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Depth and amplitude for unbounded complexes

H. -B. Foxby·S. Iyengar
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Abstract

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

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H. -B. Foxby, S. Iyengar. 2002-12-09. Depth and amplitude for unbounded complexes. https://arxiv.org/abs/math/0212125

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