On a consequence of the order ideal conjecture

Given a minimal set of generators x of an ideal I of height d   in a regular local ring (R,m,k)(R,m,k), we prove several cases for which the map Kd(x;R)⊗k→TordR(R/I,k) is the 0-map. As a consequence of the order ideal conjecture, we derive several cases for which Kd+i(x;R)⊗k→Tord+iR(R/I,k) are 0-maps for i≥0i≥0.