#### Vol. 8, No. 9, 2014

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Finiteness of unramified deformation rings

### Patrick B. Allen and Frank Calegari

Vol. 8 (2014), No. 9, 2263–2272
##### Abstract

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

##### Keywords
Galois representations
Primary: 11F80
Secondary: 11F70