Zariski's lemma

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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., $d=0$ .

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

↑ Milne, Theorem 2.6
↑ Proof: it is enough to consider a maximal ideal ${\mathfrak {m}}$ . Let $A=k[t_{1},...,t_{n}]$ and $\phi :A\to A/{\mathfrak {m}}$ be the natural surjection. By the lemma, $A/{\mathfrak {m}}=k$ and then for any $f\in {\mathfrak {m}}$ ,
$f(\phi (t_{1}),\cdots ,\phi (t_{n}))=\phi (f(t_{1},\cdots ,t_{n}))=0$ ;
that is to say, $x=(\phi (t_{1}),\cdots ,\phi (t_{n}))$ is a zero of ${\mathfrak {m}}$ .
↑ Atiyah-MacDonald 1969, Ch 5. Exercise 25
↑ Atiyah–MacDonald 1969, Ch 5. Exercise 18
↑ Atiyah–MacDonald 1969, Proposition 7.9

References

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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