[@KPX Example 6.2.6(3)] that $\delta'_i$ is of the form $z^{{\bf k}'_i}{\rm unr}(a)$ where $a\in k(x)^\times$ is the unique element such that $\varphi^{[K_0:{\mathbb{Q}}_p]}$ acts on the underlying $\varphi$-module of ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ by multiplication by $1\otimes a\in K_0\otimes_{{\mathbb{Q}}_p} k(x)$. This finishes the proof.

Note that Lemma \[paramofcrystpt\] implies that if $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ is a crystalline point, then $\omega(x)$ is algebraic ($=\delta_{\bf k}$ for ${\bf k}:=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ where the $k_{\tau,i}$ are as in Lemma \[paramofcrystpt\]). We say that a point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ such that $\omega(x)$ is algebraic is [*dominant*]{} (resp. [*strictly dominant*]{}) if $\omega(x)$ is dominant (resp. strictly dominant).

A variant of the crystalline deformation space {#variant}
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We define a certain irreducible component $Z_{{\rm tri},U}(x)$ of a sufficiently small open neighbouhood $U\subseteq X_{\rm tri}^\square({\overline{r}})$ containing a crystalline strictly dominant point $x$ (Corollary \[defnZtri(x)\]).

We fix ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in ({\mathbb{Z}}^n)^{{\mathrm{Hom}}(K,L)}$ such that $k_{\tau,i}> k_{\tau,i+1}$ for all $i,\tau$ and write $R_{\overline r}^{\square,{\bf k}{\rm -cr}}$ for the crystalline deformation ring of $\overline r$ with Hodge-Tate weights ${\bf k}$, i.e. the reduced and ${\mathbb{Z}}_p$-flat quotient of $R^\square_{\overline r}$ such that, for any finite extension $L'$ of $L$, a morphism $x:{\mathrm{Spec}}\, L'\rightarrow {\mathrm{Spec}}\, R_{{\overline{r}}}^\square$ factors through ${\mathrm{Spec}}\, R_{\overline r}^{\square, {\bf 