Survey of Kisin’s Paper Crystalline

In p-adic Hodge theory there are fully faithful functors from certain categories of p-adic representations of the Galois group GK := Gal(K/K) of a p-adic field K to certain categories of semi-linear algebra structures on finite-dimensional vector spaces in characteristic 0. For example, semistable representations give rise to weakly admissible filtered (φ,N)-modules, and Fontaine conjectured that this is an equivalence of categories. For many purposes (such as in Galois deformation theory with artinian coefficients) it is useful to have a finer theory in which p-adic vector spaces are replaced with lattices or torsion modules. Fontaine and Laffaille gave such a theory in the early 1980’s under stringent restrictions on the HodgeTate weights and absolute ramification inK. The aim of these lectures on integral p-adic Hodge theory is to explain a more recent theory, due largely to Breuil and Kisin, that has no ramification or weight restrictions. We are essentially giving a survey of [11], to which the reader should turn for more details. (If we omit discussion of a proof of a result, this should not be interpreted to mean that the proof is easy; rather, it may only mean that the techniques of the proof are a digression from the topics that seem most essential for us to discuss.)