Quasi-coherent module on a scheme

Let $X$ be a scheme (Schemes) and let $\mathcal{F}$ be a sheaf of ${\mathcal{O}}_X$-modules (Modules on a ringed space). Recall that for a commutative ring $A$ and an $A$-module $M$ the associated module sheaf $\tilde{M}$ on $\mathrm{Spec}A$ is the sheaf of ${\mathcal{O}}_{\mathrm{Spec}A}$-modules whose sections on a distinguished open $D(f)$ are $M_f$ (Module sheaf on an affine scheme).

The module $\mathcal{F}$ is quasi-coherent if every point $x\in X$ has an affine open neighbourhood $U=\mathrm{Spec}A\subseteq X$ and an $A$-module $M$ such that

$\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$

as sheaves of ${\mathcal{O}}_U$-modules. A quasi-coherent sheaf of ${\mathcal{O}}_X$-modules is also called a quasi-coherent module on $X$, and the full subcategory of ${\mathcal{O}}_X$-modules consisting of the quasi-coherent ones is written $\mathrm{QCoh}(X)$; morphisms in $\mathrm{QCoh}(X)$ are all ${\mathcal{O}}_X$-module morphisms between its objects.

Immediate consequences of the definition, used without further comment:

• The condition is invariant under isomorphism: if $\mathcal{F}\cong \mathcal{G}$ as ${\mathcal{O}}_X$-modules and $\mathcal{F}$ is quasi-coherent, then so is $\mathcal{G}$, because the isomorphism restricts over an affine open.
• The condition is local on $X$: $\mathcal{F}$ is quasi-coherent if and only if there is an open cover $X={\bigcup }_{i\in I}{U}_i$ such that each restriction $\mathcal{F}{\mathrm{\mid }}_{U_i}$ is quasi-coherent. Indeed a point of an arbitrary open subscheme $U\subseteq X$ has an affine open neighbourhood inside $U$, and an affine open subscheme of $X$ contained in $U$ is also an affine open subscheme of $U$; conversely, every point of $U_i$ lies in an affine open of $X$ inside $U_i$.
• If $\mathcal{F}$ is quasi-coherent and $V\subseteq X$ is open, then $\mathcal{F}{\mathrm{\mid }}_V$ is quasi-coherent, since $\mathcal{F}{\mathrm{\mid }}_V$ is again an ${\mathcal{O}}_V$-module and an affine open $U\subseteq V$ is an affine open of $X$.
• On an affine scheme $X=\mathrm{Spec}A$ the definition asks at each point for an affine open neighbourhood on which $\mathcal{F}$ is associated to a module; it does not ask that $\mathcal{F}$ itself be associated to a single $A$-module. The affine comparison theorem on this page shows that on an affine scheme the two conditions coincide.

The terminology follows the standard one: quasi-coherence is the local affine-module condition, and no finiteness, Noetherian or separatedness hypothesis is part of it.