New homological invariants for modules over local rings. II
[For part I see ibid. 110, 1-8 (1996; Zbl 0855.13008).] Let \(A\) be a Noetherian local ring and \(M\) a finitely generated \(A\)-module. In part I of this paper (loc. cit.), the author introduced new homological invariants \(\xi_A^i(M)\), where \(i = 0,1,2,\dots\;\). They coincide with Auslander's \(\delta\)-invariants if \(A\) is Gorenstein but \(\xi\)-invariants are defined even if \(A\) is not Cohen-Macaulay. If \(M\) is of finite projective dimension, then \(\xi\)-invariants coincide with the Betti numbers of \(M\). That is, \(\xi\)-invariants describe a part of \(M\) which is of finite projective dimension. In the present paper, the author studies some methods to compute \(\xi\)-invariants. First he gives some equations of \(\xi\)-invariants which are analogues of well-known equations of Betti numbers. Next he gives an analogue of a paper by \textit{J. Herzog} and \textit{A. Martsinkovsky} [Comment. Math. Helv. 68, 365-384 (1993; Zbl 0799.14016), lemma 2.3].
scientific article; zbMATH DE number 1526263