The associated module sheaf exists

Let $B$ be a commutative ring, $N$ a $B$-module and $g_1,\dots ,g_r\in B$ with ${\sum }_{i=1}^r{b}_i{g}_i=1$ for some $b_i\in B$. If $n\in N$ is such that $g_i^{e_i}n=0$ for some $e_i\ge 0$ and every $i$, then $n=0$: choose $E\ge {\max }_i{e}_i$ and expand $n=1^{rE}n=({\sum }_{i=1}^r{b}_i{g}_i{)}^{rE}n={\sum }_{{\nu }_1+\cdots +{\nu }_r=rE}{c}_{\nu }{b}_1^{{\nu }_1}\cdots b_r^{{\nu }_r}{g}_1^{{\nu }_1}\cdots g_r^{{\nu }_r}n,$ where every multi-index $\nu$ of total degree $rE$ has some ${\nu }_i\ge E\ge e_i$, so each summand vanishes.

Define a presheaf $\mathcal{F}$ on $X$ by letting $\mathcal{F}(U)$, for an open $U\subseteq X$, be the set of all families $(s_D)$ indexed by the distinguished opens $D=D(g)$ with $D\subseteq U$, with $s_D\in M_g$, such that ${\rho }_{g{g}^{\mathrm{\prime }}}(s_{D(g)})=s_{D(g^{\mathrm{\prime }})}$ whenever $D(g^{\mathrm{\prime }})\subseteq D(g)\subseteq U$; restrictions are $(s_D)\mapsto (s_D{\mathrm{\mid }}_{D^{\mathrm{\prime }}\subseteq U^{\mathrm{\prime }}})$. This is a presheaf of abelian groups, and for every $f$ the map $M_f\to \mathcal{F}(D(f))$, $s\mapsto ({\rho }_{fg}(s){)}_{D(g)\subseteq D(f)}$, is a bijection with inverse $(s_D)\mapsto s_{D(f)}$: the family is compatible by [F6], the composite $(s_D)\mapsto s_{D(f)}\mapsto ({\rho }_{fg}(s_{D(f)}))$ is the identity by the defining compatibility, and $s=0$ is forced by the component $D(f)$.

With $B,N,g_i$ as in step 1.1, ${\sum }_i{b}_i{g}_i=1$ and $r\ge 1$, let elements $n_i\in N_{g_i}$ satisfy $n_i=n_j$ in $N_{g_i{g}_j}$ for all $i,j$. Then there is $n\in N$ with $n\mathrm{/}1=n_i$ in $N_{g_i}$ for every $i$. Indeed, $n_i=x_i^{\mathrm{\prime }}\mathrm{/}{g}_i^{k_i}$ for some $x_i^{\mathrm{\prime }}\in N$ and $k_i\ge 0$; set $k:={\max }_i{k}_i$ and $x_i:=g_i^{k-k_i}{x}_i^{\mathrm{\prime }}$, so that $n_i=x_i\mathrm{/}{g}_i^k$ for all $i$. Compatibility of $n_i$ and $n_j$ in $N_{g_i{g}_j}$ provides, for each pair $(i,j)$, an exponent $m_{ij}\ge 0$ with $(g_i{g}_j{)}^{m_{ij}}(g_j^k{x}_i-g_i^k{x}_j)=0$ by [F3]; put $m:={\max }_{i,j}{m}_{ij}$ and $K:=k+m$. Choose $a_i\in B$ with ${\sum }_i{a}_i{g}_i^K=1$: if $K=0$, take $a_1=1$ and $a_i=0$ for $i>1$; if $K\ge 1$, then $(g_1^K,\dots ,g_r^K)=B$, since in the expansion of $1=({\sum }_i{b}_i{g}_i{)}^{rK}$ every monomial of total degree $rK$ is divisible by some $g_i^K$. Set $X_i:=g_i^{K-k}{x}_i$, so that $n_i=X_i\mathrm{/}{g}_i^K$; multiplying each witness relation by $(g_i{g}_j{)}^{m-m_{ij}}$ gives the normalised relations $g_i^K{X}_j=g_j^K{X}_i$ in $N$ for all $i,j$. Then $n:={\sum }_i{a}_i{X}_i$ satisfies $g_i^{K}n={\sum }_j{a}_j{g}_i^K{X}_j={\sum }_j{a}_j{g}_j^K{X}_i=X_i$, hence $n\mathrm{/}1=X_i\mathrm{/}{g}_i^K=n_i$ in $N_{g_i}$ for every $i$.

On each $\mathcal{F}(U)$ define $a\cdot (s_D):=(a{\mathrm{\mid }}_D\cdot s_D)$ for $a\in {\mathcal{O}}_X(U)$, using ${\mathcal{O}}_X(U)\to {\mathcal{O}}_X(D(g))=A_g$ from [F4] and the $A_g$-module structure of $M_g$ from [F6]. This is a well-defined ${\mathcal{O}}_X(U)$-module structure: each $a{\mathrm{\mid }}_D\cdot s_D$ lies in $M_g$, the families are again compatible because ring restriction and module restriction commute, the componentwise module axioms hold, and the restriction maps of $\mathcal{F}$ are ${\mathcal{O}}_X(U)$-linear after scalar restriction. Thus $\mathcal{F}$ is a presheaf of ${\mathcal{O}}_X$-modules whose sections over distinguished opens are the modules $M_f$.

