Stein factorization

In algebraic geometry, the Stein factorization, introduced by Karl Stein (1956) for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points.

One version for schemes states the following:(EGA, III.4.3.1)

Let X be a scheme, S a locally noetherian scheme and ${\displaystyle f:X\to S}$ a proper morphism. Then one can write

where ${\displaystyle g:S'\to S}$ is a finite morphism and ${\displaystyle f':X\to S'}$ is a proper morphism so that ${\displaystyle f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}}$.

...
Wikipedia