A universal deformation ring with unexpected Krull dimension

TL;DR

This paper constructs a counterexample to Gouva's Dimension conjecture, showing that the universal deformation ring can have a greater Krull dimension than previously expected, challenging established bounds.

Contribution

It provides the first known counterexample to the conjecture, demonstrating unexpected behavior in the Krull dimension of universal deformation rings.

Findings

Counterexample with larger Krull dimension

Challenges existing bounds on deformation ring dimensions

Implications for deformation theory and Galois representations

Abstract

A well known result of B. Mazur gives a lower bound for the Krull dimension of the universal deformation ring associated to an absolutely irreducible residual representation in terms of the group cohomology of the adjoint representation. The question about equality - at least in the Galois case - also goes back to B. Mazur. In the general case the question about equality is the subject of Gouv\^{e}a's "Dimension conjecture". In this note we provide a counterexample to this conjecture. More precisely, we construct an absolutely irreducible residual representation with smooth universal deformation ring of strict greater Krull dimension as expected.