Complex Bott Periodicity in algebraic geometry

We state and prove a form of Bott periodicity (for $U(n)$) in an algebraic setting (so, $GL(n)$) which makes sense over $\mathbb{Z}$, which also specializes to Bott periodicity in the usual sense (hence giving yet another proof of classical Bott periodicity). An appendix by B. Church gives a specialization of the constructions and results to motivic homotopy theory, which may be of independent interest.