Let $K$ be a field and $n \geq 1$. Then the set of isomorphism classes of vector bundles over $\mathbb{P}^n\_K$ is a semiring (i.e. almost a ring, but no additive inverses are possible). By introducing additive inverses and quotienting out exact sequences, we get the $K$-theory of $\mathbb{P}^n\_K$, which is known to be $\mathbb{Z}^{n+1}$. But is it also possible to compute exactly the semiring?


For $n=1$, there is a result by Dedekind-Weber (1892) which proves that the semiring is $\mathbb{N}[x,x^{-1}]$, where $x=\mathcal{O}(1)$ ([related topic](https://mathoverflow.net/questions/16434/using-linear-algebra-to-classify-vector-bundles-over-p1)). Some months ago, I was told that the structure is far more complicated for $n>1$. Can anybody elaborate this or even give a presentation of the semiring?


If necessary, you may assume $K = \mathbb{C}$.



Semiring of algebraic vector bundles on projective space
This semiring carries an enourmous amount of information about vector bundles on $\mathbb{P}^n$, including stuff we don't yet know. For example, you can read from it whether there are indecomposable vector bundles of any given rank; and for small rank we know very little about it (this is discussed, for example, in C. Okonek, M. Schneider, H. Spindler, "Vector bundles on complex projective spaces" , Birkhäuser (1987); I don't have access to the book here, and can't give you a more precise reference). I doubt you can can even get close to what you want.


