Finiteness of unramified deformation rings

Abstract

We prove that the universal unramified deformation ring $R^{unr}$ of a continuous Galois representation $\bar{\rho }:G_{F^{+}}\to {GL}_n\left(k\right)$ (for a totally real field $F^{+}$ and finite field $k$) is finite over $\mathcal{O}=W\left(k\right)$ in many cases. We also prove (under similar hypotheses) that the universal deformation ring $R^{univ}$ is finite over the local deformation ring $R^{loc}$.