Relative MilnorK-theory

LetR be a commutative, semi-local ring,I₁, ...,I_s ideals. In this paper, we define therelative Milnor K-groups of (R;I₁, ...,I_s),K_p^M(R;I₁, ...,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;I₁, ...,I_s) is isomorphic to the weightp portion of the relative QuillenK-groupK_p(R;I₁, ...,I_s), after inverting (p–1)!. We also define the relative group homology of GL_n(R;I₁, ...,I_s), and show thatK_p^M(R;I₁, ...,I_s) is isomorphic toH_p(GL_p(R;I₁, ...,I_s))/Im(H_p(GL_p–1 (R;I₁, ...,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:

Keywords:	MilnorK-theory  relativeK-theory  homology of GLn  Kato's conjecture  Quillen-Lichtenbaum conjectures
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