containing $\mathrm{Ann}_B(\mathfrak{n})$. It follows from [@BourbakiAC §IV.1.3 Cor.1] that $\mathfrak{p}$ is an associated prime of the $B$-module $\mathfrak{n}$ and consequently of $B$. But we have $f\in\mathfrak{p}$ which contradicts the fact that $\mathfrak{p}\cap A=0$.

We now embed this “refined” crystalline deformation space $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ into the space $X_{\rm tri}^\square({\overline{r}})$ as follows. We define a morphism of rigid spaces over $L$: $$\begin{aligned}
\label{cristotriang}
{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{{\mathrm{Sp}}\,L} T_L^{\mathrm{rig}}&\longrightarrow &{\mathfrak{X}}_{\overline r}^\square\times_{{\mathrm{Sp}}\,L}{\mathcal{T}}^n_L\\
\nonumber (r,\varphi_1,\dots,\varphi_n)&\longmapsto & (r, z^{{\bf k}_1}{\rm unr}(\varphi_1),\dots, z^{{\bf k}_n}{\rm unr}(\varphi_n)).\end{aligned}$$ This morphism is a closed embedding of reduced rigid spaces as both maps $r\mapsto r$ and $(\varphi_1,\dots,\varphi_n)\mapsto (z^{{\bf k}_1}{\rm unr}(\varphi_1),\dots, z^{{\bf k}_n}{\rm unr}(\varphi_n))$ respectively define closed embeddings ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm-cr}\!\!\hookrightarrow {\mathfrak{X}}_{\overline r}^\square$ and $T_L^{\mathrm{rig}}\hookrightarrow {\mathcal{T}}^n_L$. We claim that the restriction of the morphism $(\ref{cristotriang})$ to: $$\label{cristotriangres}
\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\hookrightarrow{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{{\mathrm{Sp}}\,L}T_L^{\mathrm{rig}}$$ factors through $X_{\rm tri}^\square(\overline r)\subset {\mathfrak{X}}_{\overline r}^\square\times_{{\mathrm{Sp}}\,L}{\mathcal{T}}^n_L$. As the source of this restriction is reduced by Lemma \[reduced\], it is enough to check it on a Zariski-dense set of points of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$.

Let $r$ be an $n$-dimensional crystalline representation of ${\mathcal G}_K$ over a finite extension $L'$ of $L$ of Hodge-Tate weights ${\bf k}$ and let $\varphi_1,\dots,\varphi_n$ be an ordering of the eigenvalues of 