Generalized fractions, Buchsbaum modules and generalized Cohen-Macaulay modules

Abstract

Let A be a (commutative Noetherian) local ring (with identity) having maximal ideal m and positive dimension. This note is concerned with, among other things, a complex of A -modules which was studied in [8] and which involves modules of generalized fractions derived from A and subsets of systems of parameters for A; in ([8], 3·5), the complex was shown to have connections with local cohomology. The complex is described as follows.

Characterizations of filter regular sequences and unconditioned strong d-sequences

“…This result is a slight generalization of the Exactness Theorem [5, 3.1]. In [9], Sharp and the third author proved, in certain situation, for a finitely generated module N over a (Noetherian) local ring R having maximal ideal m, that the i-th local cohomology module H^(N) is isomorphic to the i-th homology module of a certain complex of modules of generalized fractions. Our characterization of α-filter regular sequences, in this paper, yields improved forms of the above theorem and the results (M) may be viewed as the i-th homology module of the complex C(W, M) of A-modules which involves modules of generalized fractions derived from M and an α-filter regular sequence, and that (ii) whenever #i, , x n is an α-filter regular sequence on M, then for all 0 < i < n -1.…”

“…This result is a slight generalization of the Exactness Theorem [5, 3.1]. In [9], Sharp and the third author proved, in certain situation, for a finitely generated module N over a (Noetherian) local ring R having maximal ideal m, that the i-th local cohomology module H^(N) is isomorphic to the i-th homology module of a certain complex of modules of generalized fractions. Our characterization of α-filter regular sequences, in this paper, yields improved forms of the above theorem and the results 38 K. KHASHYARMANESH, SH.…”

Frobenius test exponents for parameter ideals in generalized Cohen–Macaulay local rings

“…The latter result is relevant to this paper, although in §4 we provide a refinement for use in the case where R has prime characteristic p. Some of the applications of modules of generalized fractions can be found in [22], [23], [24], [15], [5] and [12]. This paper provides some more.…”

“…Some of the applications of modules of generalized fractions can be found in [22], [23], [24], [15], [5] and [12]. This paper provides some more.…”

Generalized fractions and Hughes' gradetheoretic analogue of the Cousin complex

“…Now the present first author and H. Zakeri introduced a concept of module of generalized fractions in commutative algebra in [23]. Since that paper appeared, there have been several further papers, such as [3], [4], [5], [6], [8], [9], [14], [15], [18], [22], [24], [25], [26], [27] and [28], which have shown that this concept has many interactions with topics of recent and current interest in commutative algebra, especially in commutative Noetherian ring theory. In particular, there are strong links between Cousin complexes and these modules of generalized fractions: it was shown in [18, (3.4)] that, for an i4-module M such that Ass(M) has only finitely many minimal members and a filtration & of Spec(i4) which admits M, the Cousin complex C(SF, M) mentioned above is actually isomorphic to a complex of modules of generalized fractions in the sense of [23].…”