An important application of the lemma is a proof of the weak form of Hilbert's nullstellensatz: 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.
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. Thus, the lemma follows from the fact that a field is a Jacobson ring.
Two direct proofs, one of which is due to Zariski, are given in Atiyah–MacDonald. 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.
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.
- Milne, Theorem 2.6
- Proof: it is enough to consider a maximal ideal . Let and be the natural surjection. By the lemma, and then for any ,
- Atiyah-MacDonald 1969, Ch 5. Exercise 25
- Atiyah–MacDonald 1969, Ch 5. Exercise 18
- Atiyah–MacDonald 1969, Proposition 7.9
- M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN 0-201-40751-5
- James Milne, Algebraic Geometry
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