Coherent module sheaves

Let $X$ be a scheme and let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-module (Quasi-coherent module on a scheme) that is of finite type (Finite type and finitely presented module sheaves). Then $\mathcal{F}$ is coherent if for every open $U\subseteq X$ and every morphism ${\mathcal{O}}_U^n⟶\mathcal{F}{\mathrm{\mid }}_U,n\ge 0\text{ finite},$ the kernel sheaf $\ker \phi$ is of finite type (Kernel sheaves are objectwise, while cokernels and images are sheafified).

Under the Axiom of Choice (The Axiom of Choice), inherited from the associated-sheaf existence theorem at the sheaf-kernel identification below, the following affine test is equivalent to the definition. Since the conditions are local, it suffices to test affine opens $U=\mathrm{Spec}A$ on which $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ for a finitely generated $A$-module $M$: then a morphism ${\mathcal{O}}_U^n\to \tilde{M}$ is given by an $A$-linear map $\psi :A^n\to M$ (Finite type and finitely presented module sheaves) and, as localisation is exact (Localisation of modules is exact), the kernel sheaf has $\ker \phi (D(f))=\ker (\psi ){\otimes }_A{A}_f=\ker (\psi {)}_f$, so it is the associated sheaf of the finitely generated module $\ker \psi$ precisely when that kernel is finitely generated (The associated module sheaf exists).

Finite presentation does not imply coherence under the same AC assumption. Over a general base the kernel condition is a genuine additional hypothesis. Let $k$ be a field and let $A=k[x,y_1,y_2,\dots ]\mathrm{/}(x{y}_i,y_i{y}_j:i,j\ge 1);$ the classes of $1,x,x^2,\dots ,y_1,y_2,\dots$ form a $k$-basis of $A$, so $A=k[x]\oplus {⨁}_{i\ge 1}k[x]y_i$ as a $k[x]$-module, and multiplication by $x$, viewed as an $A$-linear map $\psi :A\to A$, has kernel $\ker \psi ={\mathrm{Ann}}_A(x)={⨁}_{i\ge 1}k[x]y_i$, which is not finitely generated as an $A$-module. The module $A$ is finitely presented, even free of rank one, but the associated sheaf $\tilde{A}={\mathcal{O}}_X$ on $X=\mathrm{Spec}A$ receives the morphism ${\mathcal{O}}_X\to {\mathcal{O}}_X$ corresponding to $\psi$ (Finite type and finitely presented module sheaves, The associated module sheaf exists) whose kernel is the associated sheaf of ${\mathrm{Ann}}_A(x)$ and is not of finite type. Hence ${\mathcal{O}}_X$ is not coherent on this $X$; finite presentation, and even local freeness of finite rank, do not by themselves imply coherence over an arbitrary base. The identification of coherent with finite type is a theorem about locally Noetherian schemes, proved on this page; over a locally Noetherian scheme finite locally free sheaves are therefore coherent.

Immediate consequences. Coherence is local on $X$ and invariant under isomorphism; a coherent module is of finite type by definition; restrictions of coherent modules to open subschemes are coherent; and the zero module is coherent. On a locally Noetherian scheme the kernel condition is automatic for finite-type quasi-coherent modules, and the theorem on this page proves the converse direction: there, finite type and coherent coincide (Locally Noetherian and Noetherian schemes). Over a non-Noetherian ring the two notions differ, as the $A={\mathcal{O}}_X$ example above shows, and the relation-kernel condition must be checked; the remark on this page records that warning.