  Freed-Hopkins give a mathematical ansatz for classifying gapped invertible
phases of matter with a spatial symmetry in terms of Borel-equivariant
generalized homology. We propose a slight generalization of this ansatz to
account for cases where the symmetry type mixes nontrivially with the spatial
symmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz,
we prove as a theorem a "fermionic crystalline equivalence principle," as
predicted in the physics literature. Using this and the Adams spectral
sequence, we compute classifications of some classes of phases with a point
group symmetry; in cases where these phases have been studied by other methods,
our results agree with the literature.
