Finite filtration of a generated subsheaf of the constant integer sheaf

Fix the data $U_1,\dots ,U_t$ and $s_1,\dots ,s_t$ of the statement. If $t=0$ then $K=⟨\mathrm{\varnothing }⟩=0$ by [F7] and we take $N:=0$, $F_0:=0$; this is the case covered by the boundary discussion below, so assume from now on that $t\ge 1$ and that each $U_i$ is a compact open subset of $X$ [F14]. [F7, F14, given]

For $i\le t$ and $k\in \mathbb{Z}$ put $A_{i,k}:=\{x\in U_i:(s_i{)}_x=k\cdot 1_x\}$, where $1_x$ denotes the germ at $x$ of the constant section $1$. The sheafification map identifies ${\mathbb{Z}}_X$ with the sheaf of locally constant $\mathbb{Z}$-valued functions [F1, F2, F4], and under this identification $s_i$ is a locally constant function whose value at $x$ is the integer $k$ with $(s_i{)}_x=k\cdot 1_x$; hence each $A_{i,k}$ is open in $U_i$ and therefore in $X$ [F16], and the nonempty $A_{i,k}$ cover $U_i$. Since $U_i$ is compact [F14], by [F15] finitely many of the open sets $A_{i,k}$ cover $U_i$, so only finitely many of the sets $A_{i,k}$ are nonempty. Each nonempty $A_{i,k}$ is a compact subset of $X$: if a family of open subsets of $X$ covers $A_{i,k}$, then adjoining one open set of $X$ whose trace on $U_i$ is the open subset $U_i\setminus A_{i,k}={\bigcup }_{l\mathrm{\ne }k}{A}_{i,l}$ of $U_i$ [F16] produces a family of open sets covering $U_i$, so by [F15] and compactness of $U_i$ finitely many of them cover $U_i$ and, discarding the possibly adjoined set, finitely many members of the original family cover $A_{i,k}$; hence $A_{i,k}$ is compact by [F15]. Moreover the sections $k$ over the sets $A_{i,k}$ generate the same subsheaf as the original sections: for $x\in U_i$ exactly one $k$ has $x\in A_{i,k}$, and there $(k\cdot 1_{A_{i,k}}{)}_x=k\cdot 1_x=(s_i{)}_x$ by [F17], so the subgroups generated by the germs of the two families at $x$ coincide, and [F7, F8] give $⟨s_1,\dots ,s_t⟩=⟨k\text{ over }{A}_{i,k},i\le t,k\in \mathbb{Z}⟩$. Relabel the nonempty fibres and their constant values as $(U_1,n_1),\dots ,(U_m,n_m)$ with $n_j\in \mathbb{Z}$, so that $U_j$ is compact open and $K$ is generated by the constant sections $n_j$ over $U_j$. [F1, F2, F4, F7, F8, F14, F15, F16, F17, given]

For every nonempty subset $I\subseteq \{1,\dots ,m\}$ put $U_I:={\bigcap }_{j\in I}{U}_j$ and let $d_I\ge 0$ be the unique natural number with $⟨n_j:j\in I⟩=d_I\mathbb{Z}$ in $(\mathbb{Z},+)$ [F9]. Each $U_I$ is compact open by the hypothesis on $X$ and induction on the number of elements of $I$ [F14], and the constant section $d_I$ over $U_I$ lies in the subsheaf generated by the sections $n_j$ over $U_j$: at a point $x\in U_I$ one has $d_I\in ⟨n_j:j\in I⟩\subseteq ⟨n_j:x\in U_j⟩$, so $d_I$ is a finite integer combination of the integers $n_j$ with $x\in U_j$ [F10] and therefore $(d_I\cdot 1{)}_x=d_I\cdot 1_x$ lies in the subgroup generated by the germs $(n_j\cdot 1{)}_x$ [F10]; by the stalk description of generated subsheaves [F7] the section $d_I$ over $U_I$ belongs to the generated subsheaf. Hence [F8] shows that the subsheaf generated by the enlarged family consisting of all the original sections $n_j$ over $U_j$ together with all the sections $d_I$ over $U_I$ is again $K$. For $x\in X$ write $J_x:=\{j:x\in U_j\}$; the values of the enlarged family at $x$ are exactly the integers $d_I$ with $\mathrm{\varnothing }\mathrm{\ne }I\subseteq J_x$, because $x\in U_I$ holds precisely when $I\subseteq J_x$. For such $I$ the inclusion $⟨n_j:j\in I⟩\subseteq ⟨n_j:j\in J_x⟩=d_{J_x}\mathbb{Z}$ shows that $d_I$ is a nonnegative multiple of $d_{J_x}$ whenever $d_{J_x}>0$, and $d_{J_x}$ is itself one of the values at $x$; if $J_x\mathrm{\ne }\mathrm{\varnothing }$ and not all integers $n_j$ with $j\in J_x$ are zero then $d_{J_x}>0$, and if all of them are zero then every $d_I$ with $I\subseteq J_x$ is $0$. [F7, F8, F9, F10, F14, given]

