Let $X$ be a (connected) Noetherian scheme and $K_0(X)$ denote the Grothendieck group of the category of Algebraic vector bundles (coherent sheaves that are locally free and of constant rank ( as $X$ is connected) ).
My question is: If for two Algebraic vector bundles $\mathcal F, \mathcal G$ on $X$, we have $[\mathcal F]=[\mathcal G]$ in $K_0(X)$, then is it necessarily true that $\mathcal F \oplus \mathcal O_X^{\oplus n}\cong \mathcal G \oplus \mathcal O_X^{\oplus n}$ for some integer $n\ge 0$ ?
I know this is true if $X$ is affine, but I'm not sure what happens otherwise. I'm most interested in the case where $X$ is quasi-affine.
Best Answer
This is in general false. Since you are interested in quasi-affines, let me give such an example. Take $R=k[X_1,\ldots, X_n]$ with $n$ sufficiently large (say greater than 2). Let $X=\operatorname{Spec} R-(X_1,\ldots, X_n)$. Take $M=Re_1\oplus \cdots\oplus Re_n/\sum X_ie_i$ and restrict it to $X$ to get a vector bundle $F$. Then, $[F\oplus \mathcal{O}_X]=[\mathcal{O}^n_X]$ in $K_0(X)$, but $F\not\cong\mathcal{O}_X^{n-1}$.