Homology of perfect complexes

Luchezar L. Avramov, Ragnar Olaf Buchweitz, Srikanth B. Iyengar, Claudia Miller

Research output: Contribution to journalArticle

30 Scopus citations

Abstract

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring R stronger estimates are proved for Loewy lengths of modules of finite projective dimension.

Original languageEnglish (US)
Pages (from-to)1731-1781
Number of pages51
JournalAdvances in Mathematics
Volume223
Issue number5
DOIs
StatePublished - Mar 20 2010

Keywords

  • Bernstein-Gelfand-Gelfand equivalence
  • Conormal module
  • Koszul complex
  • Loewy length
  • New Intersection Theorem
  • Perfect complex
  • Triangulated category

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Avramov, L. L., Buchweitz, R. O., Iyengar, S. B., & Miller, C. (2010). Homology of perfect complexes. Advances in Mathematics, 223(5), 1731-1781. https://doi.org/10.1016/j.aim.2009.10.009