Fix a prime $p$ and a height $n \geq 1$, then there is a closed substack $\mathcal{M}^{\geq n}$ of the stack of $\mathbb{F}_p$-formal groups consisting of formal groups having height $\geq n$. A standard problem in stable homotopy theory is to try to cook up finite spectra which map to the structure sheaf of $\mathcal{M}^{\geq n}$. You can generally only do this when $p$ is large compared with $n$. For small values of $p$ you generally have to make do with finite spectra whose image is the structure sheaf of some nilpotent thickening of $\mathcal{M}^{\geq n}$. These can always be found (a deep result of Devinatz-Hopkins-Smith). Description copied from here.

The question is: do the nilpotent thickenings arising in DHS nilpotence machinery somehow fit into the theory of crystals and crystalline cohomology (where nilpotent thickenings play a role too)? If the answer is positive, does it provide a useful geometric perspective on the nilpotence machinery?


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