Homology and the Koszul complex
Glasgow mathematical journal, Tome 12 (1971) no. 2, pp. 118-135
Voir la notice de l'article provenant de la source Cambridge University Press
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.
Moore, D. J. Homology and the Koszul complex. Glasgow mathematical journal, Tome 12 (1971) no. 2, pp. 118-135. doi: 10.1017/S0017089500001233
@article{10_1017_S0017089500001233,
author = {Moore, D. J.},
title = {Homology and the {Koszul} complex},
journal = {Glasgow mathematical journal},
pages = {118--135},
year = {1971},
volume = {12},
number = {2},
doi = {10.1017/S0017089500001233},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500001233/}
}
[1] 1.Auslander, M. and Buchsbaum, D. A., Codimension and multiplicity, Ann. of Math. 68 (1958), 625–657. Google Scholar | DOI
[2] 2.Northcott, D. G., Generalized R-sequences; to appear. Google Scholar
[3] 3.Northcott, D. G., Lessons on rings, modules and multiplicities (Cambridge University Press, 1968). Google Scholar | DOI
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