Diagonal homomorphisms into restricted products #
A family of homomorphisms φ i : Γ →* G i whose values at each γ lie in the reference
subgroup U i for all but finitely many i assembles into a homomorphism from Γ into the
restricted product Πʳ i, [G i, U i], the diagonal. The eventual-integrality evidence is an
argument of the construction: for the classical example Γ = G(K) and G i = G(K_v) it is an
arithmetic theorem about Γ, and nothing here manufactures it.
This file records the diagonal, its coordinate formula, its kernel and the resulting injectivity
criterion, its compatibility with componentwise maps, with a change of factors and with a
change of reference family, and the continuity criterion. Continuity does not follow from
continuity of the coordinate maps, because the restricted-product topology is finer than the
topology induced from Π i, G i. The criterion asks for one cofinite set S of indices at which
every γ is integral. Such a uniform set is what the restricted-product topology rewards: on the
subset {x | ∀ i ∈ S, x i ∈ U i}, which then contains the image of the diagonal, it coincides
with the topology induced from Π i, G i, so continuity there is decided by the coordinates.
The uniform integrality set cannot be dispensed with. For discrete groups with trivial reference
subgroups, infinitely many of them nontrivial, take for Γ the finitely supported elements of the
full product, topologised as a subspace of it. Its coordinate maps are continuous and each of its
elements is eventually integral, yet the diagonal into the restricted product, which is discrete,
is not continuous (continuous_eval_and_not_continuous_rationalDiagonal_range_coeMonoidHom).
The general statements have additive counterparts: addRationalDiagonal is the diagonal of a
family of additive homomorphisms Γ →+ G i, with coordinate formula addRationalDiagonal_apply.
References #
- A. Weil, Basic Number Theory.
The diagonal homomorphism into a restricted product induced by a family of homomorphisms
whose values are eventually in the reference subgroups. The eventual-integrality evidence h is
an argument, not a consequence of the construction.
Instances For
The diagonal additive homomorphism into a restricted product
induced by a family of additive homomorphisms whose values are eventually in the reference
subgroups. The eventual-integrality evidence h is an argument, not a consequence of the
construction.
Instances For
The i-th coordinate of the diagonal at γ is φ i γ.
The kernel of the diagonal is the intersection of the kernels of the coordinate maps.
The diagonal is injective exactly when the coordinate maps jointly separate points.
The diagonal is injective as soon as one coordinate map is.
A componentwise map applied after the diagonal is the diagonal of the composed coordinate
maps, whose eventual integrality follows from that of the original coordinates together with the
hypothesis hψ that ψ i eventually maps U i into V i.
A change of factors applied after the diagonal is the diagonal of the transported coordinate
maps, whose eventual integrality follows from that of the original coordinates together with the
hypothesis hψ that ψ i eventually maps U i bijectively onto V i. The change of reference
family is the case in which every ψ i is the identity.
A change of reference family applied after the diagonal is the diagonal with respect to the new family. The eventual-integrality evidence for the new family is an argument, so that the statement applies to whichever such evidence a consumer holds.
The diagonal is continuous when the coordinate maps are continuous and there is one cofinite
set of indices at which every γ is integral. Pointwise eventual integrality alone builds the
map but is not enough for continuity, because the restricted-product topology is finer than the
topology induced from the full product.
The uniform integrality set in continuous_rationalDiagonal cannot be dropped. Take
discrete groups with trivial reference subgroups, infinitely many of them nontrivial, and for Γ
the range of the coercion from the restricted product to the full product, that is, the finitely
supported elements, with the topology induced from the full product. The coordinate maps are
continuous and every element of Γ is eventually integral (it is the coercion of an element of
the restricted product), but the diagonal is not continuous:
it would make the restricted-product topology the one induced from the full product,
contradicting not_isInducing_coe_bot.