Suppose $X$ is any quasi-projective variety. let $K^0(X)$ denote the Grothendieck group of locally free sheaves.
Suppose $U$ is an open subset of $X$. Is there a localization sequence:
$$ K^0(X)\rightarrow K^0(U)\rightarrow 0. $$
I saw that this exists for group of coherent sheaves $K_0(X)$. If $X$ is not smooth then do we still have the localization sequence for $K^0$.