Finite type need not mean coherent

Assume the Axiom of Choice as inherited by the coherence theory (Coherent module sheaves, Coherent sheaves on a locally Noetherian scheme). The theorem that coherent coincides with finite type (Coherent sheaves on a locally Noetherian scheme) is a statement about locally Noetherian schemes, and its proof uses the Noetherian hypothesis on the affine charts. It is not a definition, and it does not extend to arbitrary bases: over a general scheme, finite type, and even finite presentation or finite local freeness, do not by themselves imply coherence, and the relation-kernel condition in the definition of coherence (Coherent module sheaves) is a genuine additional hypothesis that must be checked.

The warning is not formal. The definition carries an explicit example: for $A=k[x,y_1,y_2,\dots ]\mathrm{/}(x{y}_i,y_i{y}_j:i,j\ge 1)$ the module $A$ is free of rank one, hence of finite type and finitely presented, but the kernel of multiplication by $x$, viewed as the $A$-linear endomorphism $\psi :A\to A$, is ${\mathrm{Ann}}_A(x)={⨁}_{i\ge 1}k[x]y_i$, which is not finitely generated; the associated morphism ${\mathcal{O}}_X\to {\mathcal{O}}_X$ on $X=\mathrm{Spec}A$ therefore has a kernel that is not of finite type, and ${\mathcal{O}}_X$ is not coherent on $X$ (Coherent module sheaves, Coherent sheaves on a locally Noetherian scheme).

Consequences for consumers of this page:

• an implication "finite type $\Rightarrow$ coherent" may be invoked only on a locally Noetherian scheme, where it is the theorem above;
• a finitely presented module over a non-Noetherian ring needs a separate argument for the kernel condition before its associated sheaf may be called coherent;
• a locally free sheaf of finite rank over a non-Noetherian scheme need not be coherent, so coherence hypotheses must be stated explicitly when Noetherianness is dropped;
• the theorem's locally Noetherian hypothesis is used in an essential way, and the example above shows that it cannot simply be deleted.