On a minimal free resolution of the residue field over a local ring of codepth 3 of class T

Let R be any local ring with residue field k, and A the homology of the Koszul complex on a minimal set of generators of the maximal ideal of R. In this paper, we show that a minimal free resolution of k over R can be obtained from a graded minimal free resolution of k over A. More precisely, this is done by the iterated mapping cone construction, introduced by the authors in a previous work, using specific choices of ingredients. As applications, using this general perspective, we exhibit a minimal free resolution of k over a complete intersection ring of any codepth, and explicitly construct a minimal free resolution of k over a local ring of codepth 3 of class T in terms of Koszul blocks.