Simplicial Galois Deformation Functors

TL;DR

This paper advances the understanding of simplicial Galois deformation functors by introducing pseudo-characters, analyzing their deformation relations, and connecting the algebraic structures to Galois cohomology and Selmer groups.

Contribution

It introduces simplicial pseudo-characters, explores their deformation functors, and relates the cotangent complex to Galois cochains, extending the algebraic framework of simplicial Galois deformations.

Findings

Relations between deformation functors of pseudo-characters and Galois representations established.

The relative cotangent complex is linked to Galois cochains in the ordinary case.

Connection between the fundamental group of the deformation ring and Selmer groups demonstrated.

Abstract

In a recent work of Galatius and Venkatesh, the authors showed the importance of studying simplicial generalizations of Galois deformation functors. They established a precise link between the simplicial universal deformation ring $R$ prorepresenting such a deformation problem (with local conditions) and a derived Hecke algebra. Here we focus on the algebraic part of their study which we complete in two directions. First, we introduce the notion of simplicial pseudo-characters and prove relations between the (derived) deformation functors of simplicial pseudo-characters and that of simplicial Galois representations. Secondly, we define the relative cotangent complex of a simplicial deformation functor and, in the ordinary case, we relate it to the relative complex of ordinary Galois cochains. Finally, we recall how the latter can be used to relate the fundamental group of $R$ to the ordinary dual adjoint Selmer group, by a homomorphism already introduced in Galatius-Venkatesh and studied in greater generality in Tilouine-Urban.