# Crystals and nilpotence

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?