Singularities of ordinary deformation rings

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.