The Mackey double-coset formula for low-degree corestriction #
Let U be a finite-index subgroup of a group G, V any subgroup, and M a G-module. The
Mackey double-coset formula computes restriction to V of the corestriction from U:
res^G_V ∘ cor^G_U = ∑_{VsU ∈ V \ G / U} cor^V_{V ⊓ sUs⁻¹} ∘ (s)_* ∘ res^U_{U ⊓ s⁻¹Vs},
where (s)_* is conjugation by s, the compatible pair (x ↦ s⁻¹ x s, m ↦ s • m). Each summand
is built from a representative s = r D of its double coset D, for an arbitrary choice r of
representatives, and the subgroup
V ⊓ sUs⁻¹ is Tau Ceti's TauCeti.mackeySubgroup s U V, read inside V. The composite
(s)_* ∘ res^U_{U ⊓ s⁻¹Vs} is the single compatible-pair pullback along
TauCeti.mackeyToH s U V : V ⊓ sUs⁻¹ → U, y ↦ s⁻¹ y s, with coefficient map m ↦ s • m; the
summand is TauCeti.ContCohomology.explicitMackeyTerm0 in degree zero and its companions
explicitMackeyTerm1 and explicitMackeyTerm2 in degrees one and two.
The proof is a choice of transversal. The left cosets G ⧸ U split, along
TauCeti.mackeyQuotientEquiv, as the disjoint union over D ∈ V \ G / U of the V-orbits
V ⧸ (V ⊓ sUs⁻¹), and Subgroup.mackeyTransversal picks the representative v.out * r D of the
coset indexed by (D, v). For k ∈ V its transversal word is the conjugate of the transversal
word of the Mackey subgroup in V (Subgroup.lWord_mackeyTransversal):
ℓ_{(D, v)}(k) = s⁻¹ ℓ_v(k) s.
Sorting the corestriction sum over G ⧸ U by double cosets therefore gives the Mackey formula
exactly on cochains for this transversal (cochainsCor1_mackeyTransversal,
cochainsCor2_mackeyTransversal), and independence of the transversal
(TauCeti.ContCohomology.explicitCor1_eq_transversal and its degree-zero and degree-two
counterparts) turns that into the statement for the canonical corestriction
(explicitCor0_mackey, explicitCor1_mackey, explicitCor2_mackey). In positive degrees
openness of U is what makes the corestrictions continuous, and it makes each Mackey subgroup open
in V (Subgroup.isOpen_mackeySubgroup_subgroupOf); V is arbitrary.
Since the formula holds for every choice of representatives, comparing two choices that differ at a
single double coset shows that each summand depends only on the double coset VsU
(explicitMackeyTerm0_eq_of_mk_eq, explicitMackeyTerm1_eq_of_mk_eq,
explicitMackeyTerm2_eq_of_mk_eq).
Main definitions #
Subgroup.mackeyTransversal: the transversal ofG ⧸ Uadapted to the double cosetsV \ G / U.Subgroup.continuousMackeyToH: the conjugationV ⊓ sUs⁻¹ → Uas a continuous homomorphism;Subgroup.continuous_mackeyToHstates the continuity ofTauCeti.mackeyToHitself.TauCeti.ContCohomology.explicitMackeyTerm0,explicitMackeyTerm1,explicitMackeyTerm2: the summandcor^V_{V ⊓ sUs⁻¹} ∘ (s)_* ∘ resattached tosin degrees zero, one and two.
Main results #
Subgroup.lWord_mackeyTransversal: the transversal word of the adapted transversal at an element ofVis the conjugate of a transversal word of the Mackey subgroup inV.TauCeti.ContCohomology.cochainsCor1_mackeyTransversal,TauCeti.ContCohomology.cochainsCor2_mackeyTransversal: the Mackey formula on cochains.TauCeti.ContCohomology.explicitCor0_mackey,explicitCor1_mackey,explicitCor2_mackey: the Mackey double-coset formula onH⁰,H¹andH², for any choice of double-coset representatives.TauCeti.ContCohomology.explicitMackeyTerm0_eq_of_mk_eq,explicitMackeyTerm1_eq_of_mk_eq,explicitMackeyTerm2_eq_of_mk_eq: each summand is independent of the choice of representative of its double coset.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.5.6).
- K. S. Brown, Cohomology of Groups, III §9.
The transversal of G ⧸ U adapted to the double cosets V \ G / U and to a choice r of
representatives r D ∈ D: the coset indexed by (D, v) under TauCeti.mackeyQuotientEquiv is
represented by v.out * r D, the representative r D of the double coset translated by the chosen
representative v.out ∈ V of a coset of the Mackey subgroup V ⊓ sUs⁻¹, s = r D.
