Depth and amplitude for unbounded complexes

By H. Foxby, S. Iyengar. 2002

138 citations recorded by the source corpus.

We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far reaching generalization of the classical Auslander-Buchsbaum formula for the depth of finitely generated modules of finite projective dimension. We extend also Iversen's amplitude inequality to unbounded complexes. As a corollary we deduce: Given a local homomorphism Q-->R, if there is a non-zero finitely generated R-module that has finite flat dimension both over Q and over R, then the flat dimension of R over Q is finite. This last result yields a module theoretic extension of a characterization of regular local rings in characteristic p due to Kunz and Rodicio

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/53d334a9cafa03fe8c0f45b3743b7ca2d29e61cb

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Citations in the corpus

G-dimension over local homomorphisms. Applications to the Frobenius endomorphism (2003, methodology): “For the basic properties of depth, we refer to [ 17 ].”
Ascent Properties of Auslander Categories (2005, methodology): “The invariants “depth” and “width” for complexes over a local ring (S, n,l) can be computed in a number of different ways as demonstrated in [ 26 ].”
Reflexivity and rigidity for complexes, I: Commutative rings (2009, methodology): “To prove a converse we use a version of a result from [16], which incorporates a deep result in commutative algebra, namely, the New Intersection Theorem.”
Finiteness in derived categories of local rings (2004, background): “In [20],Iyengarand Sather-W agsta introducean invariant G -dim M ,called theG-dim ension ofM over .”