of modules in C’(A). In Section 4, we will show that K’(A) is generated by the elements [A/a] defined by the cyclic A-modules which are perfect of codimension r. If x, )...) x, is an A-regular sequence then the cyclic module A/(x, ,..., x,) is perfect of codimension r. For r = 0, I, and 2, the group K’(A) is generated by the elements [A/(x,,..., x,)], but this is not so for r = 3. In [2 3, Dutta, Hochster, and McLaughlin exhibit a module M which is of finite length and finite projective dimension over a Cohen-Macaulay local ring A of dimension three, such that M has negative Serre intersection multiplicity with an A-module of the form A/p, where p is prime of height one in A. Since A/p has zero intersection multiplicity with every module of the form A/(x,, x*,x,), it follows that [M] does not belong to the subgroup of K3(A) generated by the elements [A/(x,, x2, x3)]. As we will show, there is a cyclic module A/a of finite length and finite projective dimension over A, such that [M] = [A/a] in K3(A), modulo elements [A/(x,, x2, x3)]. It follows that A/a has negative intersection multiplicity with A/p. This fact is of interest in geometry. We would like to express our thanks to the referee and to Melvin Hochster, for their suggestions improving this paper. When this paper was first submitted, [2] had not appeared, and we were not aware of the 367 0021-8693/87 $3.00