Vector bundles over projective spaces. The case F1

Over the field of one element, vector bundles over n-dimensional projective spaces are considered. It is shown that all line bundles are ten- sor powers of the Hopf bundle and all vector bundles are direct sums of line bundles. This is in complete analogy to the case of the projective line over an arbitrary classical field, but drastically simpler in comparison with projective spaces of higher dimensions. Mathematics Subject Classification (2010). Primary 14D20; 14J60, 14T99.