Semiring of algebraic vector bundles on projective space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:06:21Z http://mathoverflow.net/feeds/question/20444 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20444/semiring-of-algebraic-vector-bundles-on-projective-space Semiring of algebraic vector bundles on projective space Martin Brandenburg 2010-04-05T23:58:44Z 2010-04-06T07:32:29Z <p>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?</p> <p>For $n=1$, there is a result by Dedekind-Weber (1892) which proves that the semiring is $\mathbb{N}[x,x^{-1}]$, where $x=\mathcal{O}(1)$ (<a href="http://mathoverflow.net/questions/16434/using-linear-algebra-to-classify-vector-bundles-over-p1" rel="nofollow">related topic</a>). 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?</p> <p>If necessary, you may assume $K = \mathbb{C}$.</p> http://mathoverflow.net/questions/20444/semiring-of-algebraic-vector-bundles-on-projective-space/20470#20470 Answer by Angelo for Semiring of algebraic vector bundles on projective space Angelo 2010-04-06T03:33:47Z 2010-04-06T03:33:47Z <p>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.</p>