Consistency (suspension)

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: 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.

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.

Proof

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 $k[x_1,\dots ,x_d]$ where $x_1,\dots ,x_d$ 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 $\dim A={\mathrm{tr.deg}}_{k}A$ for a finitely generated k-algebra A that is an integral domain, which is also a consequence of the normalization lemma.

Notes

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References

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