Generic smoothness for $G$-valued potentially semi-stable deformation rings

We extend Kisin’s results on the structure of characteristic $0$ Galois deformation rings to deformation rings of Galois representations valued in arbitrary connected reductive groups $G$. In particular, we show that such Galois deformation rings are complete intersection. In addition, we study explicitly the structure of the moduli space $X_{\phi ,N}$ of (framed) $(\phi ,N)$-modules when $G={\text{GL}}_n$. We show that when $G={\text{GL}}_3$ and $K_0={\text{Q}}_p$, $X_{\phi ,N}$ has a singular component, and we construct a moduli-theoretic resolution of singularities.