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 $X_{φ,N}$ of (framed) $(φ,N)$-modules when $G=\GL_n$. We show that when $G=\GL_3$ and $K_0=\Q_p$, $X_{φ,N}$ has a singular component, and we construct a moduli-theoretic resolution of singularities.

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