Affine codimension-one neighbourhoods and divisors

Reduce to a connected normal component of $X$. Its generic point lies in every nonempty affine open, so discard it from the list; if the list becomes empty, any affine open suffices. Remove repeated points. For the remaining codimension-one points put $V_i={\mathcal{O}}_{X,x_i}$, viewed as rank-one valuation rings in the common function field $L$. The $V_i$ are pairwise distinct: if two points $x_i\mathrm{\ne }{x}_j$ gave centres of the same valuation of $L$, both would dominate the same valuation ring and separatedness would force $x_i=x_j$ by [F1]. Two distinct rank-one valuation rings in $L$ are incomparable: if $V\subseteq W$ and the uniformizer $\pi$ of $V$ is invertible in $W$, then $L=V[1\mathrm{/}\pi ]\subseteq W$, so $W$ is a field, and otherwise every $x\mathrm{\notin }V$ has inverse in $\pi V$ and cannot lie in $W$; hence inclusion forces equality.

Fix $i$ and, for each $j\mathrm{\ne }i$, choose $y_{ij}\in V_i\setminus V_j$. Multiplying a sufficiently high power of $y_{ij}$ by a uniformizer of $V_i$ gives $z_{ij}$ with $v_i(z_{ij})>0$ and $v_j(z_{ij})<0$. If there is only one valuation, take $h_i$ to be its uniformizer; otherwise begin with one $z_{ij}$ and construct $h_i$ successively: to add a new index $j$, replace the preceding sum $h$ by $h+z_{ij}^M$. Choose $M$ so large that its new term has strictly smaller valuation than $h$ at $j$ and at every earlier index where $z_{ij}$ has negative valuation. At an earlier index where that valuation is nonnegative, the old negative valuation persists. Thus cancellation is excluded at every required index, and $v_i(h_i)>0$, $v_j(h_i)<0$ for all $j\mathrm{\ne }i$. Put $e_i=1\mathrm{/}(1+h_i^N)$. Increasing $N$ makes $v_i(e_i-1)$ and all $v_j(e_i)$ for $j\mathrm{\ne }i$ arbitrarily large. For prescribed targets $c_i\in L$, the sum ${\sum }_i{e}_i{c}_i$ consequently approximates $c_j$ at every $v_j$ to any fixed finite precision.

Let $C={\bigcap }_i{V}_i$. The residue map $C\to \kappa (V_i)$ is surjective: approximate a representative in $V_i$ modulo its maximal ideal and approximate zero at all other valuations, using step 2.1. Thus its kernel ${\mathfrak{m}}_i=\{c\in C:v_i(c)>0\}$ is maximal. These are all the maximal ideals: an element of $C$ is invertible exactly when all its valuations vanish, so the nonunits are the union of the ${\mathfrak{m}}_i$, and finite prime avoidance [F3] makes every maximal ideal one of the ${\mathfrak{m}}_i$. The approximation of step 2.1 separates the ${\mathfrak{m}}_i$, realizes arbitrary prescribed residues, and shows $C_{{\mathfrak{m}}_i}=V_i$: for $x\in V_i$ one approximates $1$ at $i$ and $0$ to high order at the other indices by some $t\in C$, and then $tx\in C$ with $t\mathrm{\notin }{\mathfrak{m}}_i$, so $x=(tx)\mathrm{/}t\in C_{{\mathfrak{m}}_i}$, the reverse inclusion being immediate.

