Lifting schemes to the sphere

Speaker: Robert Szafarczyk – University of Copenhagen

Abstract: Every discrete ring is a $\mathbb{Z}$-algebra. In homotopy theory we have a deeper base though, the sphere spectrum. Hence, one can ask if a given scheme arises as base change to the integers of some spectral scheme. We will present an obstruction (a necessary condition) to lift existence. In the affine case, there is a relatively simple $1$-algebraic obstruction due to Thomas Nikolaus. More precisely, if a commutative ring arises as base change of a commutative ring spectrum, then it admits what is called a delta-hat structure. For example, we can successfully obstruct number rings over $\mathbb{Q}$ and all torsion rings in this way. We extend this obstruction to schemes and apply it to group schemes and elliptic curves.