The trianguline variety for reductive groups

We study the trianguline variety for split connected reductive groups.

We generalize a theorem of Breuil, Hellmann, and Schraen about its local structure, establishing smoothness over the loci determined by various regularity conditions on the triangulation parameter, and normality at certain points outside of these smooth loci.

Along the way, we prove a crystallinity criterion for $(\varphi,\Gamma_K)$-modules with $\mathsf G$-structure.