Choose affine charts $W_i=\mathrm{Spec}{B}_i$ around $x_i$ and finite $R$-algebra generators $b_{il}$ of $B_i$. By step 3.1 write $b_{il}=a_{il}\mathrm{/}{s}_{il}$ with $a_{il},s_{il}\in C$ and $s_{il}\mathrm{\notin }{\mathfrak{m}}_i$. Let $C_0\subseteq C$ be the $R$-algebra generated by all these numerators and denominators, and put $s_i={\prod }_l{s}_{il}$, so $B_i\subseteq (C_0{)}_{s_i}$ inside $L$. Each of the finitely many generators of $C_0$ lies in $(B_i{)}_{{\mathfrak{p}}_i}=V_i$, where ${\mathfrak{p}}_i$ represents $x_i$. Writing those generators as fractions in $(B_i{)}_{{\mathfrak{p}}_i}$ and multiplying their denominators gives $t_i\in B_i\setminus {\mathfrak{p}}_i$ with $C_0\subseteq (B_i{)}_{t_i}$. Since $t_i\in (C_0{)}_{s_i}$, write $t_i=d_i\mathrm{/}{s}_i^{N_i}$ with $d_i\in C_0$; both $s_i$ and $d_i$ are units in $V_i$. The two inclusions just constructed induce mutually inverse homomorphisms, inside $L$, between $(C_0{)}_{s_i{d}_i}$ and $(B_i{)}_{t_i}[1\mathrm{/}{s}_i]$: $B_i$ lies in $(C_0{)}_{s_i}$ and $t_i$ becomes invertible after inverting $d_i$, while $C_0$ lies in $(B_i{)}_{t_i}$ and $d_i=t_i{s}_i^{N_i}$ becomes invertible after inverting $s_i$. All defining relations and both inverse identities hold because these are subrings of the same field. To identify an actual principal open of $W_i$, write $s_i=r_i\mathrm{/}{t}_i^{M_i}$ in $(B_i{)}_{t_i}$; then $r_i\mathrm{\notin }{\mathfrak{p}}_i$ and $(B_i{)}_{t_i}[1\mathrm{/}{s}_i]=(B_i{)}_{t_i{r}_i}$. Thus $U_i=D_{W_i}(t_i{r}_i)$ contains $x_i$ and is isomorphic to $D_{\mathrm{Spec}{C}_0}(s_i{d}_i)$.

The inverse morphisms $D(s_i{d}_i)\to U_i\subseteq X$ agree on every overlap: they agree at the common generic point, their source is integral, and separatedness of $X$ makes their equalizer closed. Their maps to $\mathrm{Spec}{C}_0$ likewise agree on overlaps of the $U_i$, since all are defined by the inclusion $C_0\subseteq L$. Consequently these isomorphisms glue to an isomorphism from the open $W={\bigcup }_i{U}_i\subseteq X$ onto the open ${\bigcup }_{i}D(s_i{d}_i)\subseteq \mathrm{Spec}{C}_0$. Let $J$ define its closed complement. For every prime ${\mathfrak{q}}_i$ corresponding to $x_i$, $J⊈{\mathfrak{q}}_i$; finite prime avoidance gives $f\in J\setminus {\bigcup }_i{\mathfrak{q}}_i$. Then $D(f)\subseteq W$ is affine and contains all $x_i$. A normal Noetherian scheme has finitely many disjoint open and closed integral components; applying this construction on each component with a prescribed point and taking the finite disjoint union gives (a), including any generic points discarded in step 1.1.

For (b), work on one integral component of $X$, with function field $L$, and write $U=\mathrm{Spec}A$ there. If an irreducible component of the closed boundary has generic point $z$ of codimension at least two, choose an affine normal chart $V=\mathrm{Spec}B$ containing $z$ and avoiding every other boundary component. Every height-one point of $V$ then belongs to $U$. For each $a\in A\subseteq L$, its restriction to $U\cap V$ is regular at all these points, so [F2] places $a$ in $B$. This gives a single ring homomorphism $A\to B$: sums, products, the unit and every relation are preserved inside $L$. Thus it defines an actual morphism $h:V\to U$, with no finite-generation assumption on $A$ needed. On the dense open $U\cap V$ it is the identity inclusion into $U$; separatedness of $X$ makes the composite $V\stackrel{h}{\to }U\hookrightarrow X$ equal to the inclusion $V\hookrightarrow X$ everywhere. Its image would put $z$ in $U$, a contradiction. Hence every boundary component has codimension one. If $X$ is regular, at each point its finitely many boundary prime ideals are principal in the regular local UFD. Their intersection is generated by the product of their distinct prime generators, a nonzerodivisor; these reduced ideals glue to the reduced boundary subscheme, making it an effective Cartier divisor. The same argument on the finitely many normal components proves (b) for general $X$.

Finally let $X$ be flat over a discrete valuation ring and let $U$ meet every irreducible component of the special fibre. A prime divisor $P$ contained in the boundary $X\setminus U$ and dominating the base would meet the generic fibre; a prime divisor contained in the special fibre would be a component of it, which is excluded by the fibre-density hypothesis. Hence every boundary prime meets the generic fibre, so the closure of the generic complement $X_K\setminus U_K$ contains the whole boundary and is contained in it by closedness; the boundary is therefore the closure of its generic complement and contains no special-fibre component.