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A Lipman’s type construction, glueings and complete integral closure. (English) Zbl 0648.13004
Let A be a Noetherian domain such that its integral closure \(\bar A\) is a finitely graded A-module. In this paper we provide an algorithmic construction to obtain \(\bar A\) in terms of prime ideals of \(A\): more precisely \(\bar A\) can be reached by an increasing sequence of overrings of A: \((*)\quad A_ 0\subset A_ 1\subset...\subset A_ m=\bar A.\)
The sequence is such that \(A_ i\) is obtained from \(A_{i+1}\) by a glueing of primary ideals of \(A_{i+1}\) for each i \((i=0,...,m-1)\). As a matter of fact the result holds in a more general situation which turns out to be its natural context, that is when A is (just) a Mori domain (i.e. a domain such that the ascending chain condition holds for integral divisorial ideals): in this case the above sequence stops at the complete integral closure of \(A\).

MSC:
13C99 Theory of modules and ideals in commutative rings
13G05 Integral domains
13E99 Chain conditions, finiteness conditions in commutative ring theory
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