Normalization of a classical variety by gluing affine normalizations

Chartwise normalization. By [F1] each chart $U_i$ has its affine normalization ${\nu }_i\text{\hspace{0.17em}⁣}:U_i^{\nu }\to U_i$, finite, surjective and birational, with $U_i^{\nu }$ normal and $k[U_i^{\nu }]=B_i\subseteq k(X)$.

Compatibility on overlaps. Cover $U_i\cap U_j$ by opens principal in both charts. To construct them around a point, choose $D_{U_i}(f)\subseteq U_j$, then $D_{U_j}(g)\subseteq D_{U_i}(f)$ containing the point. On $D_{U_i}(f)$ write $g=a\mathrm{/}{f}^m$ with $a\in A_i$ by [F2]; since $D_{U_j}(g)$ is contained there, it equals $D_{U_i}(fa)$. Thus each such open has coordinate ring computable as a localization of either chart; by [F2] the integral closure over $D(f)$ computed from the $i$-side is $(B_i{)}_f$ and from the $j$-side is the corresponding localization of $B_j$, and both equal the integral closure of the common ring of $D(f)$ inside the common field $k(X)$, hence agree as subrings of $k(X)$. The anti-equivalence [F4] therefore produces a canonical isomorphism $U_i^{\nu }{\mathrm{\mid }}_{D(f)}\cong U_j^{\nu }{\mathrm{\mid }}_{D(f)}$ over $D(f)$; on triple overlaps these identifications satisfy the cocycle condition because all of them are the identity of the common subring of $k(X)$, and different choices of $f$ agree by the same uniqueness.

Gluing. Since principal opens cover each overlap and the identifications of step 2.1 are compatible, the varieties $U_i^{\nu }$, together with their maps to $X$, glue along the open overlaps by the standard gluing of compatible classical charts; the maps ${\nu }_i$ agree on overlaps because they are determined by the inclusions $A_i\hookrightarrow B_i$ in the common field [F3, F4], so they glue to a morphism $\nu \text{\hspace{0.17em}⁣}:X^{\nu }\to X$. The glued variety $X^{\nu }$ is normal, since normality is local and each chart is normal [F1]; it is separated because $\nu$ is finite, the diagonal over $X$ being affine-locally cut out by the surjection $B_i{\otimes }_{A_i}{B}_i\to B_i$; and the morphism $\nu$ is finite, surjective and birational because these properties are affine-local on the target and hold on every chart by [F1] and [F2].

Independence and conclusion. If two finite affine covers are used, refine both to principal opens; on every such principal open the two affine normalizations agree as subrings of $k(X)$ by [F2], so the two glued models agree over a cover and hence are canonically identified over $X$ by [F3, F4]. Thus $X$ has a normalization $\nu \text{\hspace{0.17em}⁣}:X^{\nu }\to X$ that is finite, surjective and birational from a normal variety; its uniqueness up to unique isomorphism over $X$ is the universal property proved later on this page, and in the reducible case the disjoint union of the component normalizations is normal, finite, surjective and birational over each component, hence is a normalization of the reduced variety.