Vector bundles on $\mathbb{A}^1_k$ with doubled origin?

Question

One of the most common examples of gluing affine lines is the affine line $\mathbb{A}^1_k$ with doubled origin. Out of curiousity, is there a known classfication of the vector bundles on this space?

Answer 1

The scheme $X$ is the gluing of $U=\mathrm{Spec}(k[t])$ and $V=\mathrm{Spec}(k[s])$ along the open subschemes $U \supseteq \mathrm{Spec}(k[t^{\pm 1}]) \cong \mathrm{Spec}(k[s^{\pm 1}]) \subseteq V$, the isomorphism being $s \mapsto t$. All vector bundles on $\mathrm{Spec}(k[t])$ are trivial. It follows that a vector bundle of rank $d$ on $X$ is, up to isomorphism, the gluing of two trivial vector bundles $\mathcal{O}_U^{\oplus d}$ and $\mathcal{O}_V^{\oplus d}$ along some isomorphism between them on the intersection. This corresponds to an invertible $d \times d$-matrix over $k[t^{\pm 1}]$. Multiplying it with an invertible $d \times d$-matrix over $k[t]$ on the left or on the right will give an isomorphic vector bundle. This shows $$\mathrm{Vect}_d(X) \cong \mathrm{GL}_d(k[t]) \setminus \mathrm{GL}_d(k[t^{\pm 1}]) / \mathrm{GL}_d(k[t]).$$ You could also see this by the more general formalism of Cech cohomology. Since $k[t^{\pm 1}]$ is a PID, we have the Smith Normal Form for matrices over this ring. Taking double cosets over $k[t]$, we only have to check how many times a power of $t$ has been taken out. From this one may conclude that $\mathrm{Vect}_d(X) \cong \mathbb{Z}$.