Let $W\subseteq X$ be open with inclusion $j_W:W\hookrightarrow X$. By [F6] a section of $j_{W!}{\mathbb{Z}}_W$ over an open $Y\subseteq X$ is a section of ${\mathbb{Z}}_W$ over $Y\cap W$ with support closed in $Y$; for $x\in W$ the open neighbourhoods of $x$ contained in $W$ are cofinal among all neighbourhoods of $x$, so the colimit description of stalks [F17] gives $(j_{W!}{\mathbb{Z}}_W{)}_x=({\mathbb{Z}}_W{)}_x=\mathbb{Z}$ generated by the germ of the constant section $1$, while for $x\mathrm{\notin }W$ every section of $j_{W!}{\mathbb{Z}}_W$ vanishes on a neighbourhood of $x$, so $(j_{W!}{\mathbb{Z}}_W{)}_x=0$. For $V\subseteq U$ the morphism $\iota :j_{V!}{\mathbb{Z}}_V\to j_{U!}{\mathbb{Z}}_U$ obtained from the adjunction [F5] and the identification $(j_{U!}{\mathbb{Z}}_U){\mathrm{\mid }}_V={\mathbb{Z}}_V$ is a monomorphism, and on stalks at $x\in V$ it is the identity $\mathbb{Z}\to \mathbb{Z}$ while at $x\mathrm{\notin }V$ both stalks are $0$.

Put $N:=\max \{d_I:\mathrm{\varnothing }\mathrm{\ne }I\subseteq \{1,\dots ,m\}\}$ and, for $0\le n\le N$, let $F_n\subseteq {\mathbb{Z}}_X$ be the subsheaf generated by the sections $d_I$ over $U_I$ with $d_I\le n$ (The subsheaf generated by a family of sections). Then $F_0=0$, because the sections with $d_I\le 0$ are the zero sections of the pieces with $d_I=0$; the $F_n$ form a chain $F_0\subseteq F_1\subseteq \cdots \subseteq F_N$ of subsheaves of ${\mathbb{Z}}_X$, since by [F7, F8] adding sections already contained in a generated subsheaf does not change it; and $F_N=K$ because every $d_I\le N$ and the pieces of [step 1.3] generate $K$.

Fix $x\in X$ and $0\le n\le N$. The indices of generators of $F_n$ active at $x$ are the nonempty $I\subseteq J_x$ with $d_I\le n$ [step 2.1]. If $J_x=\mathrm{\varnothing }$, or if $d_{J_x}=0$, every such generator has zero germ and $(F_n{)}_x=0$ [F7, step 1.3]. Suppose $d_{J_x}>0$. Every positive $d_I$ for $I\subseteq J_x$ is a positive multiple of $d_{J_x}$, since $d_I\mathbb{Z}$ contains $d_{J_x}\mathbb{Z}$ [F9, step 1.3]. If $d_{J_x}>n$, no positive $d_I\le n$ is active and $(F_n{)}_x=0$. If $0<d_{J_x}\le n$, the generator indexed by $I=J_x$ is active and all other active values are multiples of it, so $(F_n{)}_x=d_{J_x}\mathbb{Z}$. Thus $(F_n{)}_x=\{\begin{array}{ll}{d}_{J_x}\mathbb{Z},& 0<d_{J_x}\le n,\\ 0,& d_{J_x}=0\text{ or }{d}_{J_x}>n\text{ or }{J}_x=\mathrm{\varnothing }\mathrm{.}\end{array}$ Zero gcd values never suppress a positive generator.

