On a Consequence of the Order Ideal Conjecture

Given a minimal set of generators $\bold{x}$ of an ideal $I$ of height d in a regular local ring ($R, m, k$) we prove several cases for which the map $K_d(\bold{x}; R) \otimes k \to \Tor_d^R (R/I, k)$ is the 0-map. As a consequence of the order ideal conjecture we derive several cases for which $K_{d+i}(\bold{x}; R) \otimes k \to \Tor_{d+i}^R (R/I, k)$ are 0-maps for $i \ge 0$.