Complex Bott Periodicity in algebraic geometry

TL;DR

This paper establishes a form of Bott periodicity within algebraic geometry over integers, providing a new proof and connecting classical topological results with algebraic and motivic homotopy theories.

Contribution

It introduces an algebraic version of Bott periodicity applicable over z, extending classical topological results and including a motivic homotopy theory perspective.

Findings

Proves algebraic Bott periodicity for $GL(n)$ over z.

Provides a new proof of classical Bott periodicity.

Includes a motivic homotopy theory specialization.

Abstract

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.