I have one technical question on norm maps on Milnor K-theory.

When $K \subset L$ is a finite extension of fields, we know (by Bass-Tate and Kato) that there exists a norm map $N_{L/K} : K^M_n (L) \to K^M _n (K)$ on the Milnor K-groups for all $n \geq 0$. For instance, when $n=1$, this coincides with the "undergraduate algebra level" norm map $L^{\times} \to K^{\times}$.

My question pertains to the situation when this finite extension of fields is replaced by a finite ring extension. For instance when $A \subset B$ is a finite étale extension of semi-local rings with infinite residue fields, Kerz (Invent. Math, 09) constructed such norm maps from $K_n ^M (B)$ to $K_n ^M (A)$.

Here I wonder if there is a generalization of it in the following slightly more general circumstances:

(1) For instance, instead of the étale assumption, if $A \subset B$ is a finite extension of semi-local rings with infinite residue fields, where $B$ is a free $A$-module of finite rank, can we possibly still obtain a norm map $K_n ^M (B) \to K_n ^M (A)$?

(2) Or its special case: when $A \subset B$ is a finite extension of semi-local rings with infinite residue fields, such that the extension is "simple", i.e. $B= A[x]/(p(x))$ for a monic irreducible polynomial $p(x) \in A[x]$?

Of course, in the case of Kerz, it is known that his étale extension assumption implies that it is automatically simple, thus $B$ is a free $A$-module of finite rank. In general, (1) is more general than (2), and (2) is more general than the case considered by Kerz. Of course, (1) generalizes the situation of Bass-Tate and Kato for finite field extensions, too.

I suspect this might hold. I hope someone who knows well about this situation could kindly give some advice on this question.

  • 2
    $\begingroup$ out of curiosity and my ignorance, how do you define the Milnor K-theory of a ring which is not a field? :) $\endgroup$
    – mathphys
    Aug 25 '17 at 11:19
  • $\begingroup$ Here, I'm still using the "usual" way, i.e. the tensor algebra on the units of the ring modulo the ideal generated by $a \otimes (1-a)$ sort of things. This definition (called naive way in Kerz's 2010 J. Algebraic Geometry paper) is normally believed to be the "correct way" for the rings I considered with infinite residue fields. $\endgroup$ Aug 25 '17 at 13:47

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