The subsheaf generated by a family of sections

Define $H(W):=\{s\in \mathcal{F}(W):s_x\in G_x\text{ for every }x\in W\}$ for every open $W\subseteq X$, the prescription of assertion 1. Each $H(W)$ is a subgroup: germs preserve addition and zero (compute these on representatives), and each $G_x$ is a subgroup, so zero belongs to $H(W)$ and $s-t$ belongs whenever $s,t$ do. This is a subpresheaf of abelian groups of $\mathcal{F}$: for $W^{\mathrm{\prime }}\subseteq W$ and $s\in H(W)$ one has $(s{\mathrm{\mid }}_{W^{\mathrm{\prime }}}{)}_x=s_x$ for $x\in W^{\mathrm{\prime }}$ [F4], so $s{\mathrm{\mid }}_{W^{\mathrm{\prime }}}\in H(W^{\mathrm{\prime }})$. [F4, given]

Let $\mathcal{H}\subseteq \mathcal{F}$ be a subsheaf and let $s\in \mathcal{F}(W)$ be a section over an open $W$ such that for every $x\in W$ the germ $s_x$ lies in the image of the germ map $\mathcal{H}(W_x)\to {\mathcal{F}}_x$ for some open neighbourhood $W_x\subseteq W$ of $x$. Consider the family $\mathcal{W}$ of all pairs $(V,h)$ with $V\subseteq W$ open and $h\in \mathcal{H}(V)\subseteq \mathcal{F}(V)$ such that $h=s{\mathrm{\mid }}_V$; this is a family of sections indexed by a set and defined by a condition, and for every $x\in W$ there is a member whose first component contains $x$: by hypothesis there are $h\in \mathcal{H}(W_x)$ and a smaller open neighbourhood $V\subseteq W_x$ of $x$ on which $h$ and $s$ agree [F3], and the restriction $h{\mathrm{\mid }}_V\in \mathcal{H}(V)$ satisfies $h{\mathrm{\mid }}_V=s{\mathrm{\mid }}_V$, so $(V,h{\mathrm{\mid }}_V)\in \mathcal{W}$. Any two members $(V,h)$ and $(V^{\mathrm{\prime }},h^{\mathrm{\prime }})$ of $\mathcal{W}$ are compatible on $V\cap V^{\mathrm{\prime }}$, because both agree there with $s$. By [F2] the family $(h{)}_{(V,h)\in \mathcal{W}}$ glues to a section $h_0\in \mathcal{H}(W)\subseteq \mathcal{F}(W)$, and $h_0{\mathrm{\mid }}_V=h=s{\mathrm{\mid }}_V$ for every member $(V,h)$; since the first components of the members cover $W$, locality in $\mathcal{F}$ [F2] gives $h_0=s$, hence $s\in \mathcal{H}(W)$. [F2, F3]

Let $W={\bigcup }_{i\in I}{W}_i$ be an open cover and let $s_i\in H(W_i)$ be compatible on the overlaps in the sense of [F2]. Each $s_i$ is a section of $\mathcal{F}$ over $W_i$ and the $s_i$ are compatible sections of $\mathcal{F}$, so by [F2] there is $s\in \mathcal{F}(W)$ with $s{\mathrm{\mid }}_{W_i}=s_i$ for all $i$. For $x\in W$ pick $i$ with $x\in W_i$; then $s_x=(s{\mathrm{\mid }}_{W_i}{)}_x=(s_i{)}_x\in G_x$ by [F4] and the definition of $H(W_i)$, hence $s\in H(W)$. Locality of $H$ is inherited from $\mathcal{F}$, so $H$ satisfies the sheaf condition and is a subsheaf of $\mathcal{F}$ by [F1]; this proves assertion 1. [F1, F2, F4, step 1.1, given]

Let $\alpha \in A$ and $x\in U_{\alpha }$. The germ $(s_{\alpha }{)}_x$ is one of the generators of the subgroup $G_x$, so $(s_{\alpha }{)}_x\in G_x$ by [F5]; hence $s_{\alpha }\in H(U_{\alpha })$ by the definition of $H$, which proves assertion 2.

