Let $X$ be a connected smooth scheme over a field and let $VB(X)$ be the exact category of vector bundles (i.e. locally free $\mathcal{O}\_X$-module whose rank is finite at every point) over $X$.


Let the abelian group $\tilde K\_0(X)$ denote the kernel of the rank morphism from the zeroth algebraic K-theory group $K\_0(X)$ to $\mathbb{Z}$.


Let $Vec\_k(X)$ be the set of isomorphism classes $[A]$ of rank $k$ vector bundles $A$ over $X$. One gets maps $i\_k:Vec\_k(X)\to Vec\_{k+1}(X)$ defined by $[A]\mapsto [A\oplus \mathbb{A}^1\_X]$, i.e. by adding a trivial bundle. The directed colimit $cVec(X)=\operatorname{colim}\_k Vec\_k(X)$ of this system has the structure of a commutative monoid by $[A]+[B]=[A\oplus B]$.



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> Is there an isomorphism $cVec(X)\cong\tilde K\_0(X)$ of monoids?
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This holds for topological K-theory over a connected CW-complex. I am a little unconvinced that this isomorphism holds since $VB(X)$ is not split-exact. However, I cannot find an argument.



Can this reduced version of algebraic K-theory be identified with this direct limit?
It's already false for $\mathbb P^1$. Let $L$ be a line bundle with no global sections except zero. Then $L\oplus L^{-1}$ is not a trivial bundle because it is not generated by global sections, and this remains true when you add a trivial bundle to it, but it is trivial in your reduced $K^0$.