If $D(f)=\mathrm{\varnothing }$, then $A_f=0$ and $M_f=0$, so locality and gluing hold uniquely, including for any finite family of empty opens. Assume now that $D(f)={\bigcup }_{i=1}^{r}D(f_i)$ is a finite cover of a nonempty distinguished open, so $r\ge 1$. Then the cover condition is ${\sum }_i{c}_i{f}_i=f^m$ for some $c_i\in A$, $m\ge 0$, equivalently ${\sum }_i{b}_i{f}_i=1$ in $A_f$; set $B:=A_f$, $N:=M_f$ and $g_i:=f_i$. (a) If $s\in M_f$ has ${\rho }_{f{f}_i}(s)=0$ in every $M_{f_i}$, then $s=0$: by [F3] some $f_i^{e_i}s=0$ in $N$, so step 1.1 applies. (b) If $s_i\in M_{f_i}=N_{g_i}$ satisfy $s_i=s_j$ in $M_{f_i{f}_j}=N_{g_i{g}_j}$ for all $i,j$, then there is $s\in M_f$ with ${\rho }_{f{f}_i}(s)=s_i$ for all $i$: step 2.1 applied to $N$ and $g_i$ produces such an $s$, and the identifications $N_{g_i}=M_{f_i}$ and $N_{g_i{g}_j}=M_{f_i{f}_j}$ are the canonical ones because $D(f_i)\subseteq D(f)$ forces $f_i\in \sqrt{(f)}$.

Now let $D(f)={\bigcup }_{i\in I}D(f_i)$ be an arbitrary cover by distinguished opens. Choose a finite subcover indexed by $J\subseteq I$ using [F5]. If $J=\mathrm{\varnothing }$, then $D(f)=\mathrm{\varnothing }$, so $M_f=0$ and every $M_{f_i}=0$; locality and gluing hold uniquely. Assume $J\mathrm{\ne }\mathrm{\varnothing }$. (a) If $s\in M_f$ has ${\rho }_{f{f}_i}(s)=0$ for all $i\in I$, then in particular it vanishes on the finite subcover indexed by $J$, so step 3.1(a) gives $s=0$. (b) If $s_i\in M_{f_i}$ satisfy $s_i=s_j$ in $M_{f_i{f}_j}$ for all $i,j\in I$, then step 3.1(b) glues the family on the finite subcover to $s\in M_f$ with ${\rho }_{f{f}_{i_j}}(s)=s_{i_j}$ for $j\in J$. For any $i\in I$ with $D(f_i)\mathrm{\ne }\mathrm{\varnothing }$, the distinguished opens $D(f_i{f}_{i_j})$, $j\in J$, cover $D(f_i)$ because $D(f_i)\subseteq D(f)={\bigcup }_{j\in J}D(f_{i_j})$. On each such overlap, compatibility gives the same image for $s_i$ and $s_{i_j}$, while the restriction of $s$ there equals the restriction of $s_{i_j}$ by construction. Thus $s_i-{\rho }_{f{f}_i}(s)$ restricts to zero on the finite cover $D(f_i)={\bigcup }_{j\in J}D(f_i{f}_{i_j})$, and step 3.1(a) gives $s_i={\rho }_{f{f}_i}(s)$. If $D(f_i)=\mathrm{\varnothing }$, then $M_{f_i}=0$ and the equality also holds. Uniqueness of $s$ follows from part (a); so the distinguished-open data satisfy the sheaf conditions on the basis, proving claim 1.

$\mathcal{F}$ satisfies locality on every open cover. Let $U={\bigcup }_{\alpha }{U}_{\alpha }$ and let $s=(s_D)\in \mathcal{F}(U)$ restrict to $0$ in every $\mathcal{F}(U_{\alpha })$. Fix a distinguished open $D\subseteq U$; the $D\cap U_{\alpha }$ that are nonempty cover $D$ and each contains a distinguished open $D^{\mathrm{\prime }}\subseteq D\cap U_{\alpha }\subseteq D$ by [F2]. The restriction of $s_D$ to $D^{\mathrm{\prime }}$ is the $D^{\mathrm{\prime }}$-component of $s{\mathrm{\mid }}_{U_{\alpha }}$, hence $0$; therefore $s_D$ vanishes in $M_{g^{\mathrm{\prime }}}$ for every such $D^{\mathrm{\prime }}$, and these $D^{\mathrm{\prime }}$ form a cover of $D$ by distinguished opens, so step 4.1(a) gives $s_D=0$; as $D$ was arbitrary, $s=0$.

