Finite type need not be locally free

Given: The Axiom of Choice; a field $k$; the polynomial ring $A=k[x]$; the scheme $X=\mathrm{Spec}A$; the module $M=A\mathrm{/}(x)$ with the class of $1$ as a generator; the associated sheaf $\mathcal{F}=\tilde{M}$; the closed point ${\mathfrak{p}}_0=(x)\in X$ and the point $\eta =(0)\in X$.

The ring $A=k[x]$ and its spectrum (The underlying space of an affine spectrum, The prime spectrum and vanishing sets, For every field $F$, $F[x]$ is a principal ideal domain, Every principal ideal domain is Noetherian, Locally Noetherian and Noetherian schemes, $F[x{]}_{(x)}$ is the ring of rational functions defined at $0$, with maximal ideal generated by $x$ and residue field $F$): $k[x]$ is a principal ideal domain, hence Noetherian, and $X$ is locally Noetherian; the points of $X$ are the prime ideals of $A$ and the basic opens are the sets $D(f)=\{\mathfrak{p}:f\mathrm{\notin }\mathfrak{p}\}$; evaluation at $0$ identifies $A\mathrm{/}(x)\cong k$, so $(x)$ is a maximal (hence prime) ideal and the local ring $A_{(x)}$ has maximal ideal $x{A}_{(x)}$ and residue field $A_{(x)}\mathrm{/}x{A}_{(x)}\cong k$; consequently every prime of $A$ containing $x$ equals $(x)$, and $x\mathrm{\notin }\mathfrak{p}$ for every prime $\mathfrak{p}\mathrm{\ne }(x)$. The chain of ideals is ordered by inclusion, $\eta$ and ${\mathfrak{p}}_0$ are distinct, and $A$ is a domain, so $\eta =(0)$ is a prime and $0\mathrm{\ne }x\in A$.

The associated sheaf (Module sheaf on an affine scheme, The associated module sheaf exists, Sections of the associated sheaf on basic opens, The stalk of an associated sheaf is the localisation, Quasi-coherent module on a scheme, Finite type and finitely presented module sheaves, Coherent sheaves on a locally Noetherian scheme): $\mathcal{F}=\tilde{M}$ is a sheaf of ${\mathcal{O}}_X$-modules whose distinguished-open sections are $\mathcal{F}(D(f))=M_f$ with restriction the canonical localisation; its stalks are ${\mathcal{F}}_{\mathfrak{p}}\cong M_{\mathfrak{p}}$; it is quasi-coherent; because $M$ is finitely generated it is of finite type; and because $X$ is locally Noetherian it is coherent.

Localisation at a prime (Localisation at a prime ideal: $R_{\mathfrak{p}}=(R\setminus \mathfrak{p}{)}^{-1}R$, The localisation relation is an equivalence relation and fraction arithmetic is well defined, Localisation commutes with quotient modules and arbitrary direct sums, $R_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{R}_{\mathfrak{p}}\cong \mathrm{Frac}(R\mathrm{/}\mathfrak{p})$ is the residue field at $\mathfrak{p}$): $A_{\mathfrak{p}}$ is the localisation of $A$ at $A\setminus \mathfrak{p}$, and every $s\mathrm{\notin }\mathfrak{p}$ is a unit of $A_{\mathfrak{p}}$ with inverse $1\mathrm{/}s$; for every submodule $N\subseteq A$ one has $(A\mathrm{/}N{)}_{\mathfrak{p}}\cong A_{\mathfrak{p}}\mathrm{/}{N}_{\mathfrak{p}}$, and localisation commutes with arbitrary direct sums, so $(A^r{)}_{\mathfrak{p}}\cong (A_{\mathfrak{p}}{)}^r$ for $r\ge 0$; the residue field is $A_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{A}_{\mathfrak{p}}\cong \mathrm{Frac}(A\mathrm{/}\mathfrak{p})$.

