Let $X$ be the scheme obtained by gluing two copies of $\Bbb P^1_k$ along $D(x)$ (basically the projective version of the line with two origins). Is there a good classification of coherent sheaves on $X$? By analogy with the case of $\Bbb P^1$ it seems clear that a coherent sheaf splits as a direct sum of a locally free part and a torsion part, but I'm worried that the extremely convenient property that a locally free sheaf on $\Bbb P^1$ is a direct sum of $\mathcal{O}(d_i)$ for various $d_i$ won't hold (the proof I know is linear algebra on the transition matrix, which I can't figure out how to adapt to this case).
Has this been written down somewhere before, or can anyone help me get to the right answer?