Stein factorisation

The factorisation separates the two ways a proper morphism can fail to be an isomorphism: \(g\) contracts connected fibres, while \(h\) is finite. Thus a proper morphism factors into a contraction followed by a finite cover.

\(f_* \OO_X\) is a coherent sheaf of \(\OO_Y\)-algebras because \(f\) is proper, and it is finite because coherent, which is where properness enters; this is why there is no Stein factorisation for a general morphism. Since \(g\) is surjective with connected fibres, the connected components of the fibre \(X_y\) are the fibres of \(g\) over the points of the finite set \(h^{-1}(y)\), so they correspond bijectively to the points of \(h^{-1}(y)\). This count is at most the degree of \(h\) over \(y\), the length \(\dim_{\kappa(y)} (f_* \OO_X \otimes \kappa(y))\), and can be smaller: for \(f = h \colon \AA^1_\CC \to \AA^1_\CC\), \(t \mapsto t^2\), which is finite with \(g = \id\), the fibre over \(0\) is the single point \(\Spec \CC[t]/(t^2)\) of length \(2\).

The immediate corollary is the connectedness statement in Zariski’s main theorem: if \(f_* \OO_X = \OO_Y\) then \(Y' = Y\), so \(h\) is the identity and the fibres of \(f\) are connected. In the other direction, for a proper morphism of varieties with \(Y\) normal and \(f\) birational, the Stein factorisation is what proves the fibres connected without a separate argument.