The Galois action on the semi-local algebra at an infinite place #
Let L/K be an extension of number fields and v an infinite place of K. An automorphism σ
of L/K acts on the semi-local algebra K_v ⊗[K] L through the second factor, by id ⊗ σ;
this is the base change TauCeti.Algebra.TensorProduct.baseChangeAutHom of σ to K_v. Under
the semi-local decomposition K_v ⊗[K] L ≃ ∏_{w ∣ v} L_w it permutes the factors: the component
at σ • w of (id ⊗ σ) z is the component of z at w, transported along the isomorphism of
completions L_w ≃ L_{σ • w} induced by σ (infiniteSemilocalEquiv_baseChangeAutHom).
When L/K is Galois, the Galois group permutes the places above v transitively, and the
stabilizer of one place w — its decomposition group — acts on L_w. The units of the
semi-local algebra are then coinduced from the decomposition group: as integral
representations of Gal(L/K),
(K_v ⊗[K] L)ˣ ≅ Coind_{D_w}^{Gal(L/K)} L_wˣ,
the map sending y to the function g ↦ ((id ⊗ g) y)_w (infiniteSemilocalUnitsCoindIso);
as at the finite places, its bijectivity is an instance of
TauCeti.resCoindToHom_bijective_of_transport.
Shapiro's lemma therefore computes the cohomology of ∏_{w ∣ v} L_wˣ from the Galois cohomology
of the archimedean local field L_w. This is the archimedean counterpart of
TauCeti.semilocalUnitsCoindIso at the finite places.
Main definitions #
TauCeti.GlobalNumberFields.infiniteSemilocalUnitsRep: the units ofK_v ⊗[K] L, as an integral representation ofAut(L/K).TauCeti.GlobalNumberFields.infiniteDecompositionUnitsRep: the units ofL_w, as an integral representation of the decomposition group ofw.TauCeti.GlobalNumberFields.infiniteSemilocalUnitsToCoind: the mapy ↦ (g ↦ ((id ⊗ g) y)_w)into the coinduced representation.TauCeti.GlobalNumberFields.infiniteSemilocalUnitsCoindIso: forL/KGalois, that map as an isomorphism of representations.
Main results #
TauCeti.GlobalNumberFields.infiniteSemilocalEquiv_baseChangeAutHom: under the semi-local decomposition,id ⊗ σcarries the factor atwto the factor atσ • wby the isomorphism of completions.
References #
- J. S. Milne, Class Field Theory, Chapter VII, §2 (the cohomology of
∏_{w ∣ v} L_wˣ). - J. Neukirch, Algebraic Number Theory, Chapter II, Proposition (8.3) and Chapter VI, §2.
The Galois action permutes the semi-local factors. If σ carries the place w above v
to w', then the component at w' of (id ⊗ σ) z is the component of z at w, transported
along the isomorphism of completions L_w ≃ L_{w'} induced by σ.
The units of the semi-local algebra K_v ⊗[K] L at an infinite place v, as an integral
representation of Aut(L/K) acting through Algebra.TensorProduct.baseChangeAutHom.
Instances For
The units of the completion L_w at an infinite place, as an integral representation of the
decomposition group of w acting through decompositionHom.
Instances For
The map y ↦ (g ↦ ((id ⊗ g) y)_w) from the semi-local units at an infinite place to the
representation coinduced from the units of L_w. It is the morphism adjoint to the projection to
the factor at w, which is equivariant for the decomposition group of w.
Instances For
The value of infiniteSemilocalUnitsToCoind at g is the component at w of
(id ⊗ g) y.
The semi-local units at an infinite place are coinduced. For L/K Galois and w a
place above the infinite place v, the units of K_v ⊗[K] L are, as an integral representation
of Gal(L/K), coinduced from the units of L_w as a representation of the decomposition group
of w.
Instances For
The coinduction isomorphism is infiniteSemilocalUnitsToCoind.