For a Commutative Noetherian ring $R$, let $G_0(R)$ denote the Grothendieck group of the abelian category of finitely generated $R$-modules i.e. it is the abelian group generated by the isomorphism classes of finitely generated $R$-modules subject to the relation : $[M]=[M_1]+[M_2]$ if there is a short exact sequence of $R$-modules $0\to M_1\to M\to M_2\to 0$.
Now consider a non-zero element $0\ne f\in R$ in a commutative Noetherian ring. Let $R_f$ be the localization of $R$ at the multiplicative set $\{f^n: n\ge 0\}$ i.e. $R_f\cong R[1/f]\cong R[X]/(fX-1)$
Then, we have two canonical exact sequences namely $G_0(R/fR)\to G_0(R)\to G_0(R_f)\to 0$ (as in Chapter II, Exercise 6.10 part (c) of Hartshorne) and also we have an exact sequence
$$\oplus_{P\in V(f)} G_0(R/P)\to G_0(R)\to G_0(R_f)\to 0$$ (for example see Lemma 28(Localization Lemma), page 37, https://home.adelphi.edu/~bstone/commalg-notes/commalg-3/algebra-notes-iii.pdf )
This raises the natural question: What is the connection between $G_0(R/fR)$ and $\oplus_{P\in V(f)} G_0(R/P)$ ?
Apart from the fact that for every $P\in V(f)$, there is a map $G_0(R/P)\to G_0(R/fR)$ induced by restriction of scalars , nothing else comes to my mind.
Please help.