# Singularities of ordinary deformation rings

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## Abstract

Let \(R^{\mathrm {univ}}\) be the universal deformation ring of a residual representation of a local Galois group. Kisin showed that many loci in \({{\mathrm{MaxSpec}}}(R^{\mathrm {univ}}[1/p])\) of interest are Zariski closed, and gave a way to study the generic fiber of the corresponding quotient of \(R^{\mathrm {univ}}\). However, his method gives little information about the quotient ring before inverting *p*. We give a method for studying this quotient in certain cases, and carry it out in the simplest non-trivial case. Precisely, suppose that \(V_0\) is the trivial two dimensional representation and let *R* be the unique \(\mathbf {Z}_p\)-flat and reduced quotient of \(R^{\mathrm {univ}}\) such that \({{\mathrm{MaxSpec}}}(R[1/p])\) consists of ordinary representations with Hodge–Tate weights 0 and 1. We describe the functor of points of (a slightly modified version of) *R* and show that the irreducible components of \({{\mathrm{Spec}}}(R)\) are normal and Cohen–Macaulay, but not Gorenstein. As a consequence, we find that certain global deformation rings are torsion-free and Cohen–Macaulay, but not Gorenstein.

## Keywords

Galois representations Deformation rings Modularity lifting## Mathematics Subject Classification

Primary 11F80 11S23## Notes

### Acknowledgements

I would like to thank Bhargav Bhatt, Brian Conrad and Mark Kisin for useful conversations. I would especially like to thank Frank Calegari and Steven Sam for comments on earlier drafts of this paper.

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