Arithmetic Breuil-Kisin-Fargues modules and several topics in p-adic Hodge theory
Let K be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered (φ,N,GK)-modules over K to the isogeny category of Breuil- Kisin-Fargues GK-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered (φ,N,GK)-modules to GK-equivariant modifications of vector bundles over the Fargues-Fontaine curve XFF , with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over XFF and the isogeny category of Breuil-Kisin-Fargues modules. We characterize the essential image of this functor and give two applications of our result. First, we give a new way of viewing the p-adic monodromy theorem of p-adic Galois representations. Also we show our theory provides a universal theory that enable us to compare many integral p-adic Hodge theories at the Ainf level.