Local universal lifting spaces of mod l Galois representations

Generic local deformation rings when

“…We explain how to deduce Theorem 2.16 from the inertially unipotent case (Corollary 2.24). The argument is essentially that of [CHT08, Corollary 2.13] and [Cho17, Proposition 2.6], albeit phrased slightly differently.…”

Untitled

“…Then one can check (and this is what is done in [Choi 2009, Proposition 2.0.5]) that this is in fact an isomorphism, and so we get the claimed isomorphism of prorepresenting objects.…”

“…The first two parts are in [Clozel et al 2008]: part (1) is Lemma 2.4.12 and part (2) is Corollary 2.4.13. Part ( 3) is the refinement to framed deformations obtained by keeping track of a basis in the construction of part (1) of the proposition, as in [Choi 2009, Proposition 2.0.5].…”

“…For n-dimensional representations the unrestricted framed deformation ring R (ρ) is always Cohen-Macaulay (in fact, a complete intersection; this is due to David Helm, building on work of Choi [2009]). It is natural to wonder whether the rings obtained by fixing the semisimplified restriction to inertia are always Cohen-Macaulay.…”