Extension-by-zero generators detect sheaf-cohomology vanishing

Let $S$ be the disjoint union of the sets $\mathcal{F}(U)$ over the compact open subsets $U\subseteq X$, a set, and for a finite subset $A\subseteq S$ let ${\mathcal{F}}_A\subseteq \mathcal{F}$ be the subsheaf generated by the sections in $A$ over their compact open domains. If $A\subseteq A^{\mathrm{\prime }}$ then ${\mathcal{F}}_A\subseteq {\mathcal{F}}_{A^{\mathrm{\prime }}}$ by the minimality half of [F4], so the assignment $A\mapsto {\mathcal{F}}_A$ is a diagram indexed by the directed partially ordered set of finite subsets of $S$ (Filtered categories and filtered colimits), and the inclusions form a cocone on it with vertex $\mathcal{F}$. For every open $W\subseteq X$ the canonical map ${\mathrm{colim}}_A{\mathcal{F}}_A(W)\to \mathcal{F}(W)$ is an isomorphism: it is injective because every ${\mathcal{F}}_A(W)$ is a subset of $\mathcal{F}(W)$ and an element of the filtered colimit of these subsets is zero in $\mathcal{F}(W)$ only if it is zero in the stage in which it appears, and it is surjective because for $s\in \mathcal{F}(W)$ the open set $W$ is compact by [F8], so $s$ is one of the sections in $S$ and $s\in {\mathcal{F}}_{\{s\}}(W)$ by [F4]. Hence the canonical morphism of sheaves ${\mathrm{colim}}_A{\mathcal{F}}_A\to \mathcal{F}$ is an isomorphism of presheaves, therefore of sheaves, and [F1] applies to the diagram $A\mapsto {\mathcal{F}}_A$ with colimit $\mathcal{F}$. [F1, F4, F8, given]

Fix a finite $A\subseteq S$ with $n$ elements and suppose $n\ge 1$; write $A=A^{\mathrm{\prime }}\cup \{s_0\}$ with $A^{\mathrm{\prime }}$ having $n-1$ elements and let $\mathcal{G}:={\mathcal{F}}_{A^{\mathrm{\prime }}}\subseteq {\mathcal{F}}_A$ be the subsheaf generated by the sections over the domains of $A^{\mathrm{\prime }}$. Let $\mathcal{Q}:={\mathcal{F}}_A\mathrm{/}\mathcal{G}$ be the cokernel sheaf of the inclusion, so that by [F9] there is a short exact sequence $0\to \mathcal{G}\to {\mathcal{F}}_A\to \mathcal{Q}\to 0$; by [F5] its long exact sequence contains the exact portion $H^q(X,\mathcal{G})\to H^q(X,{\mathcal{F}}_A)\to H^q(X,\mathcal{Q})$ for every $q$. If $H^q(X,\mathcal{G})=0$ and $H^q(X,\mathcal{Q})=0$ for all $q>d$, then $H^q(X,{\mathcal{F}}_A)=0$ for all $q>d$: the image of the first map is $0$, so the second map is injective and its target is $0$. Thus, by induction on $n$, the vanishing for all finitely generated ${\mathcal{F}}_A$ follows once it is proved for subsheaves generated by a single section over a compact open subset; the case $n=0$ is ${\mathcal{F}}_{\mathrm{\varnothing }}=0$ by [F4]. [F4, F5, F9, given]

By [step 1.1] and the second assertion of the filtered-colimit lemma [F2], $H^q(X,\mathcal{F})\cong {\mathrm{colim}}_A{H}^q(X,{\mathcal{F}}_A)$ for every $q\ge 0$. Consequently it suffices to prove $H^q(X,{\mathcal{F}}_A)=0$ for every finite $A\subseteq S$ and every $q>d$: then the colimit on the right is a filtered colimit of zero groups, as each $H^q(X,{\mathcal{F}}_A)$ vanishes and the transition maps are the induced maps on cohomology, and the group $H^q(X,\mathcal{F})$ is therefore zero. [F2, step 1.1, given]

It remains to treat a subsheaf of $\mathcal{F}$ generated by one section $s\in \mathcal{F}(U)$ over a compact open $U\subseteq X$, which by [step 1.2] is the missing input for the induction; note that $\mathcal{Q}$ in [step 1.2] is of this form, being generated by the image of $s_0$. Consider the morphism $\psi :j_{U!}{\mathbb{Z}}_U⟶\mathcal{F}$ corresponding under [F6] and [F7] to the element $s\in \mathcal{F}(U)={\mathrm{Hom}}_U({\mathbb{Z}}_U,\mathcal{F}{\mathrm{\mid }}_U)$; its adjoint transpose carries the canonical section $1\in {\mathbb{Z}}_U(U)$ to $s$, and its image is a subsheaf of $\mathcal{F}$ containing $s$, so by the minimality half of [F4] the image contains the subsheaf generated by $s$, which is ${\mathcal{F}}_A$. Conversely, each image germ at $x\in U$ is an integer multiple of $s_x$, while at $x\mathrm{\notin }U$ the source stalk vanishes; the stalk description [F4] therefore places every image section in ${\mathcal{F}}_A$. Thus the image is exactly ${\mathcal{F}}_A$, and the induced map $j_{U!}{\mathbb{Z}}_U\to {\mathcal{F}}_A$ is an epimorphism, $K:=\ker \psi$ is a subsheaf of $j_{U!}{\mathbb{Z}}_U$ (Subsheaves), and [F9] gives a short exact sequence $0\to K\to j_{U!}{\mathbb{Z}}_U\to {\mathcal{F}}_A\to 0.$ [F4, F6, F7, F9, step 1.2, given]

