Would it be a little but good exercise to construct or find out Breuil modules? My question is about p-adic Hodge-Tate theory and p-adic Galois representation.
One of the important semi-linear object in p-adic Galois representation is the $\text{Breuil Module}$. There are examples of Breuil modules.
My question-
Is it easy to give or find examples of Breuil Modules?
Would it be a little but good exercise to construct or find out examples of Breuil modules ?
Any comment will be helpful to because I am new in this area.
 A: It might be worth distinguishing here between two different but related constructions:


*

*Breuil--Kisin modules, which are finite free modules over a relatively simple base ring, namely $\mathfrak{S} = W[[u]]$ where $W$ is the Witt vectors of the residue field;

*Breuil modules, which are finite free modules over a rather more complicated ring $S$ containing $\mathfrak{S}$ (obtained from $\mathfrak{S}$ by some divided-power envelope construction).
B-K modules are simpler and easier to write down, and you can get a Breuil module from a B-K module by base-extension; so you might be well-advised to start by writing down some examples of Breuil--Kisin modules.
A nice exercise might be to try to write down some Breuil--Kisin modules of rank 2. There are some very nice examples of explicit rank 2 Wach modules (which are in many ways analogous to Breuil--Kisin modules, but only work when the base field is unramified) in a paper of Berger, Li and Zhu from 2004; it might be fun to try to translate their examples into the language of Breuil--Kisin modules, and see if you can extend them to some ramified base fields.
