Given a smooth scheme $X$ over the ring of integers of a $p$-adic field, we introduce the notion of a relative Breuil-Kisin-Fargues module $M$ on $X$. Each such $M$ simultaneously encodes the data of a lisse étale sheaf, a module with flat connection, and a crystal, whose cohomologies are then intertwined by a relative form of the $A_{inf}$ cohomology introduced in "Integral $p$-adic Hodge theory" by Bhatt-M-Scholze. They are moreover closely related to other work in relative $p$-adic Hodge theory, notably Faltings small generalised representations and his relative Fontaine Lafaille theory. Joint with Takeshi Tsuji.[-]

Any finite-dimensional p-adic representation of the absolute Galois group of a $p$-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen-Brinon. We generalize their construction to the fundamental group of a $p$-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. When the representation comes from a $Q_{p}$-representation of a $p$-adic Lie group quotient of the fundamental group, we describe its Lie algebra action in terms of the Sen operators, which is a generalization of a result of Sen-Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan.[-]