Stein factorization

Remarks

The sheaf $f_* \mathcal{O}_X$ is coherent because $f$ is proper, so $g$ is finite. Clause 3 follows from clause 2 by Zariski's connectedness theorem, which in turn rests on the theorem on formal functions. For example, if $f$ is finite, then $f$ is affine and $X$ is the relative spectrum of $f_* \mathcal{O}_X$. So $Y' = X$, $f'$ is the identity and $g = f$. At the other extreme, for $X$ proper and connected over $k$ with $\mathcal{O}(X) = k$, the map $X \to \operatorname{Spec} k$ already has $Y' = Y$. The properness hypothesis is necessary: for a non-proper $f$, the sheaf $f_* \mathcal{O}_X$ need not be coherent. For $\mathbb{A}^1_k \to \operatorname{Spec} k$, the algebra $f_* \mathcal{O}_X = k[t]$ is not finite over $k$.

Going deeper

For a student

The Stein factorization reduces many questions about proper morphisms to two cases: finite morphisms and morphisms with connected fibres. It is used in the study of contractions in birational geometry. The contractions of the minimal model program are by definition proper morphisms $f$ with $f_* \mathcal{O}_X = \mathcal{O}_Y$, exactly the maps $f'$ of the factorization. The name comes from the analogous result of Karl Stein for proper holomorphic maps of complex spaces.