When \(F\) is a field the Milnor \(K\)-groups, \(K^ M_ n(F)\), are defined as the graded algebra on \(F^*\) divided by the two-sided ideal generated by elements \(a\otimes (1-a)\). There is a natural map between Milnor and Quillen \(K\)-theory, \(s_ p: K^ M_ p(F)\to K_ p(F)\). It is shown by A. A. Suslin [Lect. Notes Math. 1046, 357-375 (1984; Zbl 0528.18007)] that, if \(F\) is infinite, \((p-1)\)! annihilates the kernel of \(s_ p\). In addition the Kato conjecture predicts that the norm residue symbol, \[ \ominus_{p,n}: K^ M_ p(F)/n\to H^ p\bigl(F; \mu^{\otimes p}_ n\bigr) \] is an isomorphism if \({1\over n}\in F\). Finally, if \(\text{gr}^*_ \gamma K_ *(F; \mathbb{Z}/\ell^ k)\) is the associated grade of the \(\gamma\)-filtration on \(\text{mod }\ell^ k\), the Quillen-Lichtenbaum conjecture predicts that the Chern class induces an isomorphism \[ c_{q,p}: \text{gr}^ q_ \gamma K_{2q-p}(F: \mathbb{Z}/\ell^ k)\to H^ p\bigl(F; \mu^{\otimes a}_{\ell^ k}\bigr) \] if \(\ell\geq 2q-p\), \(p\leq q\).
The author defines relative Milnor \(K\)-theory, \(K^ M_ *(R; I_ 1,\dots,I_ s)\) for a semi-local ring with ideals, \(I_ j\vartriangleleft R\). He shows that this generalization has the same formal properties as \(K^ M_ *(F)\) and formulates a generalized Kato conjecture. Finally he shows that the generalized Kato conjecture implies the Quillen-Lichtenbaum conjecture.