Locally factorial scheme

A scheme $X$ is locally factorial if every local ring ${\mathcal{O}}_{X,x}$, for $x\in X$, is a unique factorisation domain (A local ring is a nonzero commutative ring with a unique maximal ideal, Unique factorisation domain). This is a condition on the stalks; it does not assert that $X$ has an affine open cover whose coordinate rings are unique factorisation domains. The empty scheme is locally factorial vacuously.

Here is the componentwise form, with its hypotheses made explicit. Assume the Axiom of Choice (The Axiom of Choice), and suppose $X$ is a Noetherian normal scheme: Noetherian means that $X$ has a finite affine open cover by spectra of Noetherian rings (Locally Noetherian and Noetherian schemes), and normal means that each local ring is a normal Noetherian ring in the sense of normal noetherian ring. Then $X$ is reduced, and its irreducible components, with their reduced induced scheme structures, are integral, pairwise disjoint, and open. Consequently $X$ is locally factorial if and only if each of these components is locally factorial.

For the local-domain and reducedness inputs, fix a point $x$ and choose a chart $U=\mathrm{Spec}A$ from the finite Noetherian affine cover, with $x$ corresponding to $\mathfrak{p}\subset A$. The stalk is ${\mathcal{O}}_{X,x}\cong A_{\mathfrak{p}}$ by Affine open subschemes, Affine schemes and their coordinate rings, and The stalk of the affine structure sheaf at a prime is A_p. It is Noetherian by Every quotient and every localisation of a Noetherian ring is Noetherian. It is a local normal ring by the stated hypothesis, so the normal-ring condition at its unique maximal ideal makes ${\mathcal{O}}_{X,x}$ an integrally closed domain (A local ring is a nonzero commutative ring with a unique maximal ideal, normal noetherian ring). Thus every stalk is a domain. The nilpotent ideal sheaf has these stalkwise nilpotent elements as its germs (The reduction of a scheme), so it is zero and $X$ is reduced.

To see why the components are disjoint, let distinct irreducible components $C,D$ meet at $x$, and choose such a chart $U=\mathrm{Spec}A$ containing $x$. The nonempty intersections $C\cap U$ and $D\cap U$ are irreducible closed subsets of $U$. They are maximal there: if an irreducible closed subset of $U$ contains $C\cap U$, its closure in $X$ is irreducible, contains the dense open subset $C\cap U$ of $C$, and hence equals $C$ by maximality; since the original subset is closed in $U$, intersecting back with $U$ gives exactly $C\cap U$. The same holds for $D$. (Irreducible components as schemes, Irreducibility via nonempty open subsets, connectedness and open subspaces, Existence and basic properties of irreducible components) The affine components therefore correspond to distinct minimal primes ${\mathfrak{q}}_C$ and ${\mathfrak{q}}_D$ of $A$ contained in $\mathfrak{p}$ (Irreducible components of the spectrum correspond to minimal prime ideals). Under the prime-localisation correspondence, each remains minimal after extending to $A_{\mathfrak{p}}$: a prime below an extension contracts to a prime below the original minimal prime. The extensions remain distinct by injectivity of that correspondence, so $A_{\mathfrak{p}}\cong {\mathcal{O}}_{X,x}$ would have two distinct minimal primes (Prime ideals of a localization are exactly the primes disjoint from the denominator set). This is impossible for a domain, which has only the minimal prime $(0)$. Thus distinct components do not meet.

Finally, each chart in the finite cover has only finitely many irreducible components by A Noetherian ring has only finitely many irreducible components in its spectrum. Every irreducible component of $X$ meets a chart, and its intersection with that chart is an affine component as above; distinct global components give distinct such intersections because each is dense in its global component. There are therefore only finitely many global components. They are closed and cover $X$ by Existence and basic properties of irreducible components; since they are disjoint and finite in number, each is also open. Its reduced induced structure is integral by Irreducible components as schemes and Integral schemes. Restriction to an open subscheme preserves the stalks (Affine open subschemes), so the locally factorial condition holds on $X$ exactly when it holds on every component.

This component conclusion is asserted for the Noetherian normal case above. For arbitrary schemes outside the locally Noetherian setting, no openness or componentwise conclusion is being asserted.