TY - JOUR

T1 - Homology of perfect complexes

AU - Avramov, Luchezar L.

AU - Buchweitz, Ragnar Olaf

AU - Iyengar, Srikanth B.

AU - Miller, Claudia

N1 - Funding Information:
✩ Research partly supported by NSF grants DMS 0201904 and DMS 0803082 (L.L.A.); NSERC grant 3-642-114-80 (R.-O.B.); NSF grants DMS 0602498 and DMS 0903493 (S.B.I.); NSF grant DMS 0434528 and NSA grant H98230-06-1-0035 (C.M.). * Corresponding author. E-mail addresses: avramov@math.unl.edu (L.L. Avramov), ragnar@utsc.utoronto.ca (R.-O. Buchweitz), iyengar@math.unl.edu (S.B. Iyengar), clamille@syr.edu (C. Miller).

PY - 2010/3/20

Y1 - 2010/3/20

N2 - 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.

AB - 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.

KW - Bernstein-Gelfand-Gelfand equivalence

KW - Conormal module

KW - Koszul complex

KW - Loewy length

KW - New Intersection Theorem

KW - Perfect complex

KW - Triangulated category

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U2 - 10.1016/j.aim.2009.10.009

DO - 10.1016/j.aim.2009.10.009

M3 - Article

AN - SCOPUS:75149177047

VL - 223

SP - 1731

EP - 1781

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 5

ER -