Singularities of Ordinary Deformation Rings

Let R be the universal deformation ring of a residual representation of a local Galois group. Kisin showed that many loci in MaxSpec(R[1/p]) of interest are Zariski closed, and gave a way to study the generic fiber of the corresponding quotient of R. 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 V0 is the trivial two dimensional representation and let R be the unique Zp-flat and reduced quotient of R such that 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 irreducile components of Spec(R) are normal, Cohen–Macaulay and not Gorenstein. This has two well-known applications to global deformation rings: first, a global deformation ring employing these local conditions is torsion-free; and second, Kisin’s R[1/p] = T[1/p] theorem can be upgraded in this setting to an R = T theorem.