Formal fibers of prime ideals in polynomial rings

Summary

Let (R, m) be a Noetherian local domain of dimension n that is essentially finitely generated over a field and let Ȓ be the m-adic completion of R.Matsumura has shown that n-1 is the maximal height possible for prime ideals P of Ȓ such that P ⋂ R= (0). In this article we prove that ht P=n-1, for every prime ideal P of Ȓ that is maximal with respect to P ⋂ R= (0). We also present a related result concerning generic formal fibers of certainextensions of mixed polynomial-power series rings.

Let .(R,m) be a Noetherian local domain and let Ȓ be the m-adic completion of R. The generic formal fiber ring of R is the localization (R\(0))^-1 Ȓof Ȓ with respect to the multiplicatively closed set of nonzero elements of R. Let Gff.(R) denote the generic formal fiber ring of R. If R is essentially finitely generated over a field and dimR=n, then dim.(Gff(R)) = n-1 by the result of Matsumura [1988, Theorem 2] mentioned in the abstract. In this article we show every maximal ideal of Gff (R) has height n_1; equivalently, ht P =n-1, for every prime ideal P of Ȓ that is maximal with respect to P ⋂ R = .(0), a sharpening of Matsumura’s result.