Zariski's lemma

In algebra, Zariski's lemma, introduced by Oscar Zariski, 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 \( k[t_1, ..., t_n] \) (k algebraically closed field), then I has a zero; i.e., there is a point x in k^n such that f(x) = 0 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 \( k[x_1, \ldots , x_d] \) where \( x_1, \ldots , x_d \) are algebraically independent over k. But since K has Krull dimension zero, the polynomial ring must have dimension zero; i.e., d=0. For Zariski's original proof, see the original paper.[6]

In fact, the lemma is a special case of the general formula \( \dim A = \operatorname{tr.deg}_k A \) for a finitely generated k-algebra A that is an integral domain, which is also a consequence of the normalization lemma.

Notes

References

M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN 0-201-40751-5
James Milne, Algebraic Geometry