Finite type and finitely presented 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). For $n\ge 0$ write ${\mathcal{O}}_X^n$ for the sheaf $U\mapsto {\mathcal{O}}_X(U{)}^n$ with componentwise restriction, a sheaf of ${\mathcal{O}}_X$-modules (Modules on a ringed space); it is the free ${\mathcal{O}}_X$-module of rank $n$, and ${\mathcal{O}}_X^0=0$.

Finite type. $\mathcal{F}$ is of finite type if every point $x\in X$ has an affine open neighbourhood $U=\mathrm{Spec}A$ together with an $A$-module $M$ and an isomorphism $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ such that $M$ is finitely generated (Finitely presented modules and finitely presented algebras).

Finitely presented. $\mathcal{F}$ is finitely presented if every point $x\in X$ has an affine open neighbourhood $U=\mathrm{Spec}A$ together with an $A$-module $M$ and an isomorphism $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ such that $M$ is finitely presented.

The equivalent local form under AC. Assume the Axiom of Choice (The Axiom of Choice), inherited from the associated-sheaf existence theorem in the equivalence argument below. Because of the local nature of the conditions, $\mathcal{F}$ is of finite type if and only if $X$ can be covered by affine opens $U=\mathrm{Spec}A$ on which there is a finite family of sections $s_1,\dots ,s_n\in \mathcal{F}(U)$ generating $\mathcal{F}{\mathrm{\mid }}_U$, meaning that the induced morphism of ${\mathcal{O}}_U$-modules ${\mathcal{O}}_U^n\to \mathcal{F}{\mathrm{\mid }}_U$ is an epimorphism, equivalently that ${\mathcal{O}}_U^n⟶\mathcal{F}{\mathrm{\mid }}_U⟶0$ is an exact sequence of sheaves (Exact sequences of sheaves). Likewise $\mathcal{F}$ is finitely presented if and only if $X$ can be covered by affine opens $U$ admitting an exact sequence ${\mathcal{O}}_U^m⟶{\mathcal{O}}_U^n⟶\mathcal{F}{\mathrm{\mid }}_U⟶0$ with $m,n$ finite.

Why the two forms agree under AC. Let $U=\mathrm{Spec}A$ be affine and let $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ for an $A$-module $M$. First $\widetilde{A^n}\cong {\mathcal{O}}_U^n$, since the two sheaves have the same sections $(A_f{)}^n$ on every distinguished open and sheaves on the basis are determined by those sections (The associated module sheaf exists). Every morphism $\widetilde{A^n}\to \tilde{M}$ of ${\mathcal{O}}_U$-modules is induced by its component on global sections, an $A$-linear map $\psi :A^n\to M$: a morphism is determined by its components on distinguished opens (The associated module sheaf exists), a general $A$-linear map induces compatible maps $A_f^n\to M_f$ on distinguished opens, and these two constructions are inverse by the functoriality of the localisations recorded in Module sheaf on an affine scheme. If $M$ is generated by the images of the standard basis under $\psi$, then localisation is exact (Localisation of modules is exact), so each component $A_f^n\to M_f$ is surjective, and a morphism whose components on a basis are surjective is an epimorphism: a germ of $\tilde{M}$ at a point is represented on some distinguished open and can be lifted there. Conversely, let $\psi :A^n\to M$ correspond to an epimorphism $\phi$ and put $N=M\mathrm{/}\psi (A^n)$; the components of the composite $\widetilde{A^n}\to \tilde{M}\to \tilde{N}$ are $A_f^n\to M_f\to N_f=0$, because localisation is right exact, so this composite is the zero morphism on every distinguished open and hence is zero; it is also a composite of epimorphisms, so its target has all stalks zero, and then $N=\mathrm{\Gamma }(U,\tilde{N})=0$ by the identification of global sections (The associated module sheaf exists). Thus $\psi$ is surjective, so $M$ is generated by $n$ elements and $\phi$ exhibits the finite family of global sections $\psi (e_1),\dots ,\psi (e_n)$. Applying the same translation to the kernel of a surjection $A^n\to M$ gives the finitely presented form. The elementwise criterion for equality and vanishing in a localisation (Localisation of a module at a multiplicative subset) is what turns the componentwise statements into statements about $M$.

Immediate consequences. The conditions are local on $X$ and invariant under isomorphism of ${\mathcal{O}}_X$-modules; a finitely presented quasi-coherent module is of finite type, since a finitely presented module is generated by the images of the standard basis; restrictions to open subschemes again satisfy the corresponding condition; and the zero module is finitely presented (take $m=n=0$), hence of finite type, on every chart. No Noetherian, separatedness or finiteness hypothesis on $X$ is built into the definition, and the definition says nothing about existence of local frames, which is the stronger condition treated separately on this page.