Meaning of notations in Rost's cycle modules

Question

In chapter 2 of markus Rost's "Chow Groups with Coefficients", I encountered the notations $c_{\kappa(v)\vert F}$ and $r_{\kappa(v)\vert F}$ in formula (2.1.0) and in the homotopy property for $\mathbb A^1$ in Proposition (2.2). What are their meanings?

Answer 1

If $\varphi : F \to E$ is a field homomorphism, it induces a ring homomorphism ("restriction") $r_{E|F} : K_* F \to K_* E$ by the formula $r_{E|F} (\{a_1, \dots, a_n)\} = \{\varphi(a_1), \dots, \varphi(a_n)\}$. If $\varphi$ is finite, there is a norm homomorphism ("corestriction") $c_{E|F} : K_* E \to K_* F$. These notations are introduced at the bottom of page 327 and top of page 328, together with the top of page 330.

Next, on page 328, if $v$ is a certain type of valuation, then $\kappa (v)$ is its residue class field. In particular, if $x \in X$ then $\kappa (x)$ is the residue class field of $x$ (see top of page 321), therefore it makes sense to speak about $c_{\kappa(x), F}$.

If you let $F(u)$ be the function field of $\operatorname {Spec} F[u]$, then it makes sense to speak about $r_{F(u) | F}$ (top of page 338).