Restriction of finite-layer Tate cohomology #
For a restriction of finite normal layers K/E inside K/F, this file defines restriction
Hhatʳ(Gal(K/F), A^V) ⟶ Hhatʳ(Gal(K/E), A^V)
in every integer degree, with formation coefficients and with trivial integral coefficients. In
positive degrees it is the ordinary cohomological restriction LayerRestriction.cohomologyRes;
in degree zero it is induced by the inclusion of invariants, which on ground levels is the
inclusion A^U ⊆ A^{U'}; in degree minus one it is induced by the relative transfer; and in
degrees at most minus two it is the transfer on group homology. In degrees at most zero the smaller
Galois group is identified with the image of its inclusion into the larger one, exactly as for
LayerRestriction.tateCor.
The comparison lemmas below identify each branch with the corresponding established map. On
representatives, degree-zero restriction is the ground-level inclusion and degree-minus-one
restriction is the relative transfer of norm kernels, and restriction is functorial along a tower
F ⊆ E ⊆ E' ⊆ K in every degree. Corestriction after restriction is multiplication by the
relative degree [E : F] in every degree.
Main definitions #
TauCeti.ClassFieldTheory.LayerRestriction.tateRes: restriction of layer Tate cohomology.TauCeti.ClassFieldTheory.LayerRestriction.trivialTateRes: restriction of Tate cohomology with trivial integral coefficients.TauCeti.ClassFieldTheory.LayerRestriction.kerNormTransfer: the relative transfer of norm kernels along a restriction. It is the degree-minus-one shadow oftateRes, and the wrong-way partner ofLayerRestriction.kerNormInclusion.
Main results #
TauCeti.ClassFieldTheory.LayerRestriction.tateRes_comp_tateHIsoH_homandTauCeti.ClassFieldTheory.LayerRestriction.trivialTateRes_comp_isoGroupCohomology_hom: in positive degrees, Tate restriction is ordinary cohomological restriction.TauCeti.ClassFieldTheory.LayerRestriction.tateRes_zero_H0πandTauCeti.ClassFieldTheory.LayerRestriction.tateHZeroEquivNormQuotient_tateRes_H0π: in degree zero, restriction is the ground-level inclusion on representatives and norm quotients.TauCeti.ClassFieldTheory.LayerRestriction.trivialTateRes_zero_H0π: in degree zero, trivial-coefficient restriction is the identity on integral representatives.TauCeti.ClassFieldTheory.LayerRestriction.tateRes_neg_one_HNegOneπ: in degree minus one, restriction is the relative transferkerNormTransferon representatives.TauCeti.ClassFieldTheory.LayerRestriction.tateCor_tateRes:cor ∘ res = [E : F]in every degree.TauCeti.ClassFieldTheory.LayerRestriction.kerNormTransfer_trans_sub_mem: the relative transfer of norm kernels is transitive along a tower modulo the augmentation submodule.TauCeti.ClassFieldTheory.LayerRestriction.tateRes_trans_of_neg_one_le: Tate restriction is functorial along towers in every degree at least minus one.TauCeti.ClassFieldTheory.LayerRestriction.tateRes_trans_eq_compandTauCeti.ClassFieldTheory.LayerRestriction.tateRes_trans: Tate restriction is functorial along towers of layers in every degree.
References #
- E. Artin and J. Tate, Class Field Theory, Chapter IV, §6 and Chapter XIV, §4.
- K. S. Brown, Cohomology of Groups, Chapter III, §9 and Chapter VI, §5.
Restriction between the Tate cohomology groups of finite normal layers, in every integer degree. Positive degrees use ordinary cohomological restriction, degrees zero and minus one use the low-degree descriptions by invariants and norm kernels, and lower degrees use the transfer on group homology.
Instances For
In degree zero, layer Tate restriction is restriction of invariants to the image subgroup, followed by the range comparison.
In a positive degree, layer Tate restriction is ordinary cohomological restriction read through the canonical positive-degree comparisons.
Positive-degree Tate restriction commutes with the canonical comparison to ordinary cohomology.
Positive-degree Tate restriction commutes with the canonical comparison to ordinary cohomology.
In degree minus one, layer Tate restriction is the relative transfer on norm kernels, followed by the range comparison.
In degree -(n+2), layer Tate restriction is the transfer in group homology of degree n+1,
followed by the range comparison.
