Iterated mapping cones on the Koszul complex and their application to complete intersection rings

Let Formula: see text be a complete intersection local ring, Formula: see text be the Koszul complex on a minimal set of generators of Formula: see text, and Formula: see text be its homology algebra. We establish exact sequences involving direct sums of the components of Formula: see text and express the images of the maps of these sequences as homologies of iterated mapping cones built on Formula: see text. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field Formula: see text over Formula: see text, independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns.