The inverse deformation problem

TL;DR

This paper proves that for any complete local noetherian ring with a finite residue field, there exists a profinite group and an irreducible representation whose universal deformation ring is that ring, solving the inverse deformation problem.

Contribution

It establishes the existence of a profinite group and an irreducible representation for any given ring, providing a solution to the inverse deformation problem.

Findings

Universal deformation rings can be realized as rings of given algebraic structures.

Existence of specific profinite groups corresponding to arbitrary deformation rings.

Construction of irreducible representations with prescribed deformation properties.

Abstract

We show the inverse deformation problem has an affirmative answer: given a complete local noetherian ring $A$ with finite residue field $\pmb{k}$, we show that there is a topologically finitely generated profinite group $\Gamma$ and an absolutely irreducible continuous representation $\bar\rho:\Gamma\to GL_n(\pmb{k})$ such that $A$ is the universal deformation ring for $\Gamma,\bar\rho$.