Integral Bockstein maps and finite-order characters #
For a compact group and n ≠ 0, the connecting map of the trivial coefficient sequence
0 → ℤ → ℤ → ℤ/n → 0, with multiplication by n followed by reduction, identifies
H¹(G, ℤ/n) with the n-torsion in H²(G, ℤ). Indeed, a continuous character into the
integers is zero, and the long exact sequence identifies the image with the kernel of
multiplication by n on cohomology.
Consequently, for p ≠ 0, every p-primary class in H²(G, ℤ) comes from a finite-order
character.
This is the character interpretation of integral degree-two cohomology used to study
corestriction and the transfer. All coefficients are lifted to the universe of G, and
all cohomology groups are Mathlib's canonical continuous cohomology.
In every degree, the p-primary part of Hⁱ⁺¹(G, ℤ) vanishes as soon as Hⁱ⁺¹(G, ℤ/pᵐ) = 0
and pᵐ kills every Hⁱ(G, ℤ/pᵏ) (TauCeti.primaryComponent_continuousCohomology_int_eq_bot):
this is how the integral cohomology in the criterion for the strict cohomological dimension is
computed from finite coefficients.
The formal construction uses TauCeti.ContCohomology.DiscreteShortExact.delta and its long
exact sequence, and the character identification uses
TauCeti.ContCohomology.explicitH1AddEquivContinuousCohomology.
Compactness is essential for injectivity: for the discrete infinite cyclic group,
H¹(ℤ, ℤ/n) is nonzero when n > 1, whereas H²(ℤ, ℤ) = 0.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., the proof of (3.6.4), (i) ⇒ (ii), using the integral connecting maps.
The trivial coefficient sequence 0 → ℤ → ℤ → ℤ/n → 0, with multiplication by n
and reduction modulo n. The universe lifts make its canonical connecting maps available
over groups in any universe.
Instances For
The inclusion in the integral coefficient sequence is multiplication by n.
The projection in the integral coefficient sequence is reduction modulo n.
The trivial action on the lifted discrete integers is continuous.
The trivial action on a lifted cyclic coefficient module is continuous.
First continuous cohomology with trivial lifted integer coefficients vanishes for a compact group.
The integral Bockstein in degree i, defined by the canonical connecting map of
multiplication by n followed by reduction modulo n.
Instances For
The integral Bockstein is the connecting map of its named coefficient sequence.
The degree-one integral Bockstein is injective: continuous integer-valued characters of a compact group vanish.
The coefficient inclusion in the integral sequence acts on canonical cohomology as
multiplication by n.
A class in integral degree i + 1 comes from the integral Bockstein exactly when it is
annihilated by n.
A preimage under the integral Bockstein exists exactly for classes annihilated by n.
The degree-one integral connecting map identifies ℤ/n-valued first cohomology with the
n-torsion subgroup of integral second cohomology.
Instances For
The torsion equivalence is induced by the named integral connecting map.
The inverse torsion equivalence lifts a class to a preimage under the integral Bockstein.
The p-primary part of integral second cohomology consists exactly of classes of
finite-order characters under the integral connecting maps for powers of p.
Vanishing of the p-primary part of integral cohomology. Let G be a compact, locally
compact group, all coefficients carrying the trivial action. If Hⁱ⁺¹(G, ℤ/pᵐ) vanishes and pᵐ
kills Hⁱ(G, ℤ/pᵏ) for every k, then the p-primary component of Hⁱ⁺¹(G, ℤ) vanishes.
For modulus one the integral Bockstein vanishes, in every degree.
The integral Bockstein in degree zero vanishes for trivial coefficients.
Increasing the modulus from n to m = n * k by multiplication by k on cyclic coefficients
preserves the integral connecting class. In particular, the character descriptions of the
p-primary part are compatible as the exponent increases.
Pullback along a continuous homomorphism of compact groups commutes with the integral Bockstein, the coefficients being trivial on both sides and the coefficient maps the identity.
Pullback along a continuous homomorphism of compact groups commutes with the integral Bockstein, the coefficients being trivial on both sides and the coefficient maps the identity.
Restriction to a compact subgroup commutes with the integral Bockstein.
Restriction to a compact subgroup commutes with the integral Bockstein.
Corestriction from an open finite-index subgroup commutes with the integral Bockstein.
Corestriction from an open finite-index subgroup commutes with the integral Bockstein.
A nontrivial finite cyclic group has nonzero integral second continuous cohomology: the integral Bockstein of its identity character is nonzero.