The long exact sequence of continuous cohomology in every degree #
A short exact sequence 0 → A → B → C → 0 of discrete G-modules over a compact topological
group G induces a long exact sequence
⋯ → Hⁿ(G, A) → Hⁿ(G, B) → Hⁿ(G, C) --δ--> Hⁿ⁺¹(G, A) → Hⁿ⁺¹(G, B) → ⋯
of Mathlib's canonical continuous cohomology continuousCohomology n, in every degree n. This
file constructs the connecting map δ and proves exactness at the three repeating nodes and
naturality of δ in compatible pairs: along a continuous homomorphism φ : H →ₜ* G of compact
groups, maps of short exact sequences that are equivariant along φ carry δ over G to δ
over H. Morphisms of short exact sequences over G and restriction to a compact subgroup are the
two instances stated here; inflation is stated in
TauCeti.RepresentationTheory.Homological.ContCohomology.Inflation.ConnectingMap.
The construction starts from the short complex of homogeneous-cochain complexes
TauCeti.ContCohomology.DiscreteShortExact.continuousCochainsShortExact, which becomes a short
exact sequence of cochain complexes of ℤ-modules after forgetting topologies. TopModuleCat ℤ
is not abelian, so the snake lemma (CategoryTheory.ShortComplex.ShortExact.δ) is applied in
ModuleCat ℤ. The
forgetful functor TopModuleCat ℤ ⥤ ModuleCat ℤ is both a left and a right adjoint, so it
preserves homology, and CategoryTheory.ShortComplex.mapHomologyIso identifies the homology of the
forgotten complexes with the underlying modules of continuous cohomology. This produces δ as a
linear map. It is continuous because continuous cohomology of a discrete representation of a
compact group is discrete (TauCeti.discreteTopology_continuousCohomology), so δ is a morphism
in TopModuleCat ℤ. The same identification transports the exactness statements and the
naturality squares from ModuleCat ℤ.
The coefficient maps are the named TauCeti.ContinuousCohomology.coeffMap of the canonical
coefficient maps TauCeti.ofDiscreteModuleMap, the form in which a consumer meets them.
Main definitions #
TauCeti.ContCohomology.DiscreteShortExact.delta: the connecting mapHⁿ(G, C) ⟶ Hⁿ⁺¹(G, A)inTopModuleCat ℤ.
Main results #
TauCeti.ContCohomology.DiscreteShortExact.forget₂_map_delta:δis the snake-lemma connecting map of the forgotten cochain sequence, read throughmapHomologyIso.TauCeti.ContCohomology.DiscreteShortExact.delta_apply:δon representatives: lift a cocycle onCto a cochain onB, differentiate, and read the result as a cocycle onA.TauCeti.ContCohomology.DiscreteShortExact.isIso_delta:δis an isomorphism whenHⁿ(G, B)andHⁿ⁺¹(G, B)vanish.TauCeti.ContCohomology.DiscreteShortExact.longExact_exact₁,longExact_exact₂andlongExact_exact₃: exactness atHⁿ⁺¹(G, A),Hⁿ(G, B)andHⁿ(G, C).TauCeti.ContCohomology.DiscreteShortExact.coeffMap_proj_injectiveandcoeffMap_incl_injective:Hⁿ(G, B) → Hⁿ(G, C)is injective whenHⁿ(G, A)vanishes, andHⁿ⁺¹(G, A) → Hⁿ⁺¹(G, B)is injective whenHⁿ(G, C)vanishes.TauCeti.ContCohomology.DiscreteShortExact.coeffMap_proj_surjective:Hⁿ(G, B) → Hⁿ(G, C)is surjective whenHⁿ⁺¹(G, A)vanishes.TauCeti.ContCohomology.DiscreteShortExact.delta_map: the maps induced by compatible pairs commute withδ.TauCeti.ContCohomology.DiscreteShortExact.delta_naturality: a morphism of short exact sequences commutes withδ.TauCeti.ContCohomology.DiscreteShortExact.delta_res: restriction to a compact subgroup commutes withδ.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Springer (2008), (1.3.2) (the long exact cohomology sequence of a short exact sequence of discrete modules) and Ch. I, §5 (compatibility of the connecting maps with change of groups).
The connecting map of continuous cohomology δ : Hⁿ(G, C) ⟶ Hⁿ⁺¹(G, A) attached to a
short exact sequence 0 → A → B → C → 0 of discrete G-modules over a compact group, in every
degree n. It is the snake-lemma connecting map of the short exact sequence of homogeneous
continuous cochains (forget₂_map_delta); it is continuous because its source is discrete.
Equations
- One or more equations did not get rendered due to their size.
Instances For
After forgetting topologies, δ is the connecting map of the snake lemma for the short exact
sequence of homogeneous-cochain complexes, conjugated by the identifications
CategoryTheory.ShortComplex.mapHomologyIso of the homology of the forgotten complexes with the
underlying modules of continuous cohomology.
The connecting map on representatives. Let z₃ be a homogeneous n-cocycle with values in
C, x₂ a homogeneous n-cochain with values in B lifting it, and z₁ a homogeneous
(n + 1)-cocycle with values in A whose image in B is the differential of x₂. Then δ sends
the class of z₃ to the class of z₁. This is the continuous counterpart of Mathlib's
CategoryTheory.ShortComplex.ShortExact.δ_apply, and the form in which δ is compared with
explicit connecting maps.
