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Paper   IPM / M / 8720
School of Mathematics
  Title:   Asymptotic behaviour of certain sets of associated prime ideals of EXT-modules
  Author(s):  K. Khashyarmanesh (Joint with F. Khosh-Ahang)
  Status:   To Appear
  Journal: Manuscripta Math.
  Supported by:  IPM
  Abstract:
Let R be a commutative Noetherian ring, \fraka be an ideal of R and M be a finitely generated R-module, Melkersson and Schenzel asked whether the set AssRExtiR(R/\frakaj,M) becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s\geqslant 0 and n\geqslant 0 such that dim(SuppRHi\fraka(M)) \leqslant s for all i with i < n, then

    (i) the set ∪j > 0 SuppRExtiR(R/\frakaj, M))\geqslant s is finite for all i with i < n, and
    (ii) the set ∪j > 0 AssRExtiR(R/\frakaj, M))\geqslant s is finite for all i with i \leqslant n, where for a subset T of Spec(R), we set (T)\geqslant s: {\frakpT|dim(R/\frakp)\geqslant s}.
Also, among other things, we show that if n\geqslant 0, R is semi-local and SuppRHi\fraka(M) is finite for all i with i < n, then ∪j > 0AssRExtiR(R/\frakaj, M) is finite for all i with i\leqslant n.

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