$\mathcal{F}$ satisfies gluing. Let $U={\bigcup }_{\alpha }{U}_{\alpha }$ and let $s_{\alpha }\in \mathcal{F}(U_{\alpha })$ be compatible on overlaps $U_{\alpha }\cap U_{\beta }$. For a distinguished open $D\subseteq U$ and indices $\alpha ,\beta$ with $D\cap U_{\alpha }$, $D\cap U_{\beta }$ nonempty, any two distinguished opens $D^{\mathrm{\prime }},D^{\mathrm{\prime }\mathrm{\prime }}\subseteq D\cap U_{\alpha }\cap U_{\beta }$ satisfy $s_{\alpha ,D^{\mathrm{\prime }}}=s_{\beta ,D^{\mathrm{\prime }\mathrm{\prime }}}$ in the common module $M_g$ for $D(g)\subseteq D^{\mathrm{\prime }}\cap D^{\mathrm{\prime }\mathrm{\prime }}$, by compatibility of $s_{\alpha },s_{\beta }$ on $U_{\alpha }\cap U_{\beta }$; hence the elements $s_{\alpha ,D^{\mathrm{\prime }}}\in M_{g^{\mathrm{\prime }}}$ glue over the basis cover of $D$ formed by all distinguished $D^{\mathrm{\prime }}\subseteq D\cap U_{\alpha }$, $\alpha$ varying, to a unique $s_D\in M_D$ by step 4.1(b), and this $s_D$ is independent of the choices by step 4.1(a). The resulting family $(s_D{)}_{D\subseteq U}$ is compatible: for $D^{\mathrm{\prime }\mathrm{\prime }}\subseteq D^{\mathrm{\prime }}\subseteq U$ and a common refinement $D^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\subseteq D^{\mathrm{\prime }\mathrm{\prime }}$ of the relevant covers, both restrictions agree with $s_{\gamma ,D^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}}$ for suitable $\gamma$, so step 4.1(a) applied on $D^{\mathrm{\prime }\mathrm{\prime }}$ gives compatibility. It restricts to each $s_{\alpha }$ componentwise, so with step 5.1 and [F7], $\mathcal{F}$ is a sheaf; it is a sheaf of ${\mathcal{O}}_X$-modules by step 2.2 and the argument of step 5.1 applied to $a\cdot s-s^{\mathrm{\prime }}$ for the componentwise action.

The sheaf $\mathcal{F}$ of steps 1.2–5.1 has $\mathcal{F}(D(f))=M_f$ for every $f$ by step 1.2, so it is the required extension $\tilde{M}$ and proves claim 2, including $\tilde{M}(D(0))=M_0=0$ because $f=0$ is nilpotent and $M_0=0$; the empty open $\mathrm{\varnothing }=D(0)$ carries the empty family, so $\tilde{M}(\mathrm{\varnothing })=0$, while $\tilde{M}(X)=M_1=M$; this is claim 3.

Uniqueness: let $\mathcal{G}$ be a sheaf of ${\mathcal{O}}_X$-modules with isomorphisms ${\psi }_f:\mathcal{G}(D(f))\to M_f$ compatible with the restrictions $\rho$, and let $\mathcal{F}$ be as above. For an open $U$ and $t\in \mathcal{G}(U)$ the family $({\psi }_g(t{\mathrm{\mid }}_{D(g)}){)}_{D(g)\subseteq U}$ is an element ${\mathrm{\Phi }}_U(t)\in \mathcal{F}(U)$; the maps ${\mathrm{\Phi }}_U$ are compatible with restrictions, hence form a morphism $\mathrm{\Phi }:\mathcal{G}\to \mathcal{F}$ of sheaves of ${\mathcal{O}}_X$-modules, and on $D(f)$ the map ${\mathrm{\Phi }}_{D(f)}$ is the given isomorphism ${\psi }_f$ by step 1.2. The map ${\mathrm{\Phi }}_U$ is injective: if ${\mathrm{\Phi }}_U(t)=0$ then $t{\mathrm{\mid }}_{D(g)}=0$ for all distinguished $D(g)\subseteq U$, and these cover $U$, so $t=0$ by locality of the sheaf $\mathcal{G}$. It is surjective: given a family $(s_D)$, the components $s_D\in M_D={\psi }_D(\mathcal{G}(D))$ come from unique sections $t_D\in \mathcal{G}(D)$ which are compatible on overlaps because the $\psi$ are compatible with restrictions, so they glue by the sheaf property of $\mathcal{G}$ to $t\in \mathcal{G}(U)$, and ${\mathrm{\Phi }}_U(t)$ and $(s_D)$ have the same components. Thus each ${\mathrm{\Phi }}_U$ is a bijection, the extension is unique up to unique isomorphism, and the theorem is proved; the Axiom of Choice [F1] enters only through the published suppliers [F4] and [F5], which are stated under it, and every selection made in the proof is the extraction of finitely many witnesses (a finite subcover, finitely many exponents and coefficients) rather than an arbitrary-index choice.