Homology of perfect complexes |
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Authors: | Luchezar L Avramov Claudia Miller |
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Institution: | a Department of Mathematics, University of Nebraska, Lincoln, NE 68588, USA b Department of Computer and Mathematical Sciences, University of Toronto Scarborough, Toronto, ON M1A 1C4, Canada c Mathematics Department, Syracuse University, Syracuse, NY 13244, USA |
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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. |
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Keywords: | 13D05 13H10 13D40 13B10 13D07 13D25 18E30 |
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