| Abstract: | LetR be a commutative, semi-local ring,I
1, ...,I
s
ideals. In this paper, we define therelative Milnor K-groups of (R;I
1, ...,I
s
),K
p
M
(R;I
1, ...,I
s
), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK
p
M
(R;I
1, ...,I
s
) is isomorphic to the weightp portion of the relative QuillenK-groupK
p
(R;I
1, ...,I
s
), after inverting (p–1)!. We also define the relative group homology of GL
n
(R;I
1, ...,I
s
), and show thatK
p
M
(R;I
1, ...,I
s
) is isomorphic toH
p
(GLp(R;I
1, ...,I
s
))/Im(H
p
(GL
p–1 (R;I
1, ...,I
s
))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK
p
M
(F)/l
v
to
, and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:
|
