The inflation-restriction sequence in every positive degree #
Let S be a normal subgroup of a group G and A a representation of G. Inflation and
restriction form a complex
Hⁿ⁺¹(G ⧸ S, A^S) ⟶ Hⁿ⁺¹(G, A) ⟶ Hⁿ⁺¹(S, A),
and if Hⁱ(S, A) = 0 for 0 < i ≤ n, it is exact and inflation is injective (Milne II 1.34).
Mathlib proves the case n = 0, where there is no hypothesis, as groupCohomology.H1InfRes.
The hypotheses concern the cohomology of A restricted to S in degrees below the degree of
the complex. The file provides injectivity and exactness under these hypotheses.
When Hⁱ(S, A) vanishes also in degree n + 1, inflation is an isomorphism
(isIso_infRes_f). This is the form used for Tate's cohomological triviality criterion, where a
module is shown to be cohomologically trivial by induction along a normal series. Counting along
the exact sequence instead bounds the order of Hⁿ⁺¹(G, A) by those of its outer terms
(natCard_groupCohomology_succ_dvd_mul), the form used to bound the order of a cohomology group of
a solvable group by induction on the order of the group.
The same sequence is exact for any extension 1 → H → G → Q → 1 and representations B of Q
and C of H identified with A^H and with A restricted to H
(range_map_succ_eq_ker_map_succ). This is the form in which it applies to a tower of Galois
extensions K ⊆ L ⊆ M, where Gal(M/L) → Gal(M/K) → Gal(L/K) and the units of M fixed by
Gal(M/L) are the units of L.
Main definitions #
TauCeti.groupCohomology.infRes: the complexHⁿ⁺¹(G ⧸ S, A^S) ⟶ Hⁿ⁺¹(G, A) ⟶ Hⁿ⁺¹(S, A).
Main statements #
TauCeti.groupCohomology.mono_infRes_f: inflation is injective.TauCeti.groupCohomology.infRes_exact: the inflation-restriction sequence is exact.TauCeti.groupCohomology.isIso_infRes_f: inflation is an isomorphism whenHⁱ(S, A) = 0for0 < i ≤ n + 1.TauCeti.groupCohomology.natCard_groupCohomology_succ_dvd_mul: the order ofHⁿ⁺¹(G, A)divides the product of the orders ofHⁿ⁺¹(G ⧸ S, A^S)andHⁿ⁺¹(S, A).TauCeti.groupCohomology.map_succ_injective,TauCeti.groupCohomology.range_map_succ_eq_ker_map_succ: injectivity and exactness for an extension1 → H → G → Q → 1.
References #
- J. S. Milne, Class Field Theory, v4.03, Chapter II, Proposition 1.34.
- J.-P. Serre, Local Fields, Chapter VII, §6, Proposition 5.
ClassFieldTheory/Cohomology/Functors/InflationRestriction.leaninkbuzzard/ClassFieldTheory, commitccc3323c6750abca25b49b35106f54eb3a398509, statesinflation_restriction_monoandinflation_restriction_exactand uses the same dimension-shifting argument.
Inflation followed by restriction vanishes in every positive degree.
The inflation-restriction complex Hⁿ⁺¹(G ⧸ S, A^S) ⟶ Hⁿ⁺¹(G, A) ⟶ Hⁿ⁺¹(S, A) in degree
n + 1. In degree one it is Mathlib's groupCohomology.H1InfRes.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The inflation-restriction complex as a short complex of the two maps.
The first term of the inflation-restriction complex.
The middle term of the inflation-restriction complex.
The inflation map in the inflation-restriction complex.
The restriction map in the inflation-restriction complex.
In degree one, infRes is Mathlib's H1InfRes.
Inflation is injective (Milne II 1.34): if Hⁱ(S, A) = 0 for 0 < i ≤ n, then inflation
Hⁿ⁺¹(G ⧸ S, A^S) ⟶ Hⁿ⁺¹(G, A) is a monomorphism.
The inflation-restriction sequence is exact (Milne II 1.34): if Hⁱ(S, A) = 0 for
0 < i ≤ n, then Hⁿ⁺¹(G ⧸ S, A^S) ⟶ Hⁿ⁺¹(G, A) ⟶ Hⁿ⁺¹(S, A) is exact.
Inflation is an isomorphism when Hⁱ(S, A) = 0 for 0 < i ≤ n + 1: then
Hⁿ⁺¹(G ⧸ S, A^S) ⟶ Hⁿ⁺¹(G, A) is injective, and surjective because restriction lands in
Hⁿ⁺¹(S, A) = 0.
Counting along the inflation-restriction sequence. If Hⁱ(S, A) = 0 for 0 < i ≤ n, the
order of Hⁿ⁺¹(G, A) divides the product of the orders of Hⁿ⁺¹(G ⧸ S, A^S) and Hⁿ⁺¹(S, A).
No finiteness is assumed; in particular Hⁿ⁺¹(G, A) is finite when the two outer groups are.
Inflation along a quotient map is injective. Let π : G →* Q be surjective and let
φ : B ⟶ A identify the Q-representation B with the invariants of A under ker π. If
Hⁱ(ker π, A) = 0 for 0 < i ≤ n, then inflation Hⁿ⁺¹(Q, B) ⟶ Hⁿ⁺¹(G, A) is injective.
The inflation-restriction sequence of a group extension. Let 1 → H → G → Q → 1 be exact,
given by ι and π, let φ : B ⟶ A identify the Q-representation B with the invariants of
A under ker π, and let ψ : A ⟶ C identify A, restricted to H, with C. If
Hⁱ(H, C) = 0 for 0 < i ≤ n, then inflation and restriction
Hⁿ⁺¹(Q, B) ⟶ Hⁿ⁺¹(G, A) ⟶ Hⁿ⁺¹(H, C) form an exact sequence.