Methods in complete intersections in corank one

Let $A$ denote an affine algebra over an algebraically closed field $k$, with $\dim A=d\ge 3$. In the light of availability of cancellation theorems for stably free modules $P$ with $rank(P)=d-1$ (corank one), we try to implement the methods of complete intersections theory in corank zero, to the corank one case. Our conclusion is that cancellation theorems need to clean up some of the lack of minor generalities, for such an approach to work. However, we hypothesize and derive some of the consequences to complete intersections, of such hypotheses.