Depth and amplitude for unbounded complexes

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