For $1\le n\le N$ put $Q_n:=F_n\mathrm{/}{F}_{n-1}$ [F12, F13] and define $E_n:={\bigcup }_{d_I=n}{U}_I,W_{<n}:={\bigcup }_{0<d_I<n}{U}_I,V_n:=E_n\cap W_{<n}\mathrm{.}$ All unions are over the finite set of nonempty subsets of $\{1,\dots ,m\}$; empty unions are permitted. Each $E_n,W_{<n},V_n$ is compact open: finite unions of compact subsets are compact by [F15], and finite intersections of compact opens are compact by the hypothesis on $X$ [F14, given]. At $x\in E_n$ one has $J_x\mathrm{\ne }\mathrm{\varnothing }$ and $0<d_{J_x}\le n$ because the positive gcd $d_{J_x}$ divides a value $d_I=n$ active at $x$ [step 1.3]. If $x\in V_n$, some positive $d_I<n$ is active, so $0<d_{J_x}<n$; hence [step 3.1] gives $(Q_n{)}_x=0$. If $x\in E_n\setminus V_n$, then $d_{J_x}=n$: otherwise $0<d_{J_x}<n$ and $x\in U_{J_x}\subseteq W_{<n}$, a contradiction. Thus $(F_n{)}_x=n\mathbb{Z}$, $(F_{n-1}{)}_x=0$, and $(Q_n{)}_x=n\mathbb{Z}$. Conversely, if $x\mathrm{\notin }{E}_n$ then $d_{J_x}\mathrm{\ne }n$ whenever $J_x\mathrm{\ne }\mathrm{\varnothing }$, since $d_{J_x}=n$ would put $x$ in $U_{J_x}\subseteq E_n$; [step 3.1] therefore gives $(Q_n{)}_x=0$. The stalks of $Q_n$ are consequently $n\mathbb{Z}$ exactly on $E_n\setminus V_n$ and zero elsewhere.

The constant section $n$ of ${\mathbb{Z}}_X$ on $E_n$ belongs to $F_n(E_n)$: every point of $E_n$ belongs to some $U_I$ with $d_I=n$, and there its germ is the germ of the generator $d_I$ of $F_n$ [F7, step 2.1]. Let $\bar{n}\in Q_n(E_n)$ be its image. The group homomorphism $\mathbb{Z}\to Q_n(E_n)$ sending $1$ to $\bar{n}$ induces a morphism ${\mathbb{Z}}_{E_n}\to Q_n{\mathrm{\mid }}_{E_n}$ by the sheafification universal property [F3]: a locally constant integer-valued function acts locally as that integer times $\bar{n}$. Extension-by-zero adjunction [F5] yields ${\phi }_n:j_{E_n!}{\mathbb{Z}}_{E_n}⟶Q_n\mathrm{.}$ For $x\in E_n$ the stalk map sends $1$ to the class of $n$ in $(Q_n{)}_x$; outside $E_n$ its source stalk is zero [step 1.4]. This construction uses a single constant section, without assuming that membership in compact open subsets is locally constant.

Since $V_n\subseteq E_n$, step 1.4 gives the canonical monomorphism $j_{V_n!}{\mathbb{Z}}_{V_n}\to j_{E_n!}{\mathbb{Z}}_{E_n}$. Its composite with ${\phi }_n$ is zero: at $x\in V_n$ the target stalk $(Q_n{)}_x$ is zero by [step 4.1], and at every other point the source stalk is zero by [step 1.4]; stalkwise zero implies the morphism is zero [F11]. The resulting sequence $0⟶j_{V_n!}{\mathbb{Z}}_{V_n}⟶j_{E_n!}{\mathbb{Z}}_{E_n}\stackrel{{\phi }_n}{\to }{Q}_n⟶0$ is exact on stalks. Outside $E_n$ all three stalks vanish. On $V_n$ it is $0\to \mathbb{Z}\stackrel{\mathrm{id}}{\to }\mathbb{Z}\to 0$. On $E_n\setminus V_n$ it is $0\to 0\to \mathbb{Z}\stackrel{1\mapsto n}{\to }n\mathbb{Z}\to 0$, which is exact [steps 1.4, 4.1, 5.1]. Exactness of sheaves follows from [F11].

Take $U=E_n$ and $V=V_n$ in [steps 4.1 and 6.1]. They are compact open with $V\subseteq U$, and the chain $0=F_0\subseteq \cdots \subseteq F_N=K$ of [step 2.1] has precisely the quotient sequences required in the statement. If $N=0$ the chain is $F_0=K=0$ and there are no quotients to check. Only finitely many fibres of the original sections, subsets $I$, and finite unions and intersections of the resulting compact opens were used; no choice principle is used. ∎