# Singularities of ordinary deformation rings

- 56 Downloads
- 1 Citations

## 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.

## References

- 1.Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
- 2.Bărcănescu, Ş., Manolache, N.: Betti numbers of Segre-Veronese singularities. Rev. Roumaine Math. Pures Appl.
**26**(4), 549–565 (1981)MathSciNetMATHGoogle Scholar - 3.Conrad, B.: Structure of ordinary-crystalline deformation ring for \(\ell =p\). Unpublished notes. http://math.stanford.edu/~conrad/modseminar/pdf/L21.pdf
- 4.Calegari, F., Geraghty, D.: Modularity lifting beyond the Taylor-Wiles method, preprintGoogle Scholar
- 5.Dieudonné, J., Grothendieck, A.: Eléments de géométrie algébrique, Publ. Math. IHES, 4, 8, 11, 17, 20, 24, 28, 32, 1960–7Google Scholar
- 6.Hartshorne, R.: Residues and Duality. Lecture Notes in Mathematics, vol. 20. Springer, New York (1966)Google Scholar
- 7.Kisin, M.: Moduli of finite flat group schemes, and modularity. Ann. Math.
**170**(3), 1085–1180 (2009)MathSciNetCrossRefMATHGoogle Scholar - 8.Kisin, M.: Modularity of 2-adic Barsotti-Tate representations. Invent. Math.
**178**(3), 587–634 (2009)MathSciNetCrossRefMATHGoogle Scholar - 9.Kisin, M.: Modularity of 2-dimensional Galois representations. Curr. Dev. Math.
**2005**, 191–230 (2005)CrossRefMATHGoogle Scholar - 10.Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture. II. Invent. Math.
**178**(3), 505–586 (2009)MathSciNetCrossRefMATHGoogle Scholar - 11.Weyman, J.: Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics, vol. 149. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar