The transgression #
Let N be a closed normal subgroup of a profinite group G and M a discrete G-module. The
transgression
tg : H¹(N, M)^{G ⧸ N} → H²(G ⧸ N, M ^ N)
is the fourth arrow of the inflation-restriction-transgression five-term sequence
0 → H¹(G ⧸ N, M ^ N) → H¹(G, M) → H¹(N, M)^{G ⧸ N} → H²(G ⧸ N, M ^ N) → H²(G, M).
It is defined by lifting a cocycle c on N whose class is conjugation-invariant to a continuous
cochain f on G, differentiating, and observing that d¹ f descends to a cocycle on G ⧸ N
with values in M ^ N. This file carries out that construction on the explicit low-degree model,
proves that the resulting class is independent of every choice made, and proves exactness of the
five-term sequence at the source and the target of the transgression.
The lift #
A continuous cochain f : G → M is a transgression lift of c : N → M
(TauCeti.ContCohomology.IsTransgressionLift) when
f (g * n) = f g + g • c n and g • c (g⁻¹ n g) - c n = n • f g - f g
for all g : G and n : N. These two identities are exactly what makes d¹ f constant on the
cosets of N in each variable and N-invariant in value, so it descends to
TauCeti.ContCohomology.IsTransgressionLift.cocycle. Lifts form a group under addition, the
coboundary of m on G lifts the coboundary of m on N, and a lift of 0 descends to a
continuous cochain on G ⧸ N; together these give the independence statement
TauCeti.ContCohomology.IsTransgressionLift.cocycle_sub_mem_B2, which says that cohomologous
functions on N have lifts whose descended coboundaries differ by an explicit 2-coboundary.
None of this uses more than a topological group and a topological module.
Existence is where profiniteness enters. Given a continuous section s of G → G ⧸ N, which
TauCeti.exists_continuous_section supplies for closed N, and a continuous choice F of
elements trivialising the conjugates of c, the cochain
g ↦ F (s (g N)) + s (g N) • c ((s (g N))⁻¹ * g)
is a lift (TauCeti.ContCohomology.transgressionLift, normalised to vanish at 1). The continuous
choice of F is TauCeti.ContCohomology.exists_continuous_smul_conj_sub_eq_d0: on a compact
group the conjugate of c by g depends on g only through a coset of one open subgroup, by
uniform local constancy, and discreteness of M makes a choice on those finitely many cosets
continuous.
Main definitions #
TauCeti.ContCohomology.IsTransgressionLift: the lifting condition, andTauCeti.ContCohomology.IsTransgressionLift.cocycle: the descended coboundary of a lift.TauCeti.ContCohomology.transgressionLift: the lift attached to a continuous section.TauCeti.ContCohomology.transgressionCochainandtransgressionCocycle: the raw transgression2-cocycle onG ⧸ N.TauCeti.ContCohomology.transgression: the transgression as an additive map.
Main statements #
TauCeti.ContCohomology.transgressionCochain_apply: the raw transgression at(q, r)isd¹of the lift at(s q, s r).TauCeti.ContCohomology.transgressionCochain_sub_mem_B2: changing the section or the representative changes the raw transgression by an explicit2-coboundary.TauCeti.ContCohomology.transgression_apply: the transgression is the class of the raw transgression for every section and every representative.TauCeti.ContCohomology.transgression_explicitResConj1: transgression kills the image of restriction.TauCeti.ContCohomology.explicitInfl2_transgression: inflation kills the image of transgression.TauCeti.ContCohomology.transgression_eq_mk_cocycle: the transgression is the class of the descended coboundary of any transgression lift of a representative.TauCeti.ContCohomology.fiveTerm_exact_H1NandTauCeti.ContCohomology.fiveTerm_exact_H2Q: exactness of the five-term sequence atH¹(N, M) ^ (G ⧸ N)and atH²(G ⧸ N, M ^ N). Together withTauCeti.ContCohomology.explicitInfl1_injectiveandTauCeti.ContCohomology.explicitInfResConj_exactthis is exactness at every node.TauCeti.ContCohomology.transgression_injective_iffandTauCeti.ContCohomology.transgression_surjective_iff: the transgression is injective exactly when restriction toNvanishes, and surjective exactly when inflation toH²(G, M)vanishes.TauCeti.ContCohomology.explicitInfl2_injective_of_subsingleton: inflation intoH²(G, M)is injective whenH¹(N, M)^{G ⧸ N}vanishes.TauCeti.ContCohomology.natCard_H1_mul_natCard_H2_quotient: the order count of the five-term sequence,|H¹(G, M)| · |H²(G ⧸ N, M ^ N)| = |H¹(G ⧸ N, M ^ N)| · |H¹(N, M)^{G ⧸ N}| · |im infl₂|, and its formTauCeti.ContCohomology.natCard_H1_mul_natCard_H2_quotient_of_subsingletonwhenH²(G, M) = 0.