The connecting map is an isomorphism when the middle term is acyclic in the two adjacent
degrees: if Hⁿ(G, B) and Hⁿ⁺¹(G, B) vanish, then δ : Hⁿ(G, C) ⟶ Hⁿ⁺¹(G, A) is an
isomorphism of topological modules. This is the snake-lemma statement
CategoryTheory.ShortComplex.ShortExact.isIso_δ on the forgotten cochain complexes, transported
to the discrete cohomology modules.
Exactness at Hⁿ⁺¹(G, A): the image of the connecting map Hⁿ(G, C) ⟶ Hⁿ⁺¹(G, A) is
the kernel of the coefficient map induced by A → B.
Exactness at Hⁿ(G, B): the image of the coefficient map induced by A → B is the
kernel of the coefficient map induced by B → C. No connecting map is involved, so local
compactness of G suffices.
Exactness at Hⁿ(G, C): the image of the coefficient map induced by B → C is the
kernel of the connecting map Hⁿ(G, C) ⟶ Hⁿ⁺¹(G, A).
The connecting map kills the image of Hⁿ(G, B) → Hⁿ(G, C).
The connecting map kills the image of Hⁿ(G, B) → Hⁿ(G, C).
The coefficient map Hⁿ⁺¹(G, A) → Hⁿ⁺¹(G, B) kills the image of the connecting map.
The coefficient map Hⁿ⁺¹(G, A) → Hⁿ⁺¹(G, B) kills the image of the connecting map.
If Hⁿ(G, A) vanishes, the coefficient map Hⁿ(G, B) → Hⁿ(G, C) is injective, by exactness
at Hⁿ(G, B).
If Hⁿ(G, C) vanishes, the coefficient map Hⁿ⁺¹(G, A) → Hⁿ⁺¹(G, B) is injective, by
exactness at Hⁿ⁺¹(G, A).
If Hⁿ⁺¹(G, A) vanishes, the coefficient map Hⁿ(G, B) → Hⁿ(G, C) is surjective, by
exactness at Hⁿ(G, C): the connecting map δ : Hⁿ(G, C) ⟶ Hⁿ⁺¹(G, A) is zero, so every class
of Hⁿ(G, C) lies in its kernel, which is the image of Hⁿ(G, B).
Naturality of the connecting map in compatible pairs. Let φ : H →ₜ* G be a continuous
homomorphism of compact groups, S a short exact sequence of discrete G-modules and T one of
discrete H-modules, and let fA, fB, fC be additive maps from the terms of S to those of
T that are equivariant along φ (f (φ h • x) = h • f x) and commute with the inclusions and
the projections. Then the maps these compatible pairs induce on continuous cohomology carry the
connecting map of S to that of T:
Hⁿ(G, C) ---δ---> Hⁿ⁺¹(G, A)
| |
(φ, fC) (φ, fA)
v v
Hⁿ(H, C') --δ--> Hⁿ⁺¹(H, A')
Coefficient maps (delta_naturality, at φ = id), restriction (delta_res) and inflation
(delta_infl) are instances of this square.
Naturality of the connecting map in compatible pairs. Let φ : H →ₜ* G be a continuous
homomorphism of compact groups, S a short exact sequence of discrete G-modules and T one of
discrete H-modules, and let fA, fB, fC be additive maps from the terms of S to those of
T that are equivariant along φ (f (φ h • x) = h • f x) and commute with the inclusions and
the projections. Then the maps these compatible pairs induce on continuous cohomology carry the
connecting map of S to that of T:
Hⁿ(G, C) ---δ---> Hⁿ⁺¹(G, A)
| |
(φ, fC) (φ, fA)
v v
Hⁿ(H, C') --δ--> Hⁿ⁺¹(H, A')
Coefficient maps (delta_naturality, at φ = id), restriction (delta_res) and inflation
(delta_infl) are instances of this square.
Naturality of the connecting map. Equivariant maps fA, fB, fC from one short exact
sequence of discrete G-modules to another, commuting with the inclusions and the projections,
carry the connecting map of the first sequence to that of the second:
Hⁿ(G, C) ---δ---> Hⁿ⁺¹(G, A)
| |
fC fA
v v
Hⁿ(G, C') --δ--> Hⁿ⁺¹(G, A')
Naturality of the connecting map. Equivariant maps fA, fB, fC from one short exact
sequence of discrete G-modules to another, commuting with the inclusions and the projections,
carry the connecting map of the first sequence to that of the second:
Hⁿ(G, C) ---δ---> Hⁿ⁺¹(G, A)
| |
fC fA
v v
Hⁿ(G, C') --δ--> Hⁿ⁺¹(G, A')
Restriction commutes with the connecting map. For a compact subgroup T of G, restricting
the connecting map of S to T gives the connecting map of the restricted sequence
S.restrict T:
Hⁿ(G, C) ---δ---> Hⁿ⁺¹(G, A)
| |
res res
v v
Hⁿ(T, C) ---δ---> Hⁿ⁺¹(T, A)
The restriction of ofDiscreteModule ℤ G M to T is ofDiscreteModule ℤ T M by definition
(TauCeti.res_ofDiscreteModule), which is how the two sides compose.
Restriction commutes with the connecting map. For a compact subgroup T of G, restricting
the connecting map of S to T gives the connecting map of the restricted sequence
S.restrict T:
Hⁿ(G, C) ---δ---> Hⁿ⁺¹(G, A)
| |
res res
v v
Hⁿ(T, C) ---δ---> Hⁿ⁺¹(T, A)
The restriction of ofDiscreteModule ℤ G M to T is ofDiscreteModule ℤ T M by definition
(TauCeti.res_ofDiscreteModule), which is how the two sides compose.