Homology of perfect complexes

By L. Avramov, R. Buchweitz, S. Iyengar, C. Miller. 2006

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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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3fb25ecc8a2d4ce7d8bb3d56e49028bbb01257ba

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In Geometry and Topology

Citations in the corpus

Orlov spectra: bounds and gaps (2010, background): “Since (d) ∼= [2], we can assume that we have N(i) for any i ∈ Z.”
Lower bounds for Auslander's representation dimension (2009, methodology): “Avramov and Iyengar [5], using techniques from [ 4 ], have announced that the dimension of the stable derived category of a complete intersection local ring R is at least the codimension of R minus one.”
The radius of a subcategory of modules (2011, background): “In [6] the thickening thickX of a given subcategory X of a triangulated category is defined.”
Class and rank of differential modules (2006, background): “ontractible. Simple examples are given in Section 1. Our migration to the category of diﬀerential modules from the more familiar environment of complexes was motivated in part by the investigation in [4] of “levels” in derived categories. The treatment of diﬀerential modules presented here is intent”
Representations of Algebras and Related Topics (2011, methodology): “triangulated categories. This setup allows us to move from one category to another without too much eﬀort. Then the game plan, which is inspired by the work of Avramov, Buchweitz, Iyengar, and Miller [1], is as follows: (i) Reduce from a ﬁnite group to its elementary abelian subgroups. (ii) Use a Ko”
Stratifying triangulated categories (2009, background): “nts. 10 DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE (1) For any X∈Tand any localizing subcategory Sof T, one has X∈S ⇐⇒ΓpX∈Sfor each p∈SpecR. (2) For any X∈T, one has X∈Loc T({ΓpX|p∈SpecR}). (3) For any X∈Tand any specialization closed subset Vof SpecR, one has ΓVX∈Loc T({ΓpX|p∈V}). (4) For”
Stratifying modular representations of finite groups (2008, methodology): “Leitfaden The passage from S to kE is modelled on the work of Avramov, Buchweitz, Iyengar, and Miller [1], where it is used to establish results on numerical invariants of complexes over commutative local rings by tracking them along a chain of categories as above.”