Breuil--Kisin modules and crystalline cohomology
The theory of Breuil--Kisin modules provides a powerful classification of stable lattices in
p-adic Galois representations via certain semi-linear algebra structures over power series rings. On the other hand, the i-th integral p-adic etale cohomology of a smooth and proper scheme X over the ring of integers in a p-adic field provides a stable lattice in a p-adic Galois representation, and so has a Breuil--Kisin module attached to it. In this case, it is natural to ask if the associated Breuil--Kisin module can be described in terms of the cohomology of the scheme. In this talk, I will answer this question in the affirmative when i < p-1 and the crystalline cohomology of the special fiber of X is p-torsion-free in degrees i and i+1. This is joint work with Tong Liu.