This question is related with Exercise III.5.4 (Page 230) of Hartshorne's Algebraic Geometry. Here I start with recalling the definition of Grothendieck group $K(X)$ of a noetherian scheme $X$, which is the quotient of free abelian group generated by all coherent sheaves on $X$, by the subgroup generated by all expressions $\mathscr F' -\mathscr F + \mathscr F''$ whenever there is an exact sequence $0\to \mathscr F' \to \mathscr F \to \mathscr F'' \to 0$ (cf. Exercise II.6.10, Page 148 Hartshorne's Algebraic Geometry).

Let $k$ be a field. This exercise essentially wants to prove $K(\mathbf P_k^r) \cong \mathbb Z^r$, generated by the images of the sheaves of regular functions of $\mathbf P_k^0, \mathbf P_k^1, \ldots, \mathbf P_k^{r}$ in $K(\mathbf P_k^r)$. We can prove this claim by induction according to the hints given in the book. It is trivial when $r =0$ which corresponds to a point. Suppose the claim is true for $r-1$. Since $\mathbf P_k^{r-1}$ can be considered as a closed subscheme of $\mathbf P_k^r$ and $\mathbf P_k^r - \mathbf P_k^{r-1} \cong \mathbf A_k^{r-1}$. Hence one has an exact sequence (cf. Exercise II.6.10, page 148): $$ K(\mathbf P_k^{r-1}) \xrightarrow{\alpha} K(\mathbf P_k^r) \to K(\mathbf A_k^{r-1}) \to 0.$$ I think I can prove $\alpha$ is injective by showing it splits. So it only remains to prove $K(\mathbf A_k^{r}) \cong \mathbb Z$ for any integer $r > 0$. But this is where I get stuck. $K(\mathbf A_k^1) \cong \mathbb Z$ is easy because we fully understand the module structure over PID ($k[x]$ is PID). I doubt this method can be extended to $r > 1$. One way I tried is to prove the surjective group homomorphism $ \gamma:\ K(\mathbf A_k^{r-1}) \to \mathbb Z$ is also injective, where $\gamma$ is induced by the ranks of coherent sheaves. To prove $\mathrm{ker}(\alpha)=0$, it is enough to prove any coherent torsion sheaf $\mathscr F$ is trivial in $K(\mathbf A_k^{r})$. Here a coherent sheaf $\mathscr F$ over an integral scheme $X$ is torsion if the stalk $\mathscr F_\eta = 0$ where $\eta$ is the generic point of $X$. But this is exactly where I get stuck.

I appreciate if any one could give me some hints (answers if possible) or give a new method.