The identification of the constant sheaves ${\mathbb{Z}}_U={\mathbb{Z}}_X{\mathrm{\mid }}_U$ [F7] makes the adjoint of the identity a monomorphism $j_{U!}{\mathbb{Z}}_U\rightarrowtail {\mathbb{Z}}_X$ whose image is the subsheaf of sections of ${\mathbb{Z}}_X$ supported in $U$, so the subsheaf $K$ of [step 2.2] is a subsheaf of ${\mathbb{Z}}_X$, and the same holds for every subsheaf of $K$.

Apply the construction of [step 1.1] with $\mathcal{F}$ replaced by $K$: the disjoint union $S_K$ of the sets $K(V)$ over the compact open subsets $V\subseteq X$ is a set, and for finite $B\subseteq S_K$ the subsheaf $K_B\subseteq K$ generated by the sections in $B$ is generated by finitely many sections over compact opens and is a subsheaf of ${\mathbb{Z}}_X$ by [step 3.1]; the same argument as in [step 1.1] shows that the canonical morphism ${\mathrm{colim}}_B{K}_B\to K$ is an isomorphism of sheaves, since for every open $W$ the map ${\mathrm{colim}}_B{K}_B(W)\to K(W)$ is injective by the subset inclusions and surjective because $W$ is compact [F8] and every section of $K(W)$ lies in the subsheaf it generates [F4]. By [F1] the sheaves $K_B$ form a small filtered diagram with colimit $K$.

Let $B\subseteq S_K$ be finite and let $K_B\subseteq {\mathbb{Z}}_X$ be as in [step 4.1]. By [F3] there are $N\ge 0$ and a chain of subsheaves $0=F_0\subseteq F_1\subseteq \cdots \subseteq F_N=K_B$ of ${\mathbb{Z}}_X$ such that for each $1\le n\le N$ there are compact open subsets $V\subseteq W\subseteq X$ with a short exact sequence $0\to j_{V!}{\mathbb{Z}}_V\to j_{W!}{\mathbb{Z}}_W\to F_n\mathrm{/}{F}_{n-1}\to 0.$ We prove $H^p(X,K_B)=0$ for every $p>d$ by induction on $n$. For $n=0$ this is $H^p(X,0)=0$. For the induction step, [F9] and [F5] give the exact portion $H^p(X,F_{n-1})\to H^p(X,F_n)\to H^p(X,F_n\mathrm{/}{F}_{n-1})$ and the last group vanishes for $p>d$: applying [F5] to the short exact sequence of the previous display gives the exact portion $H^p(X,j_{W!}{\mathbb{Z}}_W)\to H^p(X,F_n\mathrm{/}{F}_{n-1})\to H^{p+1}(X,j_{V!}{\mathbb{Z}}_V)$, in which the outer groups vanish for $p>d$ by the hypothesis of the statement, so $H^p(X,F_n\mathrm{/}{F}_{n-1})=0$ for $p>d$. With $H^p(X,F_{n-1})=0$ for $p>d$ as the induction hypothesis, the first exact portion shows that $H^p(X,F_n)=0$ for $p>d$. Hence $H^p(X,K_B)=0$ for all $p>d$ and all finite $B$. [F3, F5, F9, given]

By [step 4.1] and the second assertion of the filtered-colimit lemma [F2], applied to the diagram $B\mapsto K_B$ with colimit $K$, $H^p(X,K)\cong {\mathrm{colim}}_B{H}^p(X,K_B)$ for every $p\ge 0$; by [step 5.1] every group in this colimit is zero for $p>d$, so $H^p(X,K)=0$ for every $p>d$.

In the short exact sequence $0\to K\to j_{U!}{\mathbb{Z}}_U\to {\mathcal{F}}_A\to 0$ of [step 2.2], the long exact sequence [F5] contains the exact portion $H^p(X,j_{U!}{\mathbb{Z}}_U)\to H^p(X,{\mathcal{F}}_A)\to H^{p+1}(X,K)$ for every $p$. For $p>d$ the first group vanishes by the hypothesis of the statement, since $U$ is open, and the last group vanishes by [step 6.1] because $p+1>d$; hence $H^p(X,{\mathcal{F}}_A)=0$ for every $p>d$. [F5, step 6.1, step 2.2, given]

Combining the steps: for every finite $A\subseteq S$ the sheaf ${\mathcal{F}}_A$ is generated by finitely many sections over compact opens, the induction of [step 1.2] reduces its vanishing to the case of one generator, which is [step 7.1], so $H^q(X,{\mathcal{F}}_A)=0$ for every $q>d$ and every finite $A$; then [step 2.1] identifies $H^q(X,\mathcal{F})$ with the filtered colimit of these zero groups for every $q>d$, which is zero. The proof is complete. The Axiom of Choice is used exactly through the vanishing theorem [F2] and the exactness statements [F1] that rest on derived global sections and injective resolutions, whose construction needs the Axiom of Dependent Choice [F10] supplied by AC. ∎