# soft question – What is the most common definition of Breuil-Kisin module?

What is the definition of Breuil-Kisin module ? Does the definition varies author to author ?

Notations: Consider a finite extension $$K supset mathbb{Q}_p$$ of the $$p$$-adic field, with ring of integers $$mathcal{O}_K$$. Let $$pi in mathcal{O}_K$$ be the uniformizer and let $$kappa$$ be the residue field of $$K$$. Also let, $$E(u)$$ be the Eisenstein (minimal) polynomial of the uniformizer $$pi$$ and denote $$mathfrak{S}:=W(kappa)((u))$$, where $$W(kappa)$$ is the ring of Witt vectors on the residue field $$kappa$$. Let $$varphi$$ be the Frobenius map which extends to $$mathfrak{S}$$ from $$W(kappa)$$, which itself an extension of the Frobenius on $$kappa$$. Finally let $$G$$ be a $$p$$-divisible group on $$mathcal{O}_K$$ and denote by $$BT(mathcal{O}_K)$$

$$underline{Definition}$$ 1: We define the by $$BT_{/mathfrak{S}}^{varphi}$$ the category of finite free $$mathfrak{S}:=W(kappa)((u))$$-module $$mathfrak{M}$$ equipped with an injective map $$varphi: mathfrak{M} to mathfrak{M}$$ such that the cokernel of its $$mathfrak{S}$$-linearisation $$1 otimes varphi: mathfrak{S} otimes_{mathfrak{S}} mathfrak{M} to mathfrak{M}$$ is killed by $$E(u)$$.

It is famous result conjectured by Breuil and proved by Kisin, that the categories $$BT(mathcal{O}_K)$$ and $$BT_{/mathfrak{S}}^{varphi}$$ are equivalent.

Can we say that the Defintion 1 defines Breuil-Kisin modules or at least belongs to category of Breuil-Kisin modules ?

Though I have several definitions of Breuil-Kisin modules in several articles, some of them assuming finite generated $$mathfrak{S}$$-modules and some of them assuming finite free $$mathfrak{S}$$-modules, as follows:

$$underline{Definition 2:}$$ In this article by Gee (here $$mathfrak{M}=M$$),

$$underline{Definition}$$ 3: Finally, in page 5-6 of this article by Bryden Cais, the Breuil-Kisin module is as follows (assuming finite free):

My Question/Request:

$$(1)$$ Can we say that the Defintion 1 defines Breuil-Kisin modules or at least belongs to category of Breuil-Kisin modules ?

$$(2)$$ Definition $$2$$ assumes finitely generated property while Definition $$3$$ assumes finite free property. How are they same ?

$$(3)$$ Is the Breuil-Kisin module any module defined over the ring $$mathfrak{S}:=W(kappa)((u))$$ ?

$$(4)$$ Does the definition of Breuil-Kisin module varies author to author ? If so, what is the most common definition of Breuil-Kisin module ?

Any comments on the above $$4$$ questions are appreciated.