Implementation notes #
Profiniteness and closedness of N are genuine hypotheses for producing a section: such a section
does not exist for an arbitrary topological group (the circle ℝ ⧸ ℤ has none). Total
disconnectedness is used only to produce the section; the construction for a given section needs
only compactness of G and N.
References #
The exactness arguments for the five-term sequence follow the classical cochain proofs in Neukirch--Schmidt--Wingberg (1.6.7), Ribes--Zalesskii Cor. 7.2.5(a), and Koch Thm. 3.14.
- J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.6.7).
- L. Ribes and P. Zalesskii, Profinite Groups, 2nd ed., Prop. 2.2.2 and Cor. 7.2.5.
- H. Koch, Galois Theory of p-Extensions, Thm. 3.14.
A transgression lift of a function c : N → M is a continuous cochain f : G → M
satisfying the two identities that make its coboundary descend to G ⧸ N:
f (g * n) = f g + g • c n, sofextendscalong rightN-translation;g • c (g⁻¹ n g) - c n = n • f g - f g, sof gtrivialises the difference betweencand its conjugate byg.
For a continuous 1-cocycle c on N whose class is invariant under conjugation, such lifts
exist when G is profinite and N is closed (TauCeti.ContCohomology.transgressionLift), and
the class of the descended coboundary TauCeti.ContCohomology.IsTransgressionLift.cocycle is the
transgression of the class of c.
- continuous : Continuous f
the cochain
fis continuous fextendscalong rightN-translation:f (g * n) = f g + g • c n- smul_conj_sub (g : G) (n : ↥N) : g • c ((N.inverseConjugationHom g) n) - c n = (ContCohomology.d0 (↥N) M) (f g) n
f gtrivialises the difference betweencand its conjugate byg
Instances For
The zero cochain is a transgression lift of the zero function.
Transgression lifts add.
Transgression lifts subtract.
The coboundary of m on G is a transgression lift of the coboundary of m on N.
A continuous cochain on G whose coboundary vanishes on G × N and on N × G is a
transgression lift of its own restriction to N.
A continuous 1-cocycle on G is a transgression lift of its restriction to N.
The value of a transgression lift at 1 is fixed by N.
On N, a transgression lift is the lifted function up to the constant f 1.
The function on N admitting a transgression lift is a continuous 1-cocycle.
Subtracting its value at 1 from a transgression lift gives a transgression lift of the same
function that vanishes at 1.
A transgression lift of the zero function is constant on the cosets of N.
A transgression lift of the zero function takes values fixed by N.
The coboundary of a transgression lift is unchanged by right N-translation of its second
argument.
The coboundary of a transgression lift is unchanged by right N-translation of its first
argument.
The coboundary of a transgression lift is constant, equal to f 1, on N × G.
The coboundary of a transgression lift takes values fixed by N.
The coboundary of a transgression lift is a continuous 2-cocycle on G.
The descended coboundary of a transgression lift f: the continuous 2-cocycle on
G ⧸ N with values in M ^ N whose value at (g N, h N) is d¹ f (g, h).
Equations
- hf.cocycle = TauCeti.ContCohomology.descendZ2 ⟨(TauCeti.ContCohomology.d1 G M) f, ⋯⟩ ⋯ ⋯
Instances For
The descended coboundary evaluates on quotient representatives as the coboundary of the lift.
The descended coboundary is additive in the lift.
A lift whose coboundary vanishes descends to the zero cocycle.
Independence of the lift. If c and c' differ by a 1-coboundary on N, the
descended coboundaries of any of their transgression lifts differ by a 2-coboundary on
G ⧸ N. The primitive is the difference of the two lifts, corrected by the coboundary on G of
the element trivialising c - c'; it is constant on the cosets of N and N-invariant, so it
descends to G ⧸ N.
A continuous 1-cocycle on N admitting a transgression lift has conjugation-invariant
class: the value of the lift at g trivialises the difference between the cocycle and its
conjugate by g.
If c and c' differ by a 1-coboundary on N, any of their transgression lifts have
the same class in H²(G ⧸ N, M ^ N).
Inflation kills the class of a descended coboundary: its inflation is the class of the
coboundary of a continuous cochain on G.
A continuous choice of conjugation primitives. Let N be a compact normal subgroup of a
compact group and c : N → M a continuous function to a discrete module such that, for each g,
the difference between the conjugate n ↦ g • c (g⁻¹ n g) and c is the coboundary of some
element. Then those elements can be chosen to depend continuously on g.
The conjugate depends on g only through a coset of a single open subgroup, by uniform local
constancy of (n, g) ↦ g • c (g⁻¹ n g) on the compact group N × G; a choice made on the
finitely many cosets is continuous.
