Indecomposable Vector Bundles on the Projective Line

Abstract

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]), andis 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.