On the irreducible components of some crystalline deformation rings

We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_K$ for a finite extension $K\mathrm{/}{\mathbb{Q}}_p$. This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the universal deformation ring. For $K$ unramified over ${\mathbb{Q}}_p$ and Hodge–Tate weights in $[0,p]$, we study the geometry of this space. As a consequence we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of ${\mathbb{Q}}_p$, with Hodge–Tate weights in $[0,p]$, are potentially diagonalisable.