nLab Hilbert's basis theorem

Classical affine algebraic varieties appear as sets of zeros of a set S={P α|αA}S = \{P_\alpha|\alpha\in A\} of polynomials in affine nn-dimensional space 𝔸 k n\mathbb{A}^n_k over a field kk. The coordinate algebra of 𝔸 k n\mathbb{A}^n_k is the algebra of polynomials in nn variables, k[x 1,,x n]k[x_1,\ldots,x_n], and the coordinate algebra of an affine algebraic variety is Rk[x 1,,x n]/IR \coloneqq k[x_1,\ldots,x_n]/I where ISI \coloneqq \langle S\rangle is the ideal generated by SS.

The Hilbert basis theorem (HBT) asserts that this ideal II is finitely generated; and consequently RR 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 RR is noetherian. For the case of RR a field, this is the case in the previous paragraphs.

More at Noetherian ring.


The theorem vastly generalises and subsumes Paul Gordan?‘s work on invariant theory, albeit in a non-constructive way. Emmy Noether wrote a short paper in 1920 that sidestepped the use of the HBT to construct a basis for, and so implying the finite generation of, a certain ring of invariants attached to any finite group.


Last revised on May 19, 2020 at 19:35:15. See the history of this page for a list of all contributions to it.