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Abstract
Let
T
be a complete local ring of Krull dimension at least one, and let
C 1 , C 2 , … , 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 , … , Q m , and such that, for
every
i
= 1 , 2 , … , m , the set of
maximal elements of
{ P
∈ Spec ( T ) ∣ P
∩
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