# On crystabelline deformation rings of \(\mathrm {Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)\) (with an appendix by Jack Shotton)

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## Abstract

We prove that certain crystabelline deformation rings of two dimensional residual representations of \(\mathrm {Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)\) are Cohen–Macaulay. As a consequence, this allows to improve Kisin’s \(R[1/p]=\mathbb {T}[1/p]\) theorem to an \(R=\mathbb {T}\) theorem.

## Mathematics Subject Classification

11F80 11F85## Notes

### Acknowledgements

YH was partially supported by National Natural Science Foundation of China Grants 11688101; China’s Recruitement Program of Global Experts, National Center for Mathematics and Interdisciplinary Sciences and Hua Loo-Keng Center for Mathematical Sciences of Chinese Academy of Sciences. VP was partially supported by SFB/TR45 of DFG. The project started when YH visited VP in 2013 supported by SFB/TR45 and he would like to thank the University Duisburg-Essen for the invitation and the hospitality. The authors would like to thank Jack Shotton for the appendix to the paper, as well as Toby Gee, James Newton, Shu Sasaki and Jack Thorne for their comments. We also thank the anonymous referee for their careful reading of the paper and pertinent comments.

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