Geodesic
Indecomposable Vector Bundles on the Projective Line
Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 149-154

Voir la notice de l'article provenant de la source Cambridge University Press

Let A be a commutative ring, and let Proj A[t0, ti]. By a vector bundle on X we mean a locally free sheaf of finite rank on X. Set t = t1/to. Then X is made up of two affine pieces U1 = Spec A[t], and U2 = Spec A[t-1]. Let P(R) denote the category of finitely generated projective modules over the ring R. The category of vector bundles on X is equivalent to the category of triples (P1,f1, P2), where P1 ∊ p (A[t]), P2 ∊ p(A[t-1]), and is an A[t, t-1] -isomorphism. In [2], the category of vector bundles on is denned directly in this manner, without first defining (so that one could work over a non-commutative ring). We prove that if A is a Krull ring (or a Noetherian ring with connected spec) of dimension > 0, then there is an indecomposable vector bundle of rank n on X, for every positive integer n. 
Roberts, Leslie G. Indecomposable Vector Bundles on the Projective Line. Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 149-154. doi: 10.4153/CJM-1972-013-9
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