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.