Homotopy Classification of Line Bundles Over Rigid Analytic Varieties

We construct a motivic homotopy theory for rigid analytic varieties with the rigid analytic affine line $\mathbb{A} ^1_\mathrm{rig}$ as an interval object. This motivic homotopy theory is inspired from, but not equal to, Ayoub's motivic homotopy theory for rigid analytic varieties. Working in the so constructed homotopy theory, we prove that a homotopy classification of vector bundles of rank $n$ over rigid analytic quasi-Stein spaces follows from $\mathbb{A}^1_\mathrm{rig}$-homotopy invariance of vector bundles. This $\mathbb{A}^1_\mathrm{rig}$-homotopy invariance is equivalent to a rigid analytic version of Lindel's solution to the Bass--Quillen conjecture. Moreover, we establish a homotopy classification of line bundles over rigid analytic quasi-Stein spaces. In fact, line bundles are classified by infinite projective space.