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“School of Mathematics”

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Paper   IPM / M / 16301
School of Mathematics
  Title:   on a question of Hartshorne
  Author(s):  Kamal Bahmanpour
  Status:   Published
  Journal: Collect. Math.
  Year:  2020
  Supported by:  IPM
  Abstract:
Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules HiI(M) are I-cofinite, for all finitely generated R-modules M and all i ∈ \mathbb N0, if and only if the R-modules HiI(R) are I-cofinite with dimension not exceeding 1, for all integers i ≥ 2; in addition, under these equivalent conditions it is shown that, for each minimal prime ideal \mathfrak p over I, either height\mathfrak p ≤ 1 or dimR/\mathfrak p ≤ 1, and the prime spectrum of the I-transform R-algebra DI(R) equipped with its Zariski topology is Noetherian. Also, by constructing an example we show that under the same equivalent conditions in general the ring DI(R) need not be Noetherian. Furthermore, in the case that R is a local ring, it is shown that the R-modules HiI(M) are I-cofinite, for all finitely generated R-modules M and all i ∈ \mathbb N0, if and only if for each minimal prime ideal \mathfrak P of R, either dimR/(IR+\mathfrak P) ≤ 1 or HiIR(R/\mathfrak P)=0, for all integers i ≥ 2. Finally, it is shown that if R is a semi-local ring and the R-modules HiI(M) are I-cofinite, for all finitely generated R-modules M and all i ∈ \mathbb N0, then the category of all I-cofinite modules forms an Abelian subcategory of the category of all R-modules.

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