An obstruction to lifting schemes to spectral schemes

We develop and study an obstruction for lifting schemes over the integers to spectral schemes over the sphere spectrum. This extends a result of Nikolaus for rings, which states that a necessary condition for liftability is existence of a $\hat\delta$-structure. We prove descent properties for $\hat\delta$-rings, define $\hat\delta$-schemes, and prove an analogous statement. We then apply it to concrete examples such as number rings, closed subschemes of $\mathbb P^n$, and various group schemes.