For a commutative Noetherian ring $R$, let $G_0(R)$ denote the Grothendieck group (an abelian group) of the abelian category of finitely generated $R$-modules (Note that I'm Not talking about $K_0(R)$ ) .
Now if $R,S$ are Commutative Noetherian rings, and $f: R \to S$ is a finite ring homomorphism (i.e. the $R$-module structure $r.s=f(r)s$ on $S$ makes it a finitely generated module ), then there is a transfer map $f^* : G_0(S)\to G_0(R)$ induced from viewing every finite $S$-module $M$ as an $R$-module by restrictiing scalars, and the corresponding $R$-module is still finite as $f$ is a finite map. Similarly, if $f: R\to S$ is a flat ring homomorphism (the $R$-module structure $r.s=f(r)s$ on $S$ makes it a flat module ) , then the extension of scalars $M\to M \otimes_R S$ for each $R$-module $M$ induces a base change map $f_*: G_0(R)\to G_0(S)$. Both these constructions are natural in the sense that if $f,g$ are finite (resp. flat) and the composition is well defined , then $f\circ g$ is again finite (resp. flat) and $(f \circ g)^*=g^*\circ f^*$ (resp. $(f\circ g)_*=f_*\circ g_*$) Now my questions are the following:
Let $f: R \to R$ be an injective ring homomorphism of a Noetherian domain $R$.
(1) If $f$ is a finite map, then what can we say about the transfer map $f^*: G_0(R)\to G_0(R)$ ? What is the subgroup index of the kernel/image ?
(2) If $f$ is a flat map, then what can we say about the transfer map $f_*: G_0(R)\to G_0(R)$ ? What is the subgroup index of the kernel/image ? Is $f_*$ a pure map of abelian groups ?
My thoughts: Let $i: f(R)\to R$ denote the inclusion. Then by definition, $f$ is finite (resp. flat) iff so is $i$ and moreover, $ i\circ f=i$ . So in the respective cases, we have $i_*\circ f_*=(i\circ f)_*=f_*$ and $f^*\circ i^*=(i\circ f)^*=f^*$. Apart from this, I have no idea.
Please help.