zbMATH — the first resource for mathematics

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 \(A\).

13C99 Theory of modules and ideals in commutative rings
13G05 Integral domains
13E99 Chain conditions, finiteness conditions in commutative ring theory
Full Text: DOI
[1] DOI: 10.1007/BF01214183 · Zbl 0343.13007 · doi:10.1007/BF01214183
[2] DOI: 10.2307/2373463 · Zbl 0228.13008 · doi:10.2307/2373463
[3] DOI: 10.1007/BF01796550 · Zbl 0443.13001 · doi:10.1007/BF01796550
[4] DOI: 10.2307/1970360 · Zbl 0157.08202 · doi:10.2307/1970360
[5] DOI: 10.1016/0021-8693(87)90129-3 · Zbl 0623.13009 · doi:10.1016/0021-8693(87)90129-3
[6] Japan J. Math. 8 pp 49– (1982)
[7] DOI: 10.1016/0022-4049(87)90063-6 · Zbl 0623.13008 · doi:10.1016/0022-4049(87)90063-6
[8] Ann. Sc. Norm. Sup. Pisa 24 pp 585– (1970)
[9] DOI: 10.1016/0021-8693(86)90059-1 · Zbl 0596.13002 · doi:10.1016/0021-8693(86)90059-1
[10] Nagoya Math. J. 94 pp 75– (1984) · Zbl 0526.13004 · doi:10.1017/S0027763000020845
[11] Comm. Algebra 11 (1983)
[12] B.U.M.I. (Supplemento) Algebra e Geometra, Suppl. 2 pp 243– (1980)
[13] Introduction to Commutative Algebra (1969) · Zbl 0175.03601
[14] Sur les anneaux de Mori, thèse (1976)
[15] C. R. Acad. Sc. Paris 280 pp 1571– (1975)
[16] Bull. Sc. Math. 95 pp 341– (1971)
[17] The divisor class group of a Krull domain (1973) · Zbl 0256.13001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.