factorization: $$\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\longrightarrow ({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0} \times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p} {\rm Flag})^{\mathrm{rig}}\times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p}{\mathrm{Sp}}\,L\longrightarrow T_L^{\rm rig}/\mathcal{S}_n$$ where the first map is the smooth map in the proof of Lemma \[genericopensinXcris\] below, and the second is the projection on $(\mathrm{Res}_{K_0/\mathbb{Q}_p}\mathrm{GL}_{n,K_0})_L^{\rm rig}$ followed by the base change to $L$ of the rigidification of the morphism $\mathrm{Res}_{K_0/\mathbb{Q}_p}\mathrm{GL}_{n,K_0}\rightarrow T/\mathcal{S}_n$ defined in [@HartlHellmann (9.1)]. The first map being smooth is flat and thus open by [@Bo Cor.9.4.2], and the last two are easily seen to be open.

The following (well-known) lemma was used in the proof of Lemma \[reduced\].

\[red\] Let $A$ a commutative noetherian domain and $B$ a finite flat $A$-algebra. Then the ring $B$ has no embedded component, i.e. all its associated ideals are minimal prime ideals. Moreover if $B$ is generically étale over $A$, i.e. $\mathrm{Frac}(A)\otimes_A B$ is a finite étale $\mathrm{Frac}(A)$-algebra, then the ring $B$ is reduced.

As $B$ is flat over $A$, the map $\mathrm{Spec}\,B\rightarrow\mathrm{Spec}\,A$ has an open image, and $A$ being a domain it contains the unique generic point of $\mathrm{Spec}\,A$, which implies that the natural map $A\rightarrow B$ is injective. Moreover $B$ being finite over $A$, the image of $\mathrm{Spec}\,B\rightarrow\mathrm{Spec}\,A$ is closed, hence it is $\mathrm{Spec}\,A$ since $\mathrm{Spec}\,A$ is connected. In particular $B$ is 