nLab 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 (HBT) 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.

More at Noetherian ring.

Applications

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.

References

• Emmy Noether, Der Endlichkeitsatz der Invarianten endlicher Gruppen, Mathematische Annalen, vol. 77, 1920, pages 89–92, (English translation by Colin McLarty, at arXiv:1503.07849)