Let $X$ be a Noetherian separated regular scheme and let $U$ be an open subscheme. Then does there always exist an exact sequence $K_0(X)\to K_0(U)\to 0$ ? If not, then is it at least true when $X=\text{Spec}(R)$ is affine ? And what is a good reference for this ?
(Here, $K_0(-)$ denotes the Grothendieck group of the exact category of Algebraic vector bundles)
(Edited) This is true for Noetherian seperated schemes, since surjectivity holds for $G_0$, and $K_0=G_0$. Since for any open $U$ in $X$ Noetherian, and $\mathcal F$ a coherent sheaf on $U$, there exists a coherent sheaf $\mathcal F'$ on $X$ such that $\mathcal F'|_U\cong \mathcal F$.
This is exercise $II,5.15$ in Hartshorne's Algebraic Geometry.