Let $X$ be a smooth projective $k$-variety (separated, finite type, geometrically integral $k$-scheme). Denote by $\textbf{Qcoh}\ X$ the category of quasi-coherent sheaves on $X$.
I have heard it said that $\text{K}_0(\textbf{Qcoh}\ X)$, the Grothendieck group of quasi-coherent sheaves on $X$, vanishes. It is not immediately clear to me why this should be the case. Can anybody provide some insight?
Best Answer
If $X$ is a point, $\text{Qcoh}(X)$ is the category of $k$-vector spaces. For any vector space $V$ there is a vector space $W$ and an exact sequences $0 \to V \to W \to W \to 0$, implying $[V] = 0$ in $K_0$. In general such counter-example should be easy to construct as well.
To get something meaningful you probably need to take $\text{Coh}(X)$. For example, you get $\Bbb Z$ for a point, and you should get $\Bbb Z^2$ for $\Bbb P^1$.