The section-dependent lift used by transgression. Given a compact normal subgroup N, a
continuous section s of G → G ⧸ N, and a continuous 1-cocycle c on N whose class is
conjugation-invariant, this is a continuous 1-cochain on G extending c
(transgressionLift_apply_coe) and satisfying
the identities of TauCeti.ContCohomology.IsTransgressionLift, so that its coboundary descends
to G ⧸ N. It is g ↦ F (s (g N)) + s (g N) • c ((s (g N))⁻¹ * g), normalised to vanish at
1, where F is a continuous choice of elements trivialising the conjugates of c.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The section-dependent lift is a transgression lift of c.
The section-dependent lift vanishes at 1.
The section-dependent lift extends c.
The raw transgression 2-cochain, obtained by differentiating transgressionLift and
descending to G ⧸ N, with values in M ^ N.
Equations
- TauCeti.ContCohomology.transgressionCochain G M N hN s hs_cont hs c hc = ⟨↑⋯.cocycle, ⋯⟩
Instances For
The lift-and-differentiate formula. After the inclusion M ^ N ↪ M, the raw
transgression at (q, r) is d¹ of the lift at the chosen representatives (s q, s r).
The raw transgression is a continuous 2-cocycle.
The raw transgression bundled as a continuous 2-cocycle.
Equations
- TauCeti.ContCohomology.transgressionCocycle G M N hN s hs_cont hs c hc = ⟨↑(TauCeti.ContCohomology.transgressionCochain G M N hN s hs_cont hs c hc), ⋯⟩
Instances For
Change of section and of representative is an explicit coboundary. The raw
transgressions for two continuous sections and two cohomologous representatives differ by a
continuous 2-coboundary on G ⧸ N.
The transgression tg : H¹(N, M)^{G ⧸ N} → H²(G ⧸ N, M ^ N) for a closed normal
subgroup N of a profinite group G and a discrete module M: lift a representative cocycle
through a continuous section of G → G ⧸ N, differentiate, and descend to G ⧸ N. The class
depends neither on the section nor on the representative (transgression_apply).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The transgression is the class of the raw cochain, for every continuous section and every representative cocycle.
Transgression kills restriction. The transgression of the restriction of a class in
H¹(G, M) vanishes: a cocycle on G is itself a transgression lift of its restriction, and its
coboundary is zero.
Inflation kills transgression. The inflation to H²(G, M) of a transgressed class
vanishes: it is the class of the coboundary of a continuous cochain on G.
The transgression through an arbitrary lift. The transgression of the class of c is the
class of the descended coboundary of any transgression lift of c, not only of the
section-dependent transgressionLift.
Exactness of the five-term sequence at H¹(N, M) ^ (G ⧸ N). A conjugation-invariant
class in H¹(N, M) has vanishing transgression exactly when it is the restriction of a class in
H¹(G, M).
Exactness of the five-term sequence at H²(G ⧸ N, M ^ N). A class in H²(G ⧸ N, M ^ N)
inflates to zero in H²(G, M) exactly when it is a transgression.
Injectivity of the transgression. By exactness of the five-term sequence at
H¹(N, M) ^ (G ⧸ N), the transgression is injective exactly when restriction
H¹(G, M) → H¹(N, M) ^ (G ⧸ N) is zero.
Surjectivity of the transgression. By exactness of the five-term sequence at
H²(G ⧸ N, M ^ N), the transgression is surjective exactly when inflation
H²(G ⧸ N, M ^ N) → H²(G, M) is zero, for instance when H²(G, M) vanishes.
Injectivity of inflation in degree two. When H¹(N, M)^{G ⧸ N} vanishes, for instance when
H¹(N, M) does, the transgression is zero, so by exactness of the five-term sequence at
H²(G ⧸ N, M ^ N) inflation H²(G ⧸ N, M ^ N) → H²(G, M) is injective.
The order count of the five-term sequence. The sequence
0 → H¹(G ⧸ N, M ^ N) → H¹(G, M) → H¹(N, M)^{G ⧸ N} → H²(G ⧸ N, M ^ N) → H²(G, M)
is exact through H²(G ⧸ N, M ^ N); its last map, inflation into H²(G, M), need not be
surjective, so the count ends in the image of inflation rather than in H²(G, M): the
multiplicative identity |H¹(G, M)| · |H²(G ⧸ N, M ^ N)| = |H¹(G ⧸ N, M ^ N)| · |H¹(N, M)^{G ⧸ N}| · |im (H²(G ⧸ N, M ^ N) → H²(G, M))| of natural-number
cardinalities, with Nat.card of an infinite group read as 0. It is the six-term alternating
identity AddMonoidHom.card_mul_card_mul_card_of_exact for the sequence ending in the range of
inflation.
The order count of the five-term sequence when H²(G, M) = 0:
|H¹(G, M)| · |H²(G ⧸ N, M ^ N)| = |H¹(G ⧸ N, M ^ N)| · |H¹(N, M)^{G ⧸ N}|.