Abstract
Let n be a non-negative integer, R a commutative Noetherian ring with dim(R) ≤ n + 2, a an ideal of R, and X an arbitrary R-module...
Highlights
We adopt throughout the following notation: let R denote a commutative Noetherian ring with non-zero identity, a and b ideals of R, M a finite R-module, X an arbitrary R-module which is not necessarily finite, and n a non-negative integer
Recall that X is said to be an a-weakly cofinite R-module if X is an a-torsion R-module and the set of associated prime ideals of any quotient module of ExtiR
By putting n = 1 in Corollary 3.2, we provide an affirmative answer to Question 1.3 for the case that R is a semi-local ring with dim(R) ≤ 3
Summary
We adopt throughout the following notation: let R denote a commutative Noetherian ring with non-zero identity, a and b ideals of R, M a finite (i.e., finitely generated) R-module, X an arbitrary R-module which is not necessarily finite, and n a non-negative integer. Hartshorne in [14, Proposition 7.6 and Corollary 7.7] showed that the answer to these questions is yes if R is a complete regular local ring and a is a prime ideal of R with dim. Huneke and Koh in [17, Theorem 4.1] and Delfino in [10, Theorem 3] extended Hartshorne’s result [14, Corollary 7.7] and provided affirmative answers to Questions 1.2 and 1.3 in more general local rings R and one-dimensional ideals a. As generalizations of Melkersson’s results [21, Theorems 7.4 and 7.10], we show that the answer to Questions 1.4–1.6 is yes if dim(R) ≤ n + 2. Recall that X is said to be an a-weakly cofinite R-module if X is an a-torsion R-module and the set of associated prime ideals of any quotient module of ExtiR is finite for all i (see [12, Definition 2.1] and [13, Definition 2.4]).
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