The degree-one form of a free pro-p group and changes of basis #
Let F = freeProP p X be the free pro-p group on a finite type X, with canonical generators
x_i = freeProP.of i, and let gr_1(F) = λ_1(F) ⧸ λ_2(F) be the degree-one graded piece of its
lower p-series. Two continuous characters χ, ψ : F → 𝔽_p lift to a continuous homomorphism
F → H(𝔽_p) into the Heisenberg group over 𝔽_p, x_i ↦ (χ x_i, ψ x_i, 0); on λ_1(F) its
(1, 3)-entry is a homomorphism killing λ_2(F), the Heisenberg functional
TauCeti.freeProP.heisenbergFunctional χ ψ : gr_1(F) →ₗ[𝔽_p] 𝔽_p. It is the unique linear
functional with
[⟦u⟧, ⟦v⟧] ↦ χ u · ψ v - χ v · ψ u and π ⟦u⟧ ↦ (p choose 2) · χ u · ψ u
on the brackets and p-power classes of degree-zero classes. Assembling these functionals gives
the degree-one form TauCeti.freeProP.degreeOneForm ρ, an 𝔽_p-bilinear form on the
continuous 𝔽_p-dual H¹(F, 𝔽_p) = Hom_cont(F, 𝔽_p) attached linearly to each class
ρ ∈ gr_1(F). The form is skew-symmetric, and alternating for odd p. In the basis of the dual
which is dual to the generators (TauCeti.freeProP.dualBasis), its matrix has the commutator
coordinates of ρ in the standard basis of gr_1(F) above the diagonal, their negatives below
it, and (p choose 2) times the p-power coordinates on the diagonal. These are the coordinates
Labute reads off the class ⟦r⟧ of a relator r ∈ λ_1(F) to describe the cup product on
H¹(F ⧸ ⟪r⟫, 𝔽_p) (Labute, Proposition 3); the identification of the degree-one form of ⟦r⟧
with that cup product is not proved in this file.
Its transformation law is the change-of-basis law of that matrix: a continuous homomorphism
φ : F → F' between free pro-p groups carries the form of ρ to the form of φ_* ρ pulled back
along the transpose of φ on the duals, B_{φ_* ρ}(χ, ψ) = B_ρ(χ ∘ φ, ψ ∘ φ), which in matrices is
B ↦ Pᵀ B P. Every linear automorphism of the dual is the transpose of a continuous automorphism
of F (TauCeti.freeProP.exists_continuousMulEquiv_continuousZModDualMap_eq). Hence a change of
basis of F brings the matrix of the form of ρ into any shape a basis of the dual provides, in
particular into the normal forms of the bilinear-form theory. This normalizes the form of ρ
only: for odd p the diagonal factor (p choose 2) vanishes in 𝔽_p, so the form does not see
the p-power coordinates of ρ. Those coordinates are read off by the coordinate characters
instead, and they transform under a continuous homomorphism through the values of the coordinate
characters on the images of the generators. Bringing a relator into normal form modulo λ_2(F)
(Labute, Proposition 4) combines both, and is carried out in
TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.DegreeOneForm.
Main definitions #
TauCeti.freeProP.heisenbergFunctional: the Heisenberg functionalgr_1(F) → 𝔽_pof two continuous𝔽_p-characters ofF.TauCeti.freeProP.degreeOneForm: the degree-one form, the bilinear form on the continuous𝔽_p-dual ofFattached linearly to a class ingr_1(F).
Main results #
TauCeti.freeProP.heisenbergFunctional_gradedBracket_gradedMkZero,TauCeti.freeProP.heisenbergFunctional_gradedPow_gradedMkZero,TauCeti.freeProP.heisenbergFunctional_unique: the values of the Heisenberg functional on brackets andp-power classes characterize it.TauCeti.freeProP.degreeOneForm_swap,TauCeti.freeProP.isRefl_degreeOneForm,TauCeti.freeProP.isAlt_degreeOneForm_of_ne_two,TauCeti.freeProP.isSymm_degreeOneForm_of_two: the degree-one form is skew-symmetric, hence reflexive, alternating for oddp, and symmetric forp = 2.TauCeti.freeProP.heisenbergFunctional_gradedMap,TauCeti.freeProP.degreeOneForm_gradedMap: the transformation law under a continuous homomorphism between free pro-pgroups; hence nondegeneracy of the form is invariant under topological isomorphisms (TauCeti.freeProP.nondegenerate_degreeOneForm_gradedMap_iff).TauCeti.freeProP.degreeOneForm_dualBasis_of_lt,TauCeti.freeProP.degreeOneForm_dualBasis_of_gt,TauCeti.freeProP.degreeOneForm_dualBasis_self: the matrix of the degree-one form in the dual basis of the generators is read off the coordinates of the class in the standard basis ofgr_1(F); hence atp = 2the form determines the class (TauCeti.freeProP.degreeOneForm_injective_of_two).TauCeti.freeProP.degreeOneBasis_repr_gradedBracket_inl,TauCeti.freeProP.degreeOneBasis_repr_gradedPow_gradedMkZero_inl,TauCeti.freeProP.degreeOneBasis_repr_gradedMap_inl: thep-power coordinates, which the form does not see for oddp, vanish on brackets, are read off by the coordinate characters onp-power classes, and transform under a continuous homomorphism through the values of the coordinate characters on the images of the generators.TauCeti.freeProP.degreeOneBasis_repr_gradedMk_inl,TauCeti.freeProP.degreeOneBasis_repr_gradedMk_inl_eq_zero_iff: thep-power coordinates of the class ofy ∈ λ_1(F)are the exponent sums ofydivided byp, reduced modulop; in particular they vanish exactly when the exponent sums are divisible byp ^ 2.TauCeti.freeProP.exists_continuousMulEquiv_toMatrix_degreeOneForm_gradedMap: the matrix of the degree-one form of a class in any basis of the dual is the matrix, in the dual basis of the generators, of the form of the image of the class under some continuous automorphism ofF.
