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Homology and the Koszul complex

Published online by Cambridge University Press:  18 May 2009

D. J. Moore
D. J. Moore
Affiliation:
The University, Sheffield
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Let R be a commutative ring with an identity element, E a (unitary) R-module, and x1, x2, …, xs elements of R. In these circumstances it is possible to form the Koszul complex† K(x1, x2, …, xs|E) of E with respect to x1, x2,…, xs and to investigate the implications, for E and xl, x2, …, xs, if certain of the homology modules of this complex vanish. This was first undertaken by M. Auslander and D. A. Buchsbaum [1]. Among the many results they obtain, the following [1, Proposition 2.8, p. 632] is of particular interest in connection with the present paper:

If R is Noetherian, E is finitely generated, and x1x2,…, xs belong to the Jacobson radical of R, then the statements

(a) x1, x2,…, xsis an R-sequence on E,

(b) HpK(x1, x2,…,xs⃒E) = O for all p > 0,

(c) H1K(x1, x2,…., xs,⃒E) = 0,

are all equivalent.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 12 , Issue 2 , September 1971 , pp. 118 - 135
Copyright
Copyright © Glasgow Mathematical Journal Trust 1971

References

REFERENCES

1.Auslander, M. and Buchsbaum, D. A., Codimension and multiplicity, Ann. of Math. 68 (1958), 625–657.CrossRefGoogle Scholar
2.Northcott, D. G., Generalized R-sequences; to appear.Google Scholar
3.Northcott, D. G., Lessons on rings, modules and multiplicities (Cambridge University Press, 1968).CrossRefGoogle Scholar