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The MacRae invariant. (English) Zbl 0544.13012
Commutative algebra, Symp. Durham 1981, Lond. Math. Soc. Lect. Note Ser. 72, 121-128 (1982).
[For the entire collection see Zbl 0489.00008.]
The author gives a new construction of the ideal G(M), associated by R. E. MacRae [J. Algebra 2, 153-169 (1965; Zbl 0196.310)] to every module M of finite projective dimension over the local ring R, by using the K-theory of complexes developed in his joint work with R. Fossum and B. Iversen [”A characteristic class in algebraic K- theory” (to appear); see also Preprint Ser., No.29, Aarhus Univ. (1979; Zbl 0405.18008)]. He also gives an application of the MacRae invariant to Serre’s multiplicity theory by proving the following theorem: Let M and N be finitely-generated R-modules, with \(\dim(M\otimes_ RN)=0, pd M<\infty\), and either \(\text{grad}e M=1\) or \(\dim N=1\). Then \(\dim M+\dim N\leq \dim R, \chi(M,N)\geq 0,\) and the first inequality becomes an equality if and only if the second one is strict. - It should be noted that a recent example of S. Dutta, M. Hochster, and J. MacLaughlin (to appear) shows that the last assertion of the theorem does not hold in general, over a non-regular ring R.
Reviewer: L.L.Avramov

MSC:
13H15 Multiplicity theory and related topics
13D05 Homological dimension and commutative rings
13D15 Grothendieck groups, \(K\)-theory and commutative rings