A coherent closed-point skyscraper

Setup: since $A$ is Noetherian and $X=\mathrm{Spec}A$, the scheme $X$ is locally Noetherian by [F4]; the maximal ideal $\mathfrak{m}$ is prime by [F3], so the residue field $k=A\mathrm{/}\mathfrak{m}$ is a field and $M=A\mathrm{/}\mathfrak{m}$ is generated by the class of $1$, hence is of finite type; therefore $\mathcal{F}=\tilde{M}$ is quasi-coherent by [F1] and coherent by [F4].

Stalk at the closed point: by [F1] and [F2] one has ${\mathcal{F}}_{\mathfrak{m}}\cong M_{\mathfrak{m}}\cong A_{\mathfrak{m}}\mathrm{/}\mathfrak{m}{A}_{\mathfrak{m}}=\kappa (\mathfrak{m})\cong \mathrm{Frac}(A\mathrm{/}\mathfrak{m})=k=A\mathrm{/}\mathfrak{m}$.

Stalks away from the closed point: let $\mathfrak{p}\mathrm{\ne }\mathfrak{m}$ be a prime; by [F3] $\mathfrak{m}\subseteq ̸\mathfrak{p}$, so [F2] gives $\mathfrak{m}{A}_{\mathfrak{p}}=A_{\mathfrak{p}}$ and hence ${\mathcal{F}}_{\mathfrak{p}}\cong A_{\mathfrak{p}}\mathrm{/}\mathfrak{m}{A}_{\mathfrak{p}}=0$.

Values of $\mathcal{F}$ on distinguished opens: for $f\in A$ one has $\mathcal{F}(D(f))\cong (A\mathrm{/}\mathfrak{m}{)}_f$ by [F1]. If $f\in \mathfrak{m}$ then the image of $f$ in the field $k$ is zero, so $(A\mathrm{/}\mathfrak{m}{)}_f=0$ by [F2]; if $f\mathrm{\notin }\mathfrak{m}$ then the image of $f$ in $k$ is a nonzero element of a field, hence a unit, and $(A\mathrm{/}\mathfrak{m}{)}_f\cong k$ by [F2]. Thus $\mathcal{F}(D(f))\cong k$ exactly when $\mathfrak{m}\in D(f)$, and is $0$ otherwise.

Fibre at the closed point: $\mathcal{F}(\mathfrak{m})\cong {\mathcal{F}}_{\mathfrak{m}}\mathrm{/}{\mathfrak{m}}_{\mathfrak{m}}{\mathcal{F}}_{\mathfrak{m}}$ by [F5], and by step 1.2 the stalk is the field $k=A\mathrm{/}\mathfrak{m}$ on which the maximal ideal ${\mathfrak{m}}_{\mathfrak{m}}=\mathfrak{m}{A}_{\mathfrak{m}}$ acts by zero, so the fibre is $\mathcal{F}(\mathfrak{m})\cong A\mathrm{/}\mathfrak{m}$, a one-dimensional vector space over $\kappa (\mathfrak{m})$.

