Finiteness of unramified deformation rings

We prove that the universal unramified deformation ring $R^{\mathrm{u}\mathrm{n}\mathrm{r}}$ of a continuous Galois representation $\overline{\rho}: G_{F^{+}} \rightarrow $\mathrm{G}\mathrm{L}$_n(k)$(foratotallyrealfield$F^{+}$andfinitefield$k$) is finite over $\mathcal{O}=W(k)$ in many cases. We also prove (under similar hypotheses) that the universal deformation ring $R^{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}$ is finite over the local deformation ring $R^{\mathrm{l}\mathrm{o}\mathrm{c}}$.