Inner automorphisms act trivially on continuous cohomology #
Let G be a locally compact topological group, X a smooth discrete topological representation
of G, and let g : G. The compatible pair consisting of the inner automorphism
x ↦ g⁻¹ * x * g of G and the
action X.ρ g of g on the coefficients induces an endomorphism of Hⁿ(G, X), the conjugation
map g_* of Neukirch–Schmidt–Wingberg (Chapter I, §5). This file proves that it is the identity
in every degree (TauCeti.ContinuousCohomology.map_eq_id_of_inner), for Mathlib's canonical
continuousCohomology.
The argument #
Mathlib computes Hⁿ(G, X) from the coinduced resolution Xₙ = C(G, C(G, …, C(G, X))), whose
differential is the recursion (d F) x = F - d (F x); the homogeneous n-cochains are the
G-invariant elements of Xₙ₊₁. On such an invariant element the cochain map of the compatible
pair is right translation of every argument by g,
F (x₀, …, xₙ) ↦ F (x₀ * g, …, xₙ * g),
because invariance turns g • F (g⁻¹ * x₀ * g, …) into F (x₀ * g, …)
(TauCeti.ContinuousCohomology.resolutionMap_hom_apply_eq_resolutionTranslate). Right translation
by a makes sense on the whole resolution, as the G-equivariant chain map
TauCeti.ContinuousCohomology.resolutionTranslate, (T F) x = T (F (x * a)), and it is
chain-homotopic to the identity through the prism operator
TauCeti.ContinuousCohomology.translateHomotopy,
(h F) (x₀, …, xₙ₋₁) = ∑ᵢ (-1)ⁱ F (x₀, …, xᵢ, xᵢ * a, …, xₙ₋₁ * a),
which on the curried resolution is the recursion (h F) x = T (F x (x * a)) - h (F x). The
homotopy identity d (h F) + h (d F) = T F - F
(TauCeti.ContinuousCohomology.d_translateHomotopy_add_translateHomotopy_d) holds for every
element of the resolution, invariant or not, and h is G-equivariant, so for a homogeneous
cocycle z the difference T z - z is the coboundary of the homogeneous cochain h z.
The prism operator evaluates a curried cochain along the graph x ↦ (x, x * a), and it is
continuous in F because uncurrying C(G, C(G, Y)) → C(G × G, Y) is continuous; that is where
local compactness of G is used, and it is the only hypothesis beyond Mathlib's standing ones.
Every profinite group is compact, hence locally compact.
Main definitions #
TauCeti.ContinuousCohomology.resolutionTranslate: right translation of every argument bya, as an endomorphism of each term of the coinduced resolution.TauCeti.ContinuousCohomology.translateHomotopy: the prism operator, a chain homotopy from the identity toresolutionTranslate.
Main results #
TauCeti.ContinuousCohomology.resolutionTranslate_d_apply: right translation is a chain map.TauCeti.ContinuousCohomology.d_translateHomotopy_add_translateHomotopy_d: the homotopy identityd (h F) + h (d F) = T F - F.TauCeti.ContinuousCohomology.translateHomotopy_ρ: the prism operator isG-equivariant.TauCeti.ContinuousCohomology.map_eq_id_of_inner: inner automorphisms act trivially on continuous cohomology in every degree.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Chapter I, §5,
for the conjugation maps
g_*. - K. S. Brown, Cohomology of Groups, GTM 87, Chapter III, §8, for the triviality of inner conjugation on the cohomology of a discrete group.
Right translation on the coinduced resolution #
Right translation on the coinduced resolution: the endomorphism of the n-th term
C(G, C(G, …, C(G, X))) translating every argument on the right by a,
F (x₁, …, xₙ) ↦ F (x₁ * a, …, xₙ * a), defined by the recursion (T F) x = T (F (x * a)).
Left and right translations commute, so it is G-equivariant.
Equations
- One or more equations did not get rendered due to their size.
- TauCeti.ContinuousCohomology.resolutionTranslate X a 0 = CategoryTheory.CategoryStruct.id X
Instances For
Right translation on a successor term of the resolution, at a point:
(T F) x = T (F (x * a)).
Translation by the identity acts as the identity on every term of the resolution.
Translation by a product is the composite of the two right translations.
Right translation is a chain map: it commutes with the differential of the coinduced resolution.
The prism operator #
The prism operator of right translation by a, from the (n + 1)-st to the n-th term
of the coinduced resolution. In uncurried form it is
(h F) (x₀, …, xₙ₋₁) = ∑ᵢ (-1)ⁱ F (x₀, …, xᵢ, xᵢ * a, …, xₙ₋₁ * a); on the curried resolution it
is the recursion (h F) x = T (F x (x * a)) - h (F x) with T = resolutionTranslate X a, starting
from h = 0. It is a chain homotopy from the identity to T
(d_translateHomotopy_add_translateHomotopy_d). Evaluating along the graph x ↦ (x, x * a) is
continuous in F because G is locally compact.
Equations
- One or more equations did not get rendered due to their size.
- TauCeti.ContinuousCohomology.translateHomotopy X a 0 = 0
Instances For
The prism operator on a successor term, at a point: (h F) x = T (F x (x * a)) - h (F x).
The homotopy identity d (h F) + h (d F) = T F - F: the prism operator is a chain homotopy
from the identity to right translation. It holds for every element of the resolution, invariant or
not.
The prism operator is G-equivariant, because left and right translations commute.
The prism operator preserves G-invariant elements, that is, homogeneous cochains.
On an element killed by the differential, right translation differs from the identity by the differential of the prism operator.
Inner conjugation on cohomology #
On the coinduced resolution, the map of the inner compatible pair (x ↦ g⁻¹ * x * g, X.ρ g)
is right translation by g after the action of g. In particular it is right translation by g
on G-invariant elements.
Inner automorphisms act trivially on continuous cohomology. For g : G, the compatible
pair consisting of the inner automorphism x ↦ g⁻¹ * x * g of G and the action of g on the
coefficients, which is NSW's conjugation g_*, induces the identity of Hⁿ(G, X) in every degree.
The homomorphism and the coefficient map are taken as hypotheses on their values, so that the
statement applies to any presentation of the pair. The coefficient representation is smooth
discrete: its underlying module has the discrete topology and every point stabilizer is open.