Presentations of non-commutative deformation rings via A_∞-algebras and applications to deformations of Galois representations and pseudorepresentations

2018/09/07 by Wang-Erickson, Carl
#11F80 #14A22 (primary) #14D15 #16E45 (secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1809.02484

Abstract

We introduce an A_∞-algebra structure on the Hochschild cohomology of the endomorphism bimodule of a finite-dimensional representation of an associative algebra. We prove that this structure determines a presentation for non-commutative deformations of the representation. From this, we deduce presentations of universal deformation rings of Galois representations. In turn, we apply these presentations in order to deduce universal deformation rings of Galois pseudorepresentations, supplying a a tangent and obstruction theory for pseudorepresentations. This generalizes the broadly used tangent and obstruction theory for Galois representations. We also give applications, calculating the ranks of certain Hecke algebras.

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