K(a, B, I): Ii

The 'KABF conjecture states that double relative K-theory and cyclic homology agree, at least in characteristic zero. We show that K 2 (A, B, I) maps onto HC 1 (A, B, I) whenever A-* B is a map of Q-algebras and I ~ BIB. We also reinterpret the KABI conjecture in terms of the injectivity of the inverse limit of the map from NK(A, B, I) to the inverse limit of the truncated polynomial versions of NK(A, B, I). A standard scenario for computing the algebraic K-theory of a ring A is the 'Excision situation', in which there is an ideal I of A and a ring map A ~ B so that I ~ BIB. In this situation, the double relative groups K~(A, B, I) are the obstructions to the existence of a Mayer-Vietoris sequence for the K-theory of the square