Z^j\to I^j\to Z^{j+1}\to 0$ (clause (5)); hence each short exact sequence $0\to Z^j\to I^j\to Z^{j+1}\to 0$ consists of c-soft sheaves, so by clause (5) the sequence $0\to {\mathrm{\Gamma }}_c(X,Z^j)\to {\mathrm{\Gamma }}_c(X,I^j)\to {\mathrm{\Gamma }}_c(X,Z^{j+1})\to 0$ is exact. Therefore the augmented complex $0\to {\mathrm{\Gamma }}_c(X,\mathcal{F})\to {\mathrm{\Gamma }}_c(X,I^0)\to {\mathrm{\Gamma }}_c(X,I^1)\to \cdots$ is exact, and $H_c^q(X,\mathcal{F})=H^q({\mathrm{\Gamma }}_c(X,I_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))=0$ for $q>0$ ([F1]). For $X$ compact the identification $H_c^q(X,\mathcal{F})=H^q(X,\mathcal{F})$ of [F1] gives the vanishing of sheaf cohomology.

Conclusion. Clause (1) is step 1.3, clause (2) is step 1.4, clause (3) is step 5.1, clause (4) is steps 2.2 and 2.3, and clause (5) is steps 3.1, 3.2 and 4.1. The Axiom of Choice supplies the coefficient basis of step 2.3 and the injective resolution datum and derived-functor statements of [F1]; all topological selections are finite.