Doubly relative K-theory and the relative K3

Abstract

Let R be a ring and a and b ideals of R such that an b = 0. In [4] I constructed an exact sequence a/a2@R*b/b2AK2(R,a)-*K2(R/b,a+ b/b)-0. In this article this sequence will be extended to the left to an exact sequence with two more terms as follows: K3(R,a)*K3(R/b,a+ b/b)+a/a2&~b/b2+K2(R,a) -*K2(R/b, a + b/b)+O. This is done by defining doubly relative K-groups which are intermediate between the ordinary relative K-groups. The doubly relative K, is a functor which assigns to a pair of ideals in a ring an Abelian group. In the case an b=O it will turn out that K2(R, a, b) z a/a2@RC b/b2. The extension of the exact sequence with the two relative K3 terms has also been proved independently by J.-L. Loday. He also constructs relative groups of higher order. His approach is via group cohomology, whereas in this paper nonabelian derived functors are used. In Section 5 a relativization of functors from rings to groups will be considered which yields the correct relative groups in many cases, but not for K3. However we will show that there is an exact sequence OdK$(R, a)-K3(R, a)-,a/a2@Rea/a2-+KI(R, a)+Kl(R, a)-+0 in which Kj denotes this relativization of K3 and KS the Stein relativization of K2.