Normal noetherian schemes reduce to integral components

Every affine open $U=\mathrm{Spec}A$ has $A$ noetherian: a finite cover of $X$ by noetherian affine opens refines the quasi-compact open $U$ by finitely many principal opens whose coordinate rings are localizations of noetherian rings; the ascending-chain condition on ideals can then be checked after these finitely many localizations, where equality of localized ideals implies equality on $U$. By [F3] and normality of $X$, every prime localization of $A$ is an integrally closed domain. Thus [F1] writes $A$ as a finite product of noetherian integrally closed domains. The corresponding spectra are disjoint open-and-closed subschemes of $U$; each is irreducible and reduced, since it is the spectrum of a domain. In particular, every point of $X$ has an integral affine open neighborhood.

The underlying topological space of $X$ is noetherian: each member of a finite noetherian affine cover has descending-chain condition on closed sets, and a descending chain of closed subsets of $X$ stabilizes once its restrictions to these finitely many opens stabilize. By [F2], $X$ is the union of finitely many irreducible closed components. If such a component $C$ contains $x$, choose an integral open neighborhood $U$ of $x$ from step 1.1. The nonempty open subset $C\cap U$ is dense in $C$; its closure is contained in the irreducible closure $\bar{U}$. Maximality of $C$ therefore gives $C=\bar{U}$. Consequently two components containing $x$ are equal. The finitely many components are thus pairwise disjoint, and each is open as the complement of the union of the others. The empty scheme corresponds to the empty union.

Give each open-and-closed component $C$ the restricted structure sheaf. It is irreducible by construction. It is reduced because the integral affine neighborhoods from step 1.1 cover $X$, and nilpotence and vanishing of sections can be checked locally. Its local rings are those of $X$, so they are integrally closed domains; hence $C$ is normal and integral. This proves the decomposition and the affine-ring assertion.

For a finite disjoint open cover $X={\coprod }_i{C}_i$, restriction and gluing give an equivalence between sheaves on $X$ and tuples of sheaves on the $C_i$: overlaps between different components are empty, so there are no compatibility conditions. Applied to the structure sheaf and its invertible sections, this gives functions and units componentwise; applied to locally free sheaves, it gives line bundles and finite-rank vector bundles componentwise. The rank is locally constant by local freeness; on an irreducible space two disjoint nonempty open rank loci cannot occur, so it is constant on each $C_i$.