Hilbert's basis theorem

Classical *affine algebraic varieties* appear as sets of zeros of a set $S = \{P_\alpha|\alpha\in A\}$ of polynomials in affine $n$-dimensional space $\mathbb{A}^n_k$ over a field $k$. The coordinate algebra of $\mathbb{A}^n_k$ is the algebra of polynomials in $n$ variables, $k[x_1,\ldots,x_n]$, and the coordinate algebra of an affine algebraic variety is $R \coloneqq k[x_1,\ldots,x_n]/I$ where $I \coloneqq \langle S\rangle$ is the ideal generated by $S$.

The **Hilbert basis theorem** asserts that this ideal $I$ is finitely generated; and consequently $R$ is a noetherian ring. For a proof see standard textbooks on commutative algebra or algebraic geometry (e.g. Atiyah, MacDonald); there is also a proof on wikipedia.

More generally, a finitely generated commutative algebra over a commutative noetherian ring $R$ is noetherian. For the case of $R$ a field, this is the case in the previous paragraphs.

Revised on October 17, 2009 13:55:57
by Akhil Mathew
(173.63.208.80)