Hilbert subalgebras of finitely generated algebras, II

Abstract

Recall that a Hilbert ring (also called a Jacobson ring) is a commutative ring in which every prime ideal is an intersection of maximal ideals. Such rings were considered by Goldman [4] and Krull [6] to obtain an abstract formulation of Hilbert’s Nullstellensatz. They each proved the basic theorem that any finitely generated algebra over a Hilbert ring is again a Hilbert ring. Krull also proved that every co2mtabZy generated algebra over an uncountable field is a Hilbert ring. (This was subsequently generalized by Lang [7] and Gilmer [3].) We consider here an integral domain R which is a &algebra of a finitely generated algebra over a field F, and ask whether R must be a Hilbert ring. We prove that if P is a prime ideal of R, with t.d. R/P + rank P = t.d. R, then R/P is contained in a finitely generated F-algebra (t.d. = transcendence degree over F). Hence P is an intersection of maximals. It follows that if t.d. R 3 and F is countable, then R need not be Hilbert and, indeed, we give a construction showing that R contains a non-Hilbert subalgebra. When F is uncountable, Krull’s theorem shows that R is Hilbert, regardless of its transcendence degree. (For, R has countable dimension as an F-vector space.) We give analogous results for algebras over a discrete valuation ring, and, in the Appendix, establish when all subalgebras of a finitely generated algebra are finitely generated. The author would like to thank his colleague Lance Small for a number of constructive conversations, and George Bergman for some illuminating comments on the examples. In addition he thanks the referee for pointing out the application of Nagata’s work on ideal transforms to the proof of Theorem 1.