Zariski's Main Theorem - Zariski's Main Theorem For Quasifinite Morphisms

Zariski's Main Theorem For Quasifinite Morphisms

In EGA III, Grothendieck calls the following statement which does not involve connectedness a "Main theorem" of Zariski Grothendieck (1961, Théorème 4.4.3):

If f:X→Y is a quasi-projective morphism of Noetherian schemes then the set of points that are isolated in their fiber is open in X. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over Y.

In EGA IV, Grothendieck observed that the last statement could be deduced from a more general theorem about the structure of quasi-finite morphisms, and the latter is often referred to as the "Zariski's main theorem in the form of Grothendieck". It is well known that open immersions and finite morphisms are quasi-finite. Grothendieck proved that under the hypothesis of separatedness all quasi-finite morphisms are compositions of such Grothendieck (1966, Théorème 8.12.6):

if Y is a quasi-compact separated scheme and is a separated, quasi-finite, finitely presented morphism then there is a factorization into, where the first map is an open immersion and the second one is finite.

The relation between this theorem about quasi-finite morphisms and Théorème 4.4.3 of EGA III is that if f:X→Y is a projective morphism of varieties, then the set of points that are isolated in their fiber is quasifinite over Y. Then structure theorem for quasi-finite morphisms applies and yields the desired result.