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Mathlib.Algebra.Category.Ring.Topology

Topology on Hom(R, S) #

In this file, we define topology on Hom(A, R) for a topological ring R, given by the coarsest topology that makes f ↦ f x continuous for all x : A. Alternatively, given a presentation A = ℤ[xᵢ]/I, this is the subspace topology Hom(A, R) ↪ Hom(ℤ[xᵢ], R) = Rᶥ.

Main results #

The topology on Hom(A, R) for a topological ring R, given by the coarsest topology that makes f ↦ f x continuous for all x : A (see continuous_apply). Alternatively, given a presentation A = ℤ[xᵢ]/I, This is the subspace topology Hom(A, R) ↪ Hom(ℤ[xᵢ], R) = Rᶥ (see mvPolynomialHomeo).

This is a scoped instance in CommRingCat.HomTopology.

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    theorem CommRingCat.HomTopology.continuous_apply {R A : CommRingCat} [TopologicalSpace R] (x : A) :
    Continuous fun (f : A R) => (Hom.hom f) x

    If A ≅ B, then Hom(A, R) is homeomorphc to Hom(B, R).

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      Hom(A/I, R) has the subspace topology of Hom(A, R). More generally, a surjection A ⟶ B gives rise to an embedding Hom(B, R) ⟶ Hom(A, R)

      noncomputable def CommRingCat.HomTopology.mvPolynomialHomeomorph (σ : Type v) (R A : CommRingCat) [TopologicalSpace R] [IsTopologicalRing R] :
      (of (MvPolynomial σ A) R) ≃ₜ (A R) × (σR)

      Hom(A[Xᵢ], R) is homeomorphic to Hom(A, R) × Rⁱ.

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