Lifting second cohomology along a map of formations after a refinement #
Let φ : A → A' be an equivariant map between the coefficient modules of two formations F and
F' on G, and let σ assign to every finite normal layer L a homomorphism
σ_L : H²(L, A) → ℚ/ℤ that multiplies by the relative degree under restriction. For the idele
classes of a number field, A is the ideles, A' the idele classes and σ the sum of the local
invariants. A class of H²(L, A') need not be the image under φ of a class of H²(L, A): for
the idele classes the obstruction lies in H³(L, Lˣ), and it is nonzero for the biquadratic
field ℚ(√13, √17). This file shows that the obstruction disappears after inflation to a
suitable refinement of L, once the following hold:
σvanishes exactly on the classes thatφkills;H¹of every layer ofF'vanishes;- the order of
H²(L, A')divides the degreenofL, the second inequality of class field theory in its cohomological form; - some refinement
L'ofLcarries a classywithσ_{L'}(y) = 1/n.
The argument is Tate's. Restricted to the layer L'/L, the class y has σ = n • (1/n) = 0, so
its image φ(y) restricts to zero there, and by the inflation-restriction sequence φ(y) is the
inflation of a class x of H²(L, A'). If d • x = 0 then φ(d • y) = 0, so
σ(d • y) = d • (1/n) = 0 and n ∣ d. Hence x has order at least n, and since H²(L, A') has
at most n elements, it is cyclic of order exactly n generated by x
(NormalLayer.zmultiples_eq_top_of_cohomologyInfl_eq_map,
NormalLayer.natCard_H2_eq_degree_of_natCard_H2_dvd). Every class is then a multiple k • x,
whose inflation to L' is φ(k • y): after inflation to the one refinement L', every class of
H²(L, A') lifts along φ
(NormalLayer.exists_refinement_forall_exists_map_eq_cohomologyInfl).
Main statements #
TauCeti.ClassFieldTheory.NormalLayer.exists_cohomologyInfl_eq_map: a class ofH²(L, A')whose inflation is the image of a class withσ = 1/n.TauCeti.ClassFieldTheory.NormalLayer.zmultiples_eq_top_of_cohomologyInfl_eq_map: such a class generatesH²(L, A').TauCeti.ClassFieldTheory.NormalLayer.natCard_H2_eq_degree_of_natCard_H2_dvd:H²(L, A')has exactlynelements.TauCeti.ClassFieldTheory.NormalLayer.exists_refinement_forall_exists_map_eq_cohomologyInfl: every class ofH²(L, A')lifts alongφafter inflation to one refinement ofL.
References #
- J. S. Milne, Class Field Theory, v4.03, Chapter VIII, §4 (proof of Theorem 4.7).
- J. Tate, Global class field theory, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter VII, §11.
A class with σ = 1/n on a refinement maps to an inflated class. Let L have degree n
and let y be a class on a refinement L' of L with σ_{L'}(y) = 1/n. If σ multiplies by
the relative degree under restriction, vanishes only on classes that φ kills, and H¹ of every
layer of F' vanishes, then φ(y) is the inflation of a class of H²(L, A').
H²(L, A') has exactly n elements if its order divides the degree n of L and the
inflation of some class to a refinement is the image of a class with σ = 1/n.
A class whose inflation lifts a class with σ = 1/n generates H²(L, A'), when the order
of H²(L, A') divides the degree n of L.
Every class of H²(L, A') lifts along φ after inflation to one refinement. Let L have
degree n, and suppose that σ multiplies by the relative degree under restriction and vanishes
exactly on the classes that φ kills, that H¹ of every layer of F' vanishes, that the order of
H²(L, A') divides n, and that some refinement of L carries a class with σ = 1/n. Then on
that refinement L' the inflation of every class of H²(L, A') is the image under φ of a class
of H²(L', A).