References #
- J. Labute, Classification of Demushkin groups, Canadian J. Math. 19 (1967), §3, Propositions 3 and 4.
The Heisenberg functional #
The Heisenberg group over 𝔽_p, lifted to the universe of X and given the discrete topology, is
a finite p-group, so a pair of characters defines a continuous homomorphism from F into it by
the universal property. Its (1, 3)-entry on λ_1(F) is the functional.
The Heisenberg functional of two continuous 𝔽_p-characters χ, ψ of the free pro-p
group F: the 𝔽_p-linear functional on gr_1(F) induced on λ_1(F) by the (1, 3)-entry of
the continuous homomorphism F → H(𝔽_p), x_i ↦ (χ x_i, ψ x_i, 0), into the Heisenberg group over
𝔽_p. It is characterized by its values [⟦u⟧, ⟦v⟧] ↦ χ u · ψ v - χ v · ψ u and
π ⟦u⟧ ↦ (p choose 2) · χ u · ψ u (TauCeti.freeProP.heisenbergFunctional_unique).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The Heisenberg functional on a bracket: [⟦u⟧, ⟦v⟧] ↦ χ u · ψ v - χ v · ψ u.
The Heisenberg functional on a p-power class: π ⟦u⟧ ↦ (p choose 2) · χ u · ψ u.
Uniqueness of the Heisenberg functional: a linear functional on gr_1(F) with the values
[⟦u⟧, ⟦v⟧] ↦ χ u · ψ v - χ v · ψ u and π ⟦u⟧ ↦ (p choose 2) · χ u · ψ u is the Heisenberg
functional of χ and ψ.
The Heisenberg functional is additive in its first character.
The Heisenberg functional is additive in its second character.
The Heisenberg functional of the trivial character and any character vanishes.
The Heisenberg functional of any character and the trivial character vanishes.
Naturality of the Heisenberg functional. A continuous homomorphism φ : F → F' between
free pro-p groups, with F of finite rank, carries the Heisenberg functional of χ, ψ on
gr_1(F') back to the Heisenberg functional of χ ∘ φ, ψ ∘ φ on gr_1(F).
Naturality of the Heisenberg functional, on a class.
The degree-one form #
The degree-one form of a free pro-p group F of finite rank: the 𝔽_p-bilinear form
(χ, ψ) ↦ heisenbergFunctional χ ψ ρ on the continuous 𝔽_p-dual of F, attached
𝔽_p-linearly to a class ρ ∈ gr_1(F). Its matrix in the dual basis of the generators has the
commutator coordinates of ρ above the diagonal and (p choose 2) times the p-power
coordinates on it (TauCeti.freeProP.degreeOneForm_dualBasis_of_lt,
TauCeti.freeProP.degreeOneForm_dualBasis_self); for the class of a relator r ∈ λ_1(F) these
are the coordinates Labute attaches to the one-relator group F ⧸ ⟪r⟫.
Equations
- TauCeti.freeProP.degreeOneForm = AddMonoidHom.toZModLinearMap p { toFun := TauCeti.freeProP.degreeOneFormAux✝, map_zero' := ⋯, map_add' := ⋯ }
Instances For
The degree-one form evaluates to the Heisenberg functional.
The degree-one form on a bracket: [⟦u⟧, ⟦v⟧] ↦ χ u · ψ v - χ v · ψ u.
The degree-one form on a p-power class: π ⟦u⟧ ↦ (p choose 2) · χ u · ψ u.
The degree-one form is skew-symmetric: B_ρ(ψ, χ) = -B_ρ(χ, ψ). On the p-power classes
both sides are (p choose 2) · χ u · ψ u, and 2 · (p choose 2) = p (p - 1) vanishes in 𝔽_p.
