Zariski's Main Theorem: open immersion followed by a finite morphism

The morphism $f$ of the statement is separated of finite type with finite fibres, so it is quasi-finite in the sense of [F1]; its source and target are classical varieties in the register of [F1] and [F2], with a finite affine atlas making them quasi-compact, and separatedness giving quasi-separatedness. These are exactly the hypotheses of the bridge [F2], which allows reduced reducible or empty varieties and assumes no quasi-projectivity, normality, separability, or smoothness.

Applying [F2], the relative integral closures $C_U$ of the affine coordinate rings glue to a finite classical morphism $\nu \text{\hspace{0.17em}⁣}:N\to Y$, and the evaluation maps glue to an open immersion $j\text{\hspace{0.17em}⁣}:X\hookrightarrow N$ satisfying $f=\nu \circ j$. By [F3] the map $\nu$ is finite in the page's sense, and $j$ is an open immersion, so $f=\nu \circ j$ is the asserted factorization. This is the exact classical form of Zariski's Main Theorem used below: an open immersion followed by a finite morphism.