Derived Galois deformation rings

Abstract

We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring R classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group π0R recovers Mazur's deformation ring.We give evidence that these rings R occur in the wild: For suitable Galois representations, the Langlands program predicts that π0R 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 π˚R on the homology.

Patching and the Completed Homology of Locally Symmetric Spaces

“…an example of the main theorem of [36]. Note that it is not obvious that the action of Tor R loc p * (R S , R loc p (σ )) on H * (X K 0 U p , σ • ) m is independent of the choice of non-principal ultrafilter made to carry out the patching.…”

Derived Hecke Algebra and Cohomology of Arithmetic Groups

“…We will make assumptions very close to [14, Conjecture 6.1]. We briefly summarize them and refer the reader to [14] for full details:…”

“…for (respectively) the étale cohomology of M (equivalently the group cohomology of GalpQ S {Qq with coefficients in M ), and the subset of this group consisting of classes 14 Why the coadjoint representation rather than the adjoint? They are isomorphic for G semisimple, at least away from small characteristic.…”

Hodge theory of $p$-adic varieties: a survey

“…(1) geometrically, for varieties over local fields: they can be defined for any algebraic variety over K [58], [27] under the name of syntomic cohomology groups and are used as an approximation of p-adic motivic cohomology (a refinement of p-adic étale cohomology capturing classes coming from geometry); (2) globally (over number fields): they can be globalized and extended to all H i (and not only H 1 ); see [60] for a direct construction and [44] for reinterpretations via derived Galois deformation rings.…”