Geometric Nakayama for finite-type sheaves

Let $U=\mathrm{Spec}A\subseteq W$ be an affine open neighbourhood of $x$, let $\mathfrak{p}\subseteq A$ be the prime corresponding to $x$, and write $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ with $M$ a finitely generated $A$-module by [F3]; then ${\mathcal{O}}_{X,x}\cong A_{\mathfrak{p}}$, $\kappa (x)\cong A_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{A}_{\mathfrak{p}}$, and by [F2] the fibre of the Statement is $\mathcal{F}(x)\cong M_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{M}_{\mathfrak{p}}$, while global sections satisfy $\mathrm{\Gamma }(U,\tilde{M})\cong M$ so the restrictions $s_i{\mathrm{\mid }}_U$ correspond to elements $g_i\in M$.

Nakayama at the point: for a finitely generated $A$-module $N$ and a prime $\mathfrak{p}$, one has $N_{\mathfrak{p}}=0$ if and only if $N_{\mathfrak{p}}=\mathfrak{p}{N}_{\mathfrak{p}}$. The implication from left to right is immediate; conversely, if $N_{\mathfrak{p}}=\mathfrak{p}{N}_{\mathfrak{p}}$, apply [F4] over the local ring $A_{\mathfrak{p}}$ with $I=\mathfrak{p}{A}_{\mathfrak{p}}$ to the finitely generated $A_{\mathfrak{p}}$-module $N_{\mathfrak{p}}$ to obtain $a\in \mathfrak{p}{A}_{\mathfrak{p}}$ with $(1-a)N_{\mathfrak{p}}=0$, and since $1-a\mathrm{\notin }\mathfrak{p}{A}_{\mathfrak{p}}$ is a unit of the local ring, $N_{\mathfrak{p}}=0$.

Vanishing spreads to a basic open: if $N$ is a finitely generated $A$-module with $N_{\mathfrak{p}}=0$, then $N_t=0$ for some $t\mathrm{\notin }\mathfrak{p}$. Choose finitely many generators $n_1,\dots ,n_r$ of $N$; since $n_i\mathrm{/}1=0$ in $N_{\mathfrak{p}}$, [F6] gives $t_i\mathrm{\notin }\mathfrak{p}$ with $t_i{n}_i=0$, and with $t=t_1\cdots t_r\mathrm{\notin }\mathfrak{p}$ one has $t{n}_i=0$ for all $i$, so $t$ annihilates every element of $N$ and therefore every element of $N_t$ is zero; only finitely many existential instantiations occur, so no arbitrary-index choice is used.

Let $\psi :A^n\to M$ be the $A$-linear map with $\psi (e_i)=g_i$, and let $C=\mathrm{coker}\psi =M\mathrm{/}\psi (A^n)$, a finitely generated $A$-module; by [F5] its localisation is $C_{\mathfrak{p}}\cong M_{\mathfrak{p}}\mathrm{/}{\psi }_{\mathfrak{p}}(A_{\mathfrak{p}}^n)$, and $C_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{C}_{\mathfrak{p}}\cong M_{\mathfrak{p}}\mathrm{/}({\psi }_{\mathfrak{p}}(A_{\mathfrak{p}}^n)+\mathfrak{p}{M}_{\mathfrak{p}})$. The images of $s_1,\dots ,s_n$ span $\mathcal{F}(x)$ over $\kappa (x)$ precisely when $M_{\mathfrak{p}}={\psi }_{\mathfrak{p}}(A_{\mathfrak{p}}^n)+\mathfrak{p}{M}_{\mathfrak{p}}$, that is, when $C_{\mathfrak{p}}=\mathfrak{p}{C}_{\mathfrak{p}}$; by step 1.2 applied to the finitely generated module $C$, this is equivalent to $C_{\mathfrak{p}}=0$.

Proof of claim 1: $\mathcal{F}(x)=0$ if and only if $M_{\mathfrak{p}}=\mathfrak{p}{M}_{\mathfrak{p}}$ by step 1.1, if and only if $M_{\mathfrak{p}}=0$ by step 1.2, if and only if $M_t=0$ for some $t\mathrm{\notin }\mathfrak{p}$ by step 1.3 together with the trivial implication; for such $t$ the open $D(t)$ contains $x$ and $\mathcal{F}{\mathrm{\mid }}_{D(t)}\cong \widetilde{M_t}=0$, so $\mathcal{F}$ vanishes on a neighbourhood of $x$, while conversely vanishing near $x$ gives ${\mathcal{F}}_x=0$ and hence $\mathcal{F}(x)=0$ by [F1].

Proof of claim 2: if the images of $s_1,\dots ,s_n$ span $\mathcal{F}(x)$, then $C_{\mathfrak{p}}=0$ by step 2.1, so step 1.3 applied to the finitely generated module $C$ gives $t\mathrm{\notin }\mathfrak{p}$ with $C_t=0$, that is, $M_t={\psi }_t(A_t^n)$; the restrictions of $g_1,\dots ,g_n$ to $D(t)$ therefore generate $\mathrm{\Gamma }(D(t),\tilde{M})\cong M_t$, so the induced morphism ${\mathcal{O}}_{D(t)}^n\to \mathcal{F}{\mathrm{\mid }}_{D(t)}$ is an epimorphism on the affine open $D(t)\subseteq U\subseteq W$ containing $x$.

The Axiom of Choice is inherited from [F2] and the associated-sheaf construction behind [F3]; the argument itself instantiates finitely many generators and denominators, so it makes no new arbitrary-index choice, and both claims are proved in steps 2.2 and 3.1.