Zariski's lemma

In algebra, Zariski's lemma, introduced by Oscar Zariski (1947), states that if K is a finitely generated algebra over a field k and if K is a field, then K is a finite field extension of k.

An important application of the lemma is a proof of the weak form of Hilbert's nullstellensatz:[1] if I is a proper ideal of (k algebraically closed field), then I has a zero; i.e., there is a point x in such that for all f in I.[2]

The lemma may also be understood from the following perspective. In general, a ring R is a Jacobson ring if and only if every finitely generated R-algebra that is a field is finite over R.[3] Thus, the lemma follows from the fact that a field is a Jacobson ring.

Proof

Two direct proofs, one of which is due to Zariski, are given in Atiyah–MacDonald.[4][5] The lemma is also a consequence of the Noether normalization lemma. Indeed, by the normalization lemma, K is a finite module over the polynomial ring where are algebraically independent over k. But since K has Krull dimension zero, the polynomial ring must have dimension zero; i.e., . For Zariski's original proof, see the original paper.[6]

In fact, the lemma is a special case of the general formula for a finitely generated k-algebra A that is an integral domain, which is also a consequence of the normalization lemma.

Notes

1. Milne, Theorem 2.6
2. Proof: it is enough to consider a maximal ideal . Let and be the natural surjection. By the lemma, and then for any ,
;
that is to say, is a zero of .
3. Atiyah-MacDonald 1969, Ch 5. Exercise 25
4. Atiyah–MacDonald 1969, Ch 5. Exercise 18
5. Atiyah–MacDonald 1969, Proposition 7.9
6. http://projecteuclid.org/euclid.bams/1183510605