In these notes on" Projective Modules and Complete Intersections" an account on the recent
developments in research on this subject is presented. The author's preference for the
technique of Patching isotopic isomorphisms due to Quillen, formalized by Plumsted, over
the techniques of elementary matrices is evident here. The treatment of Basic Element
theory here incorporates Plumstead's idea of the" generalized dimension functions". These
notes are highly selfcontained and should be accessible to any graduate student in …

Let A be a Noetherian ring of dimension n and P be a projective A module of rank n having
trivial determinant. It is proved that if n is even and the image of a generic element g∈ P* is
a complete intersection, then [P]=[Q Α A] in K 0 (A) for some projective A module Q of rank n-
1. Further, it is proved that if n is odd, A is Cohen-Macaulay and [P]=[Q Α A] in K 0 (A) for
some projective A module Q of rank n-1, then P has a unimodular element.