Stalks, fibres and restrictions (Fibre of a module sheaf at a point, The residue field at a point of an affine scheme, The stalk of the affine structure sheaf at a prime is A_p): the fibre of an ${\mathcal{O}}_X$-module $\mathcal{G}$ at $x\in X$ is $\mathcal{G}(x)={\mathcal{G}}_x{\otimes }_{{\mathcal{O}}_{X,x}}\kappa (x)\cong {\mathcal{G}}_x\mathrm{/}{\mathfrak{m}}_x{\mathcal{G}}_x$, a vector space over $\kappa (x)={\mathcal{O}}_{X,x}\mathrm{/}{\mathfrak{m}}_x$; a morphism of sheaves induces a $\kappa (x)$-linear map on fibres; restriction to an open $U\subseteq X$ satisfies $(\mathcal{G}{\mathrm{\mid }}_U)(x)=\mathcal{G}(x)$ for every $x\in U$; and for the structure sheaf the stalk is ${\mathcal{O}}_{X,\mathfrak{p}}\cong A_{\mathfrak{p}}$.

Free module sheaves on an affine scheme (Module sheaf on an affine scheme, The associated module sheaf exists, Sections and restrictions on distinguished opens of an affine scheme, The stalk of an associated sheaf is the localisation, Localisation commutes with quotient modules and arbitrary direct sums, Fibre of a module sheaf at a point): for a commutative ring $B$ and $r\ge 0$, the structure sheaf of $\mathrm{Spec}B$ is the associated sheaf $\tilde{B}$ of the ring, the free ${\mathcal{O}}_{\mathrm{Spec}B}$-module ${\mathcal{O}}^r$ is $\widetilde{B^r}$ (both have sections $(B_g{)}^r=(B^r{)}_g$ on the distinguished open $D(g)$, and the extension from distinguished opens is unique), and for every prime $\mathfrak{q}$ of $B$ the stalk is $({\mathcal{O}}^r{)}_{\mathfrak{q}}\cong (B^r{)}_{\mathfrak{q}}\cong (B_{\mathfrak{q}}{)}^r$, so that the fibre $({\mathcal{O}}^r)(\mathfrak{q})\cong (B_{\mathfrak{q}}{)}^r{\otimes }_{B_{\mathfrak{q}}}\kappa (\mathfrak{q})\cong \kappa (\mathfrak{q}{)}^r$ is a $\kappa (\mathfrak{q})$-vector space of dimension $r$.

Distinguished opens and localised spectra (Every point of a Zariski-open set has a distinguished-open neighbourhood inside it, The spectrum of a principal localisation is the distinguished open D(f), A principal localization identifies its spectrum with a distinguished open, Primes of a principal localization): for every open $U\subseteq X$ and every $\mathfrak{p}\in U$ there is $f\in A$ with $\mathfrak{p}\in D(f)\subseteq U$; the localisation $A\to A_f$ identifies $\mathrm{Spec}{A}_f$ with $D(f)$ as locally ringed spaces, and the primes of $A_f$ correspond to the primes of $A$ not containing $f$.

The generic point (The closure of a prime is its vanishing set, The prime spectrum and vanishing sets): the closure of a point $\mathfrak{p}\in X$ is $\overline{\{\mathfrak{p}\}}=V(\mathfrak{p})=\{\mathfrak{q}:\mathfrak{q}\supseteq \mathfrak{p}\}$; since $A$ is a domain, every prime contains $(0)$, so $V((0))=X$ and $\overline{\{(0)\}}=X$.

Locally free sheaves (Locally free sheaves of finite rank, Modules on a ringed space): $\mathcal{E}$ is locally free of rank $r$ near $x$ if some open neighbourhood $U$ of $x$ satisfies $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$, where ${\mathcal{O}}_U^r$ is the free ${\mathcal{O}}_U$-module of rank $r$, with ${\mathcal{O}}_U^0=0$; $\mathcal{E}$ is finite locally free if this holds at every point with some rank; a locally free sheaf is quasi-coherent.

The Axiom of Choice is inherited from the associated-sheaf existence theorem, the closure description of [F7] and the affine equivalence used through [F3] and [F5]; no further choice is made in the computation below (The Axiom of Choice).

Proof technique: direct; compute the stalk and fibre of $\widetilde{A\mathrm{/}(x)}$ at all points of $\mathrm{Spec}k[x]$, show that the generic point lies in every nonempty open set, and use a free chart at the closed point $(x)$ to force two different ranks $r=0$ and $r=1$.