Derived Galois deformation rings

Abstract

We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring $\mathcal{R}$ classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group ${\pi }_0\mathcal{R}$ recovers Mazur's deformation ring. We give evidence that these rings $\mathcal{R}$ occur in the wild: For suitable Galois representations, the Langlands program predicts that ${\pi }_0\mathcal{R}$ should act on the homology of an arithmetic group. We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of ${\pi }_{\ast }\mathcal{R}$ on the homology.

This is the submitted version. A few minor changes from version 1. Metadata and added acknowledgement in v3

References in corpus (3)

• Higher Topos Theory
• Homotopy limits and colimits and enriched homotopy theory
• Absolute algebra and Segal's Gamma sets

Cited by in corpus (19)

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• Representation homology of topological spaces
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• Derived deformation rings allowing congruences]{Derived deformation rings allowing congruences
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• Period Rings with Big Coefficients and Application II
• Critical Lambda-adic modular forms and bi-ordinary complexes
• Topics on Geometric and Representation Theoretic Aspects of Period Rings I