Values of $\mathcal{F}$ on arbitrary opens: let $U\subseteq X$ be open; the distinguished opens inside $U$ form an open cover of $U$, and by [F1] the restriction $\mathcal{F}(D(g))\to \mathcal{F}(D(h))$ for $D(h)\subseteq D(g)\subseteq U$ is the canonical localisation map $M_g\to M_h$, which under the identifications of step 1.4 is the identity of $k$ whenever $g,h\mathrm{\notin }\mathfrak{m}$, and is the map $k\to 0$ or $0\to 0$ otherwise. If $\mathfrak{m}\in U$, choose $D(f_0)\subseteq U$ with $f_0\mathrm{\notin }\mathfrak{m}$, which exists because the distinguished opens form a basis of the topology; for $c\in k$ the elements $c\in \mathcal{F}(D(g))$ for $g\mathrm{\notin }\mathfrak{m}$ and $0\in \mathcal{F}(D(g))$ for $g\in \mathfrak{m}$ form a compatible family, since $D(g)\cap D(h)=D(gh)$ and $gh\mathrm{\notin }\mathfrak{m}$ holds exactly when both $g,h\mathrm{\notin }\mathfrak{m}$ ($\mathfrak{m}$ is prime by [F3]), so they glue to a unique $\phi (c)\in \mathcal{F}(U)$; the map $\phi$ is injective because $\phi (c){\mathrm{\mid }}_{D(f_0)}=c$, and it is surjective because for $s\in \mathcal{F}(U)$ with $c=s{\mathrm{\mid }}_{D(f_0)}$ and any distinguished $D(g)\subseteq U$ the two sections $s{\mathrm{\mid }}_{D(g)}$ and $\phi (c){\mathrm{\mid }}_{D(g)}$ of $\mathcal{F}(D(g))$ agree after restriction to $D(f_{0}g)\subseteq D(g)$: if $g\mathrm{\notin }\mathfrak{m}$ then both equal $c$ in $\mathcal{F}(D(f_{0}g))\cong k$ by the identity computed above, and if $g\in \mathfrak{m}$ then $f_{0}g\in \mathfrak{m}$ as well and both are $0$; in the case $g\mathrm{\notin }\mathfrak{m}$ the restriction $\mathcal{F}(D(g))\to \mathcal{F}(D(f_{0}g))$ is the identity of $k$ and hence injective, and in the case $g\in \mathfrak{m}$ the group $\mathcal{F}(D(g))$ is $0$ by step 1.4, so in both cases $s{\mathrm{\mid }}_{D(g)}=\phi (c){\mathrm{\mid }}_{D(g)}$; since a section of the sheaf $\mathcal{F}$ over $U$ is determined by its restrictions to the cover by distinguished opens, $s=\phi (c)$ and $\mathcal{F}(U)\cong k$. If $\mathfrak{m}\mathrm{\notin }U$, then $g\in \mathfrak{m}$ for every distinguished $D(g)\subseteq U$ (else $\mathfrak{m}\in D(g)\subseteq U$), so the restriction of any $s\in \mathcal{F}(U)$ to each member of the covering family is zero by step 1.4 and $s=0$ by separatedness; hence $\mathcal{F}(U)=0$. For opens $U^{\mathrm{\prime }}\subseteq U$ both containing $\mathfrak{m}$ the restriction $\mathcal{F}(U)\to \mathcal{F}(U^{\mathrm{\prime }})$ carries ${\phi }_U(c)$ to ${\phi }_{U^{\mathrm{\prime }}}(c)$ because the gluing is given by the same formula over the distinguished opens inside $U^{\mathrm{\prime }}$, so under these isomorphisms it is the identity of $k$; consequently $\mathcal{F}\cong i_{\ast }(A\mathrm{/}\mathfrak{m})$, compatibly with all restrictions.

Conclusion: by step 1.1 the sheaf $\mathcal{F}=(A\mathrm{/}\mathfrak{m}{)}^{\sim }$ is coherent, by steps 1.2 and 1.3 its stalk is $A\mathrm{/}\mathfrak{m}$ at $\mathfrak{m}$ and zero at every other point, by step 2.1 its fibre at the closed point $\mathfrak{m}$ is $A\mathrm{/}\mathfrak{m}$, and by step 2.2 it is the skyscraper sheaf $i_{\ast }(A\mathrm{/}\mathfrak{m})=i_{\ast }(\mathcal{F}(\mathfrak{m}))$, the direct image of its one-point fibre under the inclusion of the closed point of [F7]; the example is verified.

Choice accounting: the ring $A$, the maximal ideal $\mathfrak{m}$, the module $M=A\mathrm{/}\mathfrak{m}$ and the point $\mathfrak{m}$ are given data, and no chart, section or isomorphism is selected by an infinite simultaneous choice; the skyscraper identification is canonical on every open by the components exhibited in step 2.2, and the only Axiom of Choice is the inherited one recorded in [F8].