Fix $x\in X$. An element of $H_x$ is the class of a section $s\in H(W)$ over an open neighbourhood $W$ of $x$ [F3], and its image under the germ map into ${\mathcal{F}}_x$ is $s_x$, which lies in $G_x$ by the definition of $H(W)$; two sections of $H$ over neighbourhoods of $x$ with the same germ in ${\mathcal{F}}_x$ agree on a smaller neighbourhood [F3] and therefore define the same element of $H_x$, so this is well defined and $H_x\subseteq G_x$ inside ${\mathcal{F}}_x$. Conversely let $g\in G_x$. The finite integer combinations of the generating germs form a subgroup containing those germs, and every subgroup containing them contains all such combinations. Thus [F5] identifies this subgroup with $G_x$, and $g$ is a finite integer combination $g={\sum }_j{n}_j(s_{{\alpha }_j}{)}_x$ of germs of generators with $x\in U_{{\alpha }_j}$, $1\le j\le m$. Put $W:={\bigcap }_j{U}_{{\alpha }_j}$ (with $W=X$ for the empty combination), an open neighbourhood of $x$, and $t:={\sum }_j{n}_j{s}_{{\alpha }_j}{\mathrm{\mid }}_W\in \mathcal{F}(W)$. The stalk is the filtered colimit of the section groups over neighbourhoods of $x$ [F3], so the germ of an integer combination of sections over a common open set is the corresponding integer combination of germs, and $t_x=g$. For every $y\in W$ one has $y\in U_{{\alpha }_j}$ for all $j$, so $t_y={\sum }_j{n}_j(s_{{\alpha }_j}{)}_y\in G_y$; hence $t\in H(W)$ and $g=t_x\in H_x$. Therefore $H_x=G_x$ inside ${\mathcal{F}}_x$, which proves assertion 3.

Let $\mathcal{H}\subseteq \mathcal{F}$ be a subsheaf of abelian groups with $s_{\alpha }\in \mathcal{H}(U_{\alpha })$ for every $\alpha \in A$. For $x\in X$ and $\alpha$ with $x\in U_{\alpha }$ the germ $(s_{\alpha }{)}_x\in {\mathcal{F}}_x$ is the image of the section $s_{\alpha }\in \mathcal{H}(U_{\alpha })$ under the germ map [F4], so $G_x\subseteq {\mathcal{H}}_x$ because ${\mathcal{H}}_x$ is a subgroup of ${\mathcal{F}}_x$ containing all these germs [F5]. Now let $s\in H(W)$. For every $x\in W$ the germ $s_x$ lies in $G_x\subseteq {\mathcal{H}}_x$, so the hypothesis of [step 1.2] holds for $\mathcal{H}$ and $s$, and [step 1.2] gives $s\in \mathcal{H}(W)$. Hence $H\subseteq \mathcal{H}$ as subsheaves of $\mathcal{F}$ by [F1], which is the first half of assertion 4.

Conversely let $\mathcal{H}\subseteq \mathcal{F}$ be a subsheaf with ${\mathcal{H}}_x=G_x$ for every $x\in X$. If $s\in \mathcal{H}(W)$ for an open $W$, then $s_x\in {\mathcal{H}}_x=G_x$ for every $x\in W$, so $s\in H(W)$ and $\mathcal{H}\subseteq H$ by [F1]. For the reverse inclusion, if $s\in H(W)$, then every germ $s_x$ belongs to $G_x={\mathcal{H}}_x$, so [step 1.2] gives $s\in \mathcal{H}(W)$. Hence $H\subseteq \mathcal{H}$, and $\mathcal{H}=H$, proving the second half of assertion 4 even when $\mathcal{H}$ is initially only a subsheaf of sets.

Let $(t_{\beta }\in \mathcal{F}(V_{\beta }){)}_{\beta \in B}$ be a further family with $t_{\beta }\in H(V_{\beta })$ for every $\beta$, and let $G_x^{\mathrm{\prime }}\le {\mathcal{F}}_x$ be the subgroup generated by the germs at $x$ of the combined family. For $x\in X$ one has $G_x\subseteq G_x^{\mathrm{\prime }}$, and every new generator $(t_{\beta }{)}_x$ with $x\in V_{\beta }$ lies in $G_x$ because $t_{\beta }\in H(V_{\beta })$; hence $G_x^{\mathrm{\prime }}=G_x$ for every $x$. The subsheaf $\mathcal{H}:=H$ satisfies ${\mathcal{H}}_x=G_x=G_x^{\mathrm{\prime }}$ for every $x$ by [step 2.3], so [step 2.5] applied to the combined family gives that the subsheaf generated by the combined family equals $H$, which is assertion 5. ∎