## Generically finite morphisms

I’m going to take a quick break from representation theory and discuss my solution of Exercise II.3.7 from Hartshorne’s book, which is pretty technical but I think has an interesting proof. For one thing, it applies Noether normalization to the generic fiber of a morphism, which is a very non-classical sort of move.

Hartshorne defines a morphism of irreducible schemes to be *generically finite* provided that the set-theoretic preimage of the generic point is finite. The name for this term might suggest another definition, namely that there exists a nonempty open subscheme over which is finite. The exercise is that under some assumptions, namely that and are integral and is dominant and of finite type, Hartshorne’s definition implies this one

First let us assume and are affine and prove that the function field of is a finite extension of the function field of . The key here is to consider the scheme-theoretic fiber , which is the spectrum of . Note that if we write , we can identify the localization with , whence is a domain. Now by the finite type assumption is a finitely generated -algebra, so Noether normalization provides an injection which makes into a finitely generated module for . Geometrically this corresponds to a surjection , and since is homeomorphic to the finite space this forces . But this means is a finite-dimensional -algebra, and a finite-dimensional domain over a field is a field. Thus is actually the fraction field of and finite over .

To finish the affine case, choose a finite set of generators for over , which by the previous paragraph satisfy some polynomials with coefficients in . By clearing denominators, we see that these polynomials can actually be taken to have coefficients in . Write for the product of the leading coefficients of these polynomials, so the localization is integral over . But is a finitely generated -algebra, hence a finitely generated -module. Thus we can take to be the desired finite morphism.

Now we reduce from the general situation to the case that and are affine: choose an open affine and use the finite type hypothesis to cover with finitely many open affines . From the proof of the affine case, we can find principal open sets such that each is finite. In particular, is affine, so let us write . Also each is principal in , hence affine, say . By the usual trick we can find such that for each and for all . Now since the are finitely generated modules for , each satisfies a monic polynomial with coefficients in , which may be taken to have nonzero constant term . Put , so any prime ideal of which contains also contains , hence . Thus , from which it follows that , and is the desired finite morphism.