BREUIL-KISIN MODULES // ABSOLUTE PRISMS // FROBENIUS DESCENT
A Breuil-Kisin module is a finite free A_{inf}-module M with a φ-semilinear map φ_M: M → M whose cokernel is killed by a power of ξ (the kernel of θ). These classify lattices in crystalline representations.
The absolute prismatic cohomology △_X is defined without a base prism — it lives over the final object Z_p. For X smooth proper over O_K, the absolute prismatic cohomology △_X is a perfect complex of Breuil-Kisin modules.
△_X = RΓ((X)_△^{abs}, O_△)
Frobenius descent says: a vector bundle with Frobenius structure on the prismatic site is the same as a p-adic local system after inverting I. The category of prismatic F-crystals is equivalent to Z_p-local systems on the generic fiber X_η.