Approximation properties for some non-Noetherian local rings

Hans Schoutens

Vol. 131 (1988), No. 2, 331–359

DOI: 10.2140/pjm.1988.131.331

Abstract

In this paper we study Artin approximation in power series rings in several variables over complete rank-one valuation rings. In particular we prove that the completion of the algebraic elements has the approximation property over the ring of algebraic power series.

Moreover, for an important class of complete rank-one valuation rings, e.g. the ring of complex p-adic integers, we prove that the ring of algebraic power series is equal to the henselisation of the polynomial ring and that each algebraic power series has coefficients lying in a finitely generated R-algebra, where R is discrete valuation rings.

Mathematical Subject Classification 2000

Primary: 14B12

Secondary: 13J05

Milestones

Received: 9 September 1986

Published: 1 February 1988