Modules of finite length and finite projective dimension (Q1401461)

The authors show that the localization exact sequence from \(K\)-theory given by \textit{R. W. Thomason} and \textit{T. Trobaugh} [in: The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 247--435 (1990; Zbl 0731.14001)], which is a generalization of the exact sequence used by Levine, can be used to provide more examples of modules of finite length and finite projective dimension, and for some rings, to describe the intersection properties of such modules completely (see theorem 1). Further, using the methods of \textit{H. Gillet} and \textit{C. Soulé} [Invent. Math. 90, 243--277 (1987; Zbl 0632.14009)], it leads to a new proof that the vanishing theorem holds for pairs of modules \(M, N\) of finite projective dimension over a Cohen-Macaulay local ring \(R\) whose punctured spectrum is regular (see theorem 2).