The degree-one form is reflexive, being skew-symmetric: B_ρ(χ, ψ) = 0 implies
B_ρ(ψ, χ) = 0.
The degree-one form is alternating for odd p: the diagonal factor (p choose 2) is
divisible by p.
The degree-one form is symmetric at p = 2: skew-symmetry is symmetry when -1 = 1.
The transformation law of the degree-one form. A continuous homomorphism φ : F → F'
between free pro-p groups of finite rank carries the degree-one form of ρ ∈ gr_1(F) to the
degree-one form of φ_* ρ pulled back along the transpose of φ on the continuous duals:
B_{φ_* ρ}(χ, ψ) = B_ρ(χ ∘ φ, ψ ∘ φ). In matrices, a change of generators by P acts on the
matrix of the form by B ↦ Pᵀ B P.
Nondegeneracy of the degree-one form is invariant under topological isomorphisms of free
pro-p groups: the form of e_* ρ is the form of ρ transported along the transpose of e,
which is a linear automorphism of the continuous duals.
Coordinates of the degree-one form #
The degree-one form reads off the commutator coordinates: for i < j, the value of the
form of ρ on the i-th and j-th coordinate characters is the coefficient of [⟦x_i⟧, ⟦x_j⟧] in
the expansion of ρ in the standard basis of gr_1(F).
The degree-one form reads off the commutator coordinates, below the diagonal: for
j < i, the value of the form of ρ on the i-th and j-th coordinate characters is the
negative of the coefficient of [⟦x_j⟧, ⟦x_i⟧] in the expansion of ρ in the standard basis of
gr_1(F).
The degree-one form reads off the p-power coordinates: the value of the form of ρ on
the i-th coordinate character twice is (p choose 2) times the coefficient of π ⟦x_i⟧ in the
expansion of ρ in the standard basis of gr_1(F).
At p = 2 the degree-one form determines the class: the diagonal entries of its matrix
are the 2-power coordinates and the entries above the diagonal the commutator coordinates.
The degree-one form of a class without p-power part is alternating, for every p
including p = 2: such a class is a combination of bracket classes [⟦x_i⟧, ⟦x_j⟧], on which
B(χ, χ) = χ x_i · χ x_j - χ x_j · χ x_i = 0.
The p-power coordinates #
For odd p the degree-one form does not see the p-power coordinates of a class. They are
read off by the coordinate characters instead: the coefficient of π x'_k in π ⟦g⟧ is the value
of the k-th coordinate character at g, and under a continuous homomorphism the p-power
coordinates transform through the values of the coordinate characters on the images of the
generators, the brackets contributing nothing.
Brackets have no p-power coordinates: the coefficient of π x'_k in the bracket of two
degree-zero classes vanishes. On two generator classes the bracket is ± [x'_a, x'_b] with
a ≠ b, a basis vector other than π x'_k, or zero; the general case follows by bilinearity.
The p-power coordinates of a p-power class: the coefficient of π x'_k in π ⟦g⟧ is
the value at g of the k-th coordinate character. The function x ↦ coord_{π x'_k} (π x) is
linear on gr_0(F), because the defect of additivity of π is a bracket, which has no p-power
coordinates.
The p-power coordinates of a power of a generator: the class of x_i ^ (p * n) in
gr_1(F) has coefficient n at π x'_i and 0 at the other π x'_k.
The p-power coordinates through the exponent sums, vanishing form: the coefficient of
π x'_k in the class of y ∈ λ_1(F) vanishes exactly when p ^ 2 divides the k-th exponent sum
of y.
The p-power coordinates are the exponent sums divided by p, modulo p: if the k-th
exponent sum of y ∈ λ_1(F) is p * c, then the coefficient of π x'_k in the class of y is the
reduction of c modulo p.
The transformation law of the p-power coordinates. A continuous homomorphism
φ : F → F' between free pro-p groups of finite rank carries a class with p-power coordinates
c_i to a class with p-power coordinates c'_k = Σ_i c_i · χ_k(φ x_i), where χ_k is the
k-th coordinate character of F'; the brackets contribute nothing.
Normal forms after an automorphism #
A basis of the dual is the dual basis of the generators after an automorphism. For every
class ρ ∈ gr_1(F) and every basis η of the continuous 𝔽_p-dual of F, there is a continuous
automorphism e of F such that the degree-one form of e_* ρ on the dual basis of the
generators is the degree-one form of ρ on η.
Any matrix of the degree-one form is attained in the dual basis of the generators after an
automorphism. For every class ρ ∈ gr_1(F) and every basis η of the continuous 𝔽_p-dual of
F, there is a continuous automorphism e of F such that the matrix of the degree-one form of
e_* ρ in the dual basis of the generators is the matrix of the degree-one form of ρ in η. So
a normal form for the matrix of the form, such as a symplectic basis, is realized by a change of
generators of F.