n-generator ideals in Prüfer domains
In this paper the author proves the following two theorems: Theorem 1. For every integer \(n\geq 1\), there is a Prüfer domain R of Krull dimension n and an ideal \(I_ n\) of R with \(\mu (I_ n)=n+1.\) - Theorem 2. There is a Prüfer domain R such that for every integer \(n>0\), there is an ideal \(I_ n\) of R with \(\mu (I_ n)=n+1.\) This shows that a theorem of R. Heitman (which says that in the situation of theorem 1, or any finitely generated ideal, \(\mu (I)\leq n+1))\) is the best possible. The basic construction depends on the following observation due to \textit{A. Dress} [Arch. Math. 16, 388-392 (1965; Zbl 0137.402)]. If R is any domain such that for any \(a,b\in R\) with \(a^ 2+b^ 2\neq 0, a^ 2(a^ 2+b^ 2)^{-1}\) and \(ab(a^ 2+b^ 2)^{- 1}\) belong to R then R is a Prüfer domain. Further, if \(\sqrt{-1}\) is in the quotient field of R, then R is a field: so by starting with any domain, one can attach elements as above to get eventually a Prüfer domain. The author considers a suitable finitely generated algebra \(A_ n\) over the reals with the property that the underlying real algebraic variety of \(A_ n\) is \({\mathbb{P}}^ n\), the real projective n-space. The ideal in question would then correspond to the canonical line bundle of \({\mathbb{P}}^ n\). One gets the ring R by the above construction. The lemma which allows comparison of \(A_ n\) and R is the following: (Lemma 9) Let \(A\subset B=A[a^ 2(a^ 2+b^ 2)^{-1},ab(a^ 2+b^ 2)^{-1}],\) in the above situation. Then (though B is not integral over A) the corresponding map on real algebraic varieties is proper. Now using Stiefel-Whitney classes of the canonical bundle of \({\mathbb{P}}^ n\) the author obtains necessary obstructions for the number of generators of the ideal to be small. There are more theorems all of which indicate that Prüfer domains behave like arbitrary commutative rings and not any better.
scientific article; zbMATH DE number 3869483