Instances For
The adapted transversal represents the coset indexed by (D, w) by w.out * r D.
Subgroup.mackeyTransversal is a transversal: it picks a representative of every coset.
The transversal word of the adapted transversal. At an element k of V and the coset
indexed by (D, w), the transversal word of Subgroup.mackeyTransversal is the conjugate by
r D of the transversal word of the Mackey subgroup V ⊓ (r D) U (r D)⁻¹ in V, taken at the
canonical transversal Quotient.out.
For an open subgroup U of a topological group, every Mackey subgroup V ⊓ sUs⁻¹ is open in
V.
The conjugation TauCeti.mackeyToH s U V : V ⊓ sUs⁻¹ → U, y ↦ s⁻¹ y s, as a continuous
homomorphism.
Instances For
Subgroup.continuousMackeyToH evaluates as TauCeti.mackeyToH.
The homomorphism underlying Subgroup.continuousMackeyToH is TauCeti.mackeyToH.
The conjugation TauCeti.mackeyToH s U V : V ⊓ sUs⁻¹ → U, y ↦ s⁻¹ y s, is continuous.
Conjugation by s and the action of s on the coefficients form a compatible pair from U
to the Mackey subgroup V ⊓ sUs⁻¹.
TauCeti.ContCohomology.smul_mackeyToH_smul for the continuous conjugation
Subgroup.continuousMackeyToH, the form taken by the compatible-pair maps in positive degrees.
Degree zero #
The summand of the degree-zero Mackey formula attached to s:
cor^V_{V ⊓ sUs⁻¹} ∘ (s)_* ∘ res, sending a U-invariant m to the norm of s • m from the
Mackey subgroup up to V.
Instances For
The degree-zero Mackey summand is the norm, over the canonical transversal of the Mackey
subgroup in V, of the translate s • m.
The Mackey double-coset formula in degree zero (NSW (1.5.6)): restricting to V the
corestriction of a U-invariant is the sum, over the double cosets V \ G / U, of the
corestrictions from the Mackey subgroups of its conjugates, for any choice r of representatives
of the double cosets.
The degree-zero Mackey summand depends only on the double coset VsU of s.
Degree one #
The Mackey formula for the degree-one corestriction cochain. At the adapted transversal
Subgroup.mackeyTransversal, the restriction to V of the corestriction of a 1-cochain f on
U is, on the nose, the sum over the double cosets V \ G / U of the corestrictions, from the
Mackey subgroups to V at the canonical transversal, of the conjugates
y ↦ s • f (s⁻¹ y s) of f.
The summand of the degree-one Mackey formula attached to s:
cor^V_{V ⊓ sUs⁻¹} ∘ (s)_* ∘ res : H¹(U, M) → H¹(V, M).
Instances For
The degree-one Mackey summand sends the class of a continuous 1-cocycle to the class of the
corestriction, at the canonical transversal, of its conjugate.
The Mackey double-coset formula in degree one (NSW (1.5.6)): for an open subgroup U of
finite index and any subgroup V,
res^G_V ∘ cor^G_U = ∑_{VsU} cor^V_{V ⊓ sUs⁻¹} ∘ (s)_* ∘ res on H¹(U, M), for any choice r of
representatives s = r D of the double cosets.
The degree-one Mackey summand depends only on the double coset VsU of s.
Degree two #
The Mackey formula for the degree-two corestriction cochain, the degree-two counterpart of
TauCeti.ContCohomology.cochainsCor1_mackeyTransversal: at the adapted transversal the identity
holds on the nose.
The summand of the degree-two Mackey formula attached to s:
cor^V_{V ⊓ sUs⁻¹} ∘ (s)_* ∘ res : H²(U, M) → H²(V, M).
Instances For
The degree-two Mackey summand sends the class of a continuous 2-cocycle to the class of the
corestriction, at the canonical transversal, of its conjugate.
The degree-two Mackey summand is the corestriction from the Mackey subgroup of the pullback
along conjugation by s.
The Mackey double-coset formula in degree two (NSW (1.5.6)): for an open subgroup U of
finite index and any subgroup V,
res^G_V ∘ cor^G_U = ∑_{VsU} cor^V_{V ⊓ sUs⁻¹} ∘ (s)_* ∘ res on H²(U, M), for any choice r of
representatives s = r D of the double cosets.
The degree-two Mackey summand depends only on the double coset VsU of s.