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Comparison of graded and ungraded Cousin complexes

Published online by Cambridge University Press:  20 January 2009

Henrike Petzl  and
Rodney Y. Sharp
Henrike Petzl
Affiliation:
Pure Mathematics Section, School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield, S3 7RH
Rodney Y. Sharp
Affiliation:
Pure Mathematics Section, School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield, S3 7RH
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Abstract

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Let R = ⊕n∈zRn be a ℤ-graded commutative Noetherian ring and let M be a ℤ-graded R-module. S. Goto and K. Watanabe introduced the graded Cousin complex *C(M)* for M, a complex of graded R-modules. Also one can ignore the grading on M and construct the Cousin complex C(M)* for M, discussed in earlier papers by the second author. The main results in this paper are that *C(M)* can be considered as a subcomplex of C(M)* and that the resulting quotient complex is always exact. This sheds new light on the known facts that, when M is non-zero and finitely generated, C(M)* is exact if and only if *C(M)* is (and this is the case precisely when M is Cohen-Macaulay).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 2 , June 1998 , pp. 289 - 301
Copyright
Copyright © Edinburgh Mathematical Society 1998

References

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