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Let $F$ be a CM field and let $(\overline{r}_{\pi,\lambda})_{\lambda}$ be thecompatible system of residual $\mathcal{G}_n$-valued representations of$\operatorname{Gal}_{F}$ attached to a RACSDC automorphic representation $\pi$of $\operatorname{GL}_n(\mathbb{A})$, as studied by Clozel, Harris and Taylorand others. Under mild assumptions, we prove that the fixed-determinantuniversal deformation rings attached to $\overline{r}_{\pi,\lambda}$ areunobstructed for all places $\lambda$ in a subset of Dirichlet density $1$,continuing the investigations of Mazur, Weston and Gamzon. During the proof, wedevelop a general framework for proving unobstructedness (which could be usefulfor other applications in future) and an $R=T$-theorem, relating the universalcrystalline deformation ring of $\overline{r}_{\pi,\lambda}$ and a certainunitary fixed-type Hecke algebra.

Unobstructedness of Galois deformation rings associated to RACSDC automorphic representations