Layer Tate restriction is generic subgroup restriction followed by the inverse range comparison, in every degree.
In degree zero, restriction is the ground-level inclusion. Read through the identification
of Hhat⁰ with the norm quotient A^U / N(A^V), restricting the class of an element of the ground
level A^U of K/F gives the class of the same element in the ground level A^{U'} of K/E.
In degree zero, restriction is the ground-level inclusion on representatives. The
representative of a class in the ground level A^U of K/F gives the class of the same element in
the ground level A^{U'} of K/E.
In degree zero, restriction is the ground-level inclusion on zero-dimensional classes. The
zero-dimensional Tate class of an element of the ground level A^U of K/F restricts to the
zero-dimensional Tate class of the same element in the ground level A^{U'} of K/E.
The relative transfer of norm kernels along a restriction: the transfer of the image of
Gal(K/E) in Gal(K/F), read back into the smaller layer through repIso. On representatives,
degree -1 restriction is this map (tateRes_neg_one_HNegOneπ); it is the wrong-way partner of
LayerRestriction.kerNormInclusion.
Instances For
The norm-kernel transfer is the relative transfer of the image subgroup, read back through the identification of coefficient modules.
In degree minus one, layer Tate restriction is the relative transfer on representatives.
Towers #
The norm-kernel transfer is transitive along a tower, modulo the augmentation submodule.
The transfer is transitive only up to the choice of coset representatives, and that choice is
invisible in degree -1 Tate cohomology, which is the norm kernel modulo exactly this
submodule.
Tate restriction is functorial along a tower in every degree at least minus one.
Restricting from K/F to K/E and then to K/E' agrees with direct restriction from K/F to
K/E'.
Unlike the nonnegative degrees, degree -1 is not a formal consequence
of functoriality of some change-of-group map: there restriction is the relative transfer, which is
transitive only modulo the augmentation submodule — exactly the submodule degree -1 Tate
cohomology divides by. See kerNormTransfer_trans_sub_mem.
Tate restriction is functorial along a tower, in every integer degree, as an identity of
morphisms: restricting from K/F to K/E' is restricting from K/F to K/E and then from K/E
to K/E'.
Tate restriction is functorial along a tower, in every integer degree, as an identity of
morphisms: restricting from K/F to K/E' is restricting from K/F to K/E and then from K/E
to K/E'.
Tate restriction is functorial along a tower, in every integer degree. Restricting from
K/F to K/E and then to K/E' agrees with direct restriction from K/F to K/E'.
Corestriction after restriction is multiplication by the relative degree [E : F], in
every Tate degree.
Corestriction after restriction is multiplication by the relative degree [E : F], in
every Tate degree.
Corestriction after restriction is multiplication by the relative degree [E : F], in
every Tate degree.
Trivial coefficients #
Restriction on the Tate cohomology of finite normal layers with trivial integral coefficients, in every integer degree. It uses cohomological restriction in positive degrees, the low-degree maps in degrees zero and minus one, and the homological transfer below them.
Instances For
In degree zero, trivial-coefficient Tate restriction is restriction of invariants to the image subgroup, followed by the range comparison.
In degree zero, trivial-coefficient restriction is the identity on integral
representatives. The class of an integer in Hhat⁰(Gal(K/F), ℤ) restricts to the class of the
same integer in Hhat⁰(Gal(K/E), ℤ). Corestriction instead multiplies representatives by the
relative degree [E : F] (trivialTateCor_zero_H0π).
In a positive degree, trivial-coefficient Tate restriction is ordinary cohomological restriction to the image subgroup, followed by the range comparison.
Positive-degree trivial-coefficient Tate restriction is ordinary cohomological restriction to the image subgroup, read through the canonical comparisons with ordinary cohomology.
Positive-degree trivial-coefficient Tate restriction is ordinary cohomological restriction to the image subgroup, read through the canonical comparisons with ordinary cohomology.
In degree minus one, trivial-coefficient Tate restriction is the relative transfer on norm kernels, followed by the range comparison.
In degree -(n+2), trivial-coefficient Tate restriction is the transfer in group homology of
degree n+1, followed by the range comparison.
Restriction with trivial integral coefficients is generic subgroup restriction followed by its inverse range comparison.
In degree -2, trivial-coefficient Tate restriction is first-homology transfer followed by
the range comparison.