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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.
Nguyen et al. (Thu,) studied this question.