Weitere Bemerkungen zu einem problem der Schnittheorie und über ein Maβ von A. Seidenberg für die Imperfektheit

Generalized fractions, Buchsbaum modules and generalized Cohen-Macaulay modules

“…We can now combine (2-3) and (2)(3)(4) (ii) Entirely similar comments apply in the case of quasi-Buchsbaum modules and (2-5) (ii).…”

“…(Here, H^A) is the ith local cohomology module of A with respect to m; it should be noted that, by ([8], 3-5), H^(A) £ U(A)j l ll 1 A.) These results lead to a characterization of generalized Cohen-Macaulay local rings in terms of modules of generalized fractions, and also to a description of the invariant I(A) of a Buchsbaum local ring A ( [4], Satz 2) in terms of the complex C{°ll{A),A). 1.…”

“…Our next aim is to show that, when the conditions of the first part of (2-3) are satisfied, each local cohomology module H x m (M) (for 0 ^ i ^ n-1) actually 'appears' in the complex C? (y, M) in the sense that it is isomorphic to ker e i /im e*- 1 (1)(2)(3)(4)(5) to see that J ? » k < 5 S i < i )…”

Idealizations of maximal Buchsbaum modules over a Buchsbaum ring

“…is surjective for all p ^ d) and H^i^) = (0) for p = 0, i, ..., d-1. Considering the exact sequence of A -modules 0 -t f 4 -> j ; _ 1 _^t _ 1 _ » o (17) we obtain a commutative diagram with an exact bottom row for each p as follows: 4/m,…”

On Buchsbaum rings obtained by gluing

“…Then it is well-known that A is a Buchsbaum ring (c.f. [6], p. 469, Beispiel). One may follow this fact by using Theorem (2.5), since A is obtained from B by gluing over the maximal ideal.…”