Acyclicity over local rings with radical cube zero

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Abstract

This paper studies infinite acyclic complexes of finitely generated free modules over a commutative noetherian local ring (R, m) with m3 = 0. Conclusive results are obtained on the growth of the ranks of the modules in acyclic complexes, and new sufficient conditions are given for total acyclicity. Results are also obtained on the structure of rings that are not Gorenstein and admit acyclic complexes; part of this structure is exhibited by every ring R that admits a non-free finitely generated module M with Extn R(M, R) = 0 for a few n > 0.

Original languageEnglish
Pages (from-to)1439-1454
Number of pages16
JournalIllinois Journal of Mathematics
Volume51
Issue number4
DOIs
StatePublished - 2008

Keywords

  • Betti numbers
  • Complete resolutions
  • Infinite syzygies
  • Infinite syzygy
  • Minimal free resolutions
  • Totally acyclic complexes
  • Totally reflexive modules

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