Modules over twisted group rings and vector bundles over the anisotropic real conic

We prove, in an elementary way, that a locally free sheaf of finite rank over the anisotropic real conic is the direct sum of indecomposable locally free sheaves of rank 1 or 2. Our proof is purely algebraic, and is based on a classification of graded ℂ[X, Y]-modules endowed with a certain action of the cyclic group ℤ/4ℤ.