Automatic continuity into finite-dimensional algebras #
An algebra homomorphism from a noetherian nonarchimedean Banach algebra into a finite-dimensional topological algebra is continuous. The source kernel is closed, so its quotient is a Hausdorff finite-dimensional vector space. This permits continuity to be tested on finite-dimensional quotients without assuming continuity of the original homomorphism.
References #
- [Bosch, Güntzer, Remmert, Non-Archimedean Analysis][bosch-guntzer-remmert], §§3.7.2 and 3.7.5.
theorem
TauCeti.continuous_algHom_of_finiteDimensional
(K : Type u_1)
[NontriviallyNormedField K]
[CompleteSpace K]
{A : Type u_2}
[NormedCommRing A]
[NormedAlgebra K A]
[IsUltrametricDist A]
[CompleteSpace A]
[IsNoetherianRing A]
{B : Type u_3}
[Ring B]
[Algebra K B]
[TopologicalSpace B]
[IsTopologicalAddGroup B]
[ContinuousSMul K B]
[Module.Finite K B]
(f : A →ₐ[K] B)
:
Continuous ⇑f
An algebra homomorphism from a noetherian ultrametric Banach algebra into a finite-dimensional topological algebra is automatically continuous. The target need not carry a norm, be complete, or be Hausdorff.