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The Evans-Griffith syzygy theorem and Bass numbers


Author: Winfried Bruns
Journal: Proc. Amer. Math. Soc. 115 (1992), 939-946
MSC: Primary 13D02; Secondary 13C14, 13C15, 13D25
DOI: https://doi.org/10.1090/S0002-9939-1992-1088439-0
MathSciNet review: 1088439
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Abstract: Let $ (R,\mathfrak{m})$ be a Noetherian local ring containing a field. The syzygy theorem of Evans and Griffith (see The syzygy problem, Ann. of Math. (2) 114 (1981), 323-353) says that a nonfree $ m$th syzygy module $ M$ over $ R$ which has finite projective dimension must have rank $ \geq m$. This theorem is an assertion about the ranks of the homomorphisms in certain acyclic complexes. It is the aim of this paper to demonstrate that the condition of acyclicity can be relaxed in a natural way. We shall use the generalization thus obtained to show that the Bass numbers of a module satisfy restrictions analogous to those which the syzygy theorem imposes on Betti numbers.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1088439-0
Article copyright: © Copyright 1992 American Mathematical Society

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