Functions extend across codimension two

It suffices to prove that restriction ${\mathcal{O}}_X(V)\to {\mathcal{O}}_U(U\cap V)$ is an isomorphism on a basis of affine opens $V$: the kernel and cokernel sheaves then have zero stalks. By [F1], each such $V$ is a finite disjoint union of affine spectra of noetherian integrally closed domains. These factor spectra form a basis as $V$ varies, and restriction to a factor preserves its structure sheaf. Thus it is enough to consider $V=\mathrm{Spec}A$ for such a domain $A$. Its generic point and each point corresponding to a height-one prime of $A$ belong to $U$: codimension is unchanged when passing to an open neighborhood, since chains of generalizations of a point in an open set remain in that open set.

Put $W=U\cap V$ and let $K=\mathrm{Frac}(A)$. Since $W$ contains the generic point, it is dense in the integral scheme $V$. A section $s$ on $W$ determines an element $z\in K$ by its generic stalk. For every height-one prime $\mathfrak{p}$, the point $\mathfrak{p}$ lies in $W$, so the germ of $s$ there belongs to $A_{\mathfrak{p}}$ and maps to $z$ in $K$. Hence $z$ lies in every height-one localization, and [F2] gives $z\in A$. The section defined by $z$ agrees with $s$ on $W$: on any principal open $D(f)\subseteq W$, both are elements of the domain $A_f$ and have the same image in $K$, so they are equal; such opens cover $W$. Thus restriction from $A$ to ${\mathcal{O}}_U(W)$ is surjective. It is injective because $W$ contains the generic point and $A\hookrightarrow K$ is injective. This also covers the case that $A$ is a field.

Steps 1.1 and 2.1 show that the restriction morphism is an isomorphism on a basis and therefore an isomorphism of sheaves of rings. Taking sections over $X$ gives $\mathrm{\Gamma }(X,{\mathcal{O}}_X)\cong \mathrm{\Gamma }(U,{\mathcal{O}}_U)$. A ring isomorphism carries precisely the invertible elements to invertible elements, with inverses preserved, so it also gives an isomorphism of global unit groups.