Hilbert's Nullstellensatz - Proof and Generalization

Proof and Generalization

There are many known proofs of the theorem. One proof is the following:

1. Note that it is enough to prove Zariski's lemma: a finitely generated algebra over a field k that is a field is a finite field extension of k.
2. Prove Zariski's lemma.

The proof of Step 1 is elementary. Step 2 is deeper. It follows, for example, from the Noether normalization lemma. See Zariski's lemma for more. Here we sketch the proof of Step 1. Let (k algebraically closed field), I an ideal of A and V the common zeros of I in . Clearly, . Let . Then for some prime ideal in A. Let and its maximal ideal. By Zariski's lemma, is a finite extension of k; thus, is k since k is algebraically closed. Let be the images of under the natural map . It follows that and .

The Nullstellensatz will also follow trivially once one systematically developed the theory of a Jacobson ring, a ring in which a radical ideal is an intersection of maximal ideals. This idea is due to Bourbaki. Let be a Jacobson ring. If is a finitely generated R-algebra, then is a Jacobson ring. Further, if is a maximal ideal, then is a maximal ideal of R, and is a finite extension field of .

Another generalization states that a faithfully flat morphism locally of finite type with X quasi-compact has a quasi-section, i.e. there exists affine and faithfully flat and quasi-finite over X together with an X-morphism