Integral second cohomology of the profinite integers #
The inverse of the connecting isomorphism of 0 → ℤ → ℚ → ℚ/ℤ → 0, followed by
evaluation of rational characters at the canonical generator, gives H²(zHat, ℤ) ≃+ ℚ/ℤ.
Thus second integral cohomology is nonzero, although second cohomology with torsion coefficients
vanishes.
The cohomology carrier is Mathlib's continuous cohomology. The integers and rationals are
lifted to the universe of zHat, and the rational circle may be any discrete model in that
universe. All three coefficient actions are trivial. The normalization theorem specifies
the value of every class obtained by the connecting map from a continuous character.
The construction combines TauCeti.ContCohomology.rationalIntegralDeltaIso with evaluation
of rational characters of zHat; it does not construct a new cohomology theory.
References #
- J.-P. Serre, Local Fields, Chapter XIII, §1, Propositions 1–2.
- J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.3.2), (1.6.1) and (1.7.7).
Second integral continuous cohomology of the profinite integers is the rational circle. The inverse sends a rational-circle value to the connecting class of the unique continuous character taking that value at the canonical generator.
Instances For
The integral degree-two equivalence evaluates the corresponding rational character at the canonical generator.
The connecting class of a continuous rational-circle cocycle is normalized by its value at the generator. This fixes the sign of the identification with the rational circle.
Conversely, the connecting class of a character is the inverse image of its value at the generator.
The class corresponding to 1/n has additive order n. In particular, second
integral cohomology contains elements of every finite order.
Second integral continuous cohomology of the profinite integers is nontrivial:
the class of invariant 1/2 has order two.