Transitivity of discrete coinduction #
Let G be a topological group, U ≤ G a subgroup with compact closure and V ≤ U a subgroup of
U. For a discrete V-module A, coinducing first to U and then to G is the same as
coinducing to G in one step:
Coind_U^G (Coind_V^U A) ≅ Coind_V^G A, f ↦ (g ↦ f g 1),
with inverse φ ↦ (g ↦ (u ↦ φ (u * g))). This is the coinduced form of the transitivity
Ind_V^G = Ind_U^G ∘ Ind_V^U of induction, and it is the identification of coefficient modules that
the dimension-shifting proof of Shapiro's lemma in every degree runs on: the acyclic module
Coind_1^G A of the trivial subgroup of G is the coinduction from U of the acyclic module
Coind_1^U A of the trivial subgroup of U.
Since TauCeti.DiscreteCoind coinduces from a subgroup of the ambient group, the inner subgroup is
a subgroup V of the subtype U, and the one-step coinduction is from the subgroup W of G
with the same elements, that is W = V.map U.subtype. The module A then carries an action of V
and an action of W, and the statement requires them to agree on elements with the same underlying
element of G; both hypotheses are explicit arguments rather than a definitional identification
of the two subgroups, whose types differ. The case used by dimension shifting is V = ⊥ and
W = ⊥, where every action of the trivial group is trivial and the agreement is automatic.
The topological input is that U is relatively compact in G, IsCompact (closure U): a locally
constant function on G is then locally constant under right translation uniformly in the
translating element u ∈ U (TauCeti.exists_isOpen_forall_mem_mul_right_eq), which is what makes
g ↦ (u ↦ φ (u * g)) locally constant. In a compact group, as in the profinite setting, every
subgroup is relatively compact.
Main definitions #
TauCeti.DiscreteCoind.transEquiv: the additive equivalenceCoind_U^G (Coind_V^U A) ≃+ Coind_W^G A, which isG-equivariant (TauCeti.DiscreteCoind.transEquiv_smul) and compatible with evaluation at1(TauCeti.DiscreteCoind.eval_transEquiv).TauCeti.DiscreteCoind.transIso: the same identification as an isomorphism of the topologicalG-representations attached to the two discrete modules byTauCeti.ofDiscreteModule.TauCeti.DiscreteCoind.transIsoBot: its caseV = W = ⊥,Coind_U^G (Coind_1^U A) ≅ Coind_1^G A, the identification dimension shifting uses.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Springer (2008),
Ch. I §6, where the coinduced module is written
Ind(see the footnote on p. 61). - L. Ribes, P. Zalesskii, Profinite Groups, Section 6.10.
Transitivity of discrete coinduction: Coind_U^G (Coind_V^U A) ≃+ Coind_W^G A for
V ≤ U ≤ G with U relatively compact in G, where W is V regarded as a subgroup of G.
The forward map evaluates the inner coinduced function at 1, f ↦ (g ↦ f g 1); the inverse is
φ ↦ (g ↦ (u ↦ φ (u * g))).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The transitivity equivalence evaluates the inner coinduced function at 1.
The inverse of the transitivity equivalence translates the argument by U.
The transitivity equivalence is G-equivariant for the right-translation actions.
The inverse of the transitivity equivalence is G-equivariant.
The transitivity equivalence is compatible with the counits: evaluating the one-step coinduced
function at 1 is evaluating the two-step one at 1 twice.
Transitivity of coinduction as an isomorphism of topological representations: the
canonical objects of TopRep ℤ G attached to Coind_U^G (Coind_V^U A) and to Coind_W^G A are
isomorphic, by TauCeti.DiscreteCoind.transEquiv on underlying modules. Applying continuous
cohomology to it identifies Hⁿ(G, Coind_U^G (Coind_V^U A)) with Hⁿ(G, Coind_W^G A).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The transitivity isomorphism acts on underlying modules as transEquiv.
The inverse of the transitivity isomorphism acts on underlying modules as
transEquiv.symm.
The trivial subgroup #
Transitivity of coinduction for the trivial subgroup:
Coind_U^G (Coind_1^U A) ≅ Coind_1^G A as topological G-representations, the case V = W = ⊥ of
TauCeti.DiscreteCoind.transIso. Dimension shifting uses this identification of coefficient
modules together with a separate acyclicity result for Coind_1^G A to pass Shapiro's lemma from
one degree to the next.
Equations
- TauCeti.DiscreteCoind.transIsoBot U A hU = TauCeti.DiscreteCoind.transIso ⋯ ⋯ hU
Instances For
The trivial-subgroup transitivity isomorphism evaluates the inner coinduced function at 1.
The inverse of the trivial-subgroup transitivity isomorphism translates the argument by U.