Zariski's main theorem

Remarks

The hypothesis says that $S$ is quasi-finite at $\mathfrak{q}$: the point $\mathfrak{q}$ is isolated in its fibre over $\mathfrak{p}$. This is Grothendieck's algebraic form of the theorem. In words, near a point where it is quasi-finite, a finite type algebra looks like a localization of an integral algebra. In geometric language, a quasi-finite separated morphism of finite type between Noetherian schemes factors as an open immersion followed by a finite morphism. Zariski's original form follows: a birational morphism of varieties with finite fibres onto a normal variety is an open immersion. For example, $k[x, x^{-1}]$ is quasi-finite over $k[x]$ at every prime, and indeed it is the localization of $k[x]$ at $g = x$. Without the hypothesis the conclusion fails: for $R = k$ a field, $S = k[x]$ and $\mathfrak{q} = (x)$, the prime $\mathfrak{q}$ contains $(0)$, which also lies over $(0)$; here $S' = k$, and $k_g = k$ is never $k[x]_g$.

Going deeper

For a student

The theorem has many forms: the algebraic one stated here, the factorisation of quasi-finite separated morphisms, and Zariski's statement on birational maps to normal varieties. It is a basic tool in algebraic geometry, for example in the study of étale and unramified morphisms. The role of integral closure is central: the finite algebra in the factorisation is built from the integral closure of $R$ in $S$.