Free resolutions and Koszul homology Original Research Article

Let (R, m) be a regular local ring, and M an R-module. The minimal free resolution F of M has a natural m-adic filtration. In this paper we establish an isomorphism of the spectral sequence which is associated to the filtered complex F, and the spectral sequence associated to a double complex derived from the Koszul complex of M with respect to a minimal system of generators of m. We use this result to describe explicitly the maps in the resolution of terms of Koszul cycles whenever the resolution of the module is pure.