Completions of reduced local rings with prescribed minimal prime ideals

Loepp, Susan; Perpetua, Byron

doi:10.2140/involve.2016.9.101

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Completions of reduced local rings with prescribed minimal prime ideals

Susan Loepp and Byron Perpetua

Vol. 9 (2016), No. 1, 101–118

DOI: 10.2140/involve.2016.9.101

Abstract

Let $T$ be a complete local ring of Krull dimension at least one, and let $C_{1}, C_{2}, \dots, C_{m}$ each be countable sets of prime ideals of $T$ . We find necessary and sufficient conditions for $T$ to be the completion of a reduced local ring $A$ such that $A$ has exactly $m$ minimal prime ideals $Q_{1}, Q_{2}, \dots, Q_{m}$ , and such that, for every $i = 1, 2, \dots, m$ , the set of maximal elements of ${P \in Spec (T) ∣ P \cap A = Q_{i}}$ is the set $C_{i}$ .

Keywords

completions of local rings, minimal prime ideals

Mathematical Subject Classification 2010

Primary: 13B35, 13F25, 13J05, 13J10

Milestones

Received: 21 August 2014

Revised: 27 October 2014

Accepted: 9 January 2015

Published: 17 December 2015

Communicated by Scott T. Chapman

Authors

Susan Loepp
Department of Mathematics and Statistics Williams College 18 Hoxsey Street Bronfman Science Center Williamstown, MA 01267 United States

Byron Perpetua
Department of Mathematics and Statistics Williams College c/o Susan Loepp, Bronfman Science Center 18 Hoxsey Street Williamstown,MA 01267 United States

Vol. 9, No. 1, 2016