Automatic continuity for affinoid algebras

Every algebra homomorphism from a noetherian ultrametric Banach algebra into an affinoid algebra carrying a complete compatible ultrametric algebra norm is continuous. No continuity of a Tate presentation is assumed. Finite quotients by powers of all maximal ideals separate the target, including its nilpotents; continuity into these quotients closes the graph. Consequently such homomorphisms also preserve power-boundedness.

References

• [Bosch, Güntzer, Remmert, Non-Archimedean Analysis][bosch-guntzer-remmert], §§3.7.5 and 6.1.3.

theorem TauCeti.IsAffinoidAlgebra.isClosed_ideal {K : Type u_1} [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] {A : Type u_2} [NormedCommRing A] [NormedAlgebra K A] [IsUltrametricDist A] [CompleteSpace A] (hA : IsAffinoidAlgebra K A) (I : Ideal A) : IsClosed ↑I

Every ideal of an affinoid algebra carrying a complete compatible ultrametric algebra norm is closed.

theorem TauCeti.IsAffinoidAlgebra.continuous_algHom {K : Type u_1} [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] {A : Type u_2} [NormedCommRing A] [NormedAlgebra K A] [IsUltrametricDist A] [CompleteSpace A] (hA : IsAffinoidAlgebra K A) {B : Type u_3} [NormedCommRing B] [NormedAlgebra K B] [IsUltrametricDist B] [CompleteSpace B] [IsNoetherianRing B] (f : B →ₐ[K] A) : Continuous ⇑f

Every homomorphism from a noetherian ultrametric Banach algebra to a completely normed affinoid algebra is continuous. The affinoid presentation is purely algebraic.

theorem TauCeti.IsAffinoidAlgebra.isPowerBounded_map {K : Type u_1} [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] {A : Type u_2} [NormedCommRing A] [NormedAlgebra K A] [IsUltrametricDist A] [CompleteSpace A] (hA : IsAffinoidAlgebra K A) {B : Type u_3} [NormedCommRing B] [NormedAlgebra K B] [IsUltrametricDist B] [CompleteSpace B] [IsNoetherianRing B] (f : B →ₐ[K] A) {b : B} (hb : Huber.IsPowerBounded b) : Huber.IsPowerBounded (f b)

Every homomorphism from a noetherian ultrametric Banach algebra to a completely normed affinoid algebra maps power-bounded elements to power-bounded elements.