Vol. 91, No. 1, 1980

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Asymptotic prime divisors and going down

Stephen Joseph McAdam

Vol. 91 (1980), No. 1, 179–186

Let I be an ideal in a commutative Noetherian domain R, and let Î be the integral closure of I. It is known that the sequences of sets of primes Ass(R∕In) and Ass(R∕În) both eventually become constants, those constants denoted A(I) and Â(I) respectively. The main result of this paper is that if T is an integral extension of a local domain (R,M), and if I is an ideal of R such that T∕IT contains a height 0 maximal ideal, then M Â(I). This fact is then used to study when Â(P) = {P} for P a prime of R. (This is a variation of the question when does Pn = P(n) for all large n?) It is shown that if Â(P) = {P}, then “going down to P” holds. Finally, the main argument is used to produce an example of an n-dimensional local domain, (R,M) such that for any P SpecR −{0}, and any m 2, M Ass(R∕Pm). Also the analytic spread of any such P is n.

Mathematical Subject Classification 2000
Primary: 13E05
Secondary: 13B20, 13A17
Received: 30 November 1979
Published: 1 November 1980
Stephen Joseph McAdam