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Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology

Adic topology #

Given a commutative ring R and an ideal I in R, this file constructs the unique topology on R which is compatible with the ring structure and such that a set is a neighborhood of zero if and only if it contains a power of I. This topology is non-archimedean: every neighborhood of zero contains an open subgroup, namely a power of I.

It also studies the predicate IsAdic which states that a given topological ring structure is adic, proving a characterization and showing that raising an ideal to a positive power does not change the associated topology.

Finally, it defines WithIdeal, a class registering an ideal in a ring and providing the corresponding adic topology to the type class inference system.

Main definitions and results #

Implementation notes #

The I-adic topology on a ring R has a contrived definition using I^n • ⊤ instead of I to make sure it is definitionally equal to the I-topology on R seen as an R-module.

theorem Ideal.adic_basis {R : Type u_1} [CommRing R] (I : Ideal R) :
SubmodulesRingBasis fun (n : ) => I ^ n

The adic ring filter basis associated to an ideal I is made of powers of I.

Equations
  • I.ringFilterBasis = .toRingFilterBasis
Instances For

    The adic topology associated to an ideal I. This topology admits powers of I as a basis of neighborhoods of zero. It is compatible with the ring structure and is non-archimedean.

    Equations
    • I.adicTopology = .topology
    Instances For
      theorem Ideal.hasBasis_nhds_zero_adic {R : Type u_1} [CommRing R] (I : Ideal R) :
      (nhds 0).HasBasis (fun (_n : ) => True) fun (n : ) => (I ^ n)

      For the I-adic topology, the neighborhoods of zero has basis given by the powers of I.

      theorem Ideal.hasBasis_nhds_adic {R : Type u_1} [CommRing R] (I : Ideal R) (x : R) :
      (nhds x).HasBasis (fun (_n : ) => True) fun (n : ) => (fun (y : R) => x + y) '' (I ^ n)
      theorem Ideal.adic_module_basis {R : Type u_1} [CommRing R] (I : Ideal R) (M : Type u_2) [AddCommGroup M] [Module R M] :
      I.ringFilterBasis.SubmodulesBasis fun (n : ) => I ^ n
      def Ideal.adicModuleTopology {R : Type u_1} [CommRing R] (I : Ideal R) (M : Type u_2) [AddCommGroup M] [Module R M] :

      The topology on an R-module M associated to an ideal M. Submodules $I^n M$, written I^n • ⊤ form a basis of neighborhoods of zero.

      Equations
      • I.adicModuleTopology M = (I.ringFilterBasis.moduleFilterBasis ).topology
      Instances For
        def Ideal.openAddSubgroup {R : Type u_1} [CommRing R] (I : Ideal R) (n : ) :

        The elements of the basis of neighborhoods of zero for the I-adic topology on an R-module M, seen as open additive subgroups of M.

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        Instances For
          def IsAdic {R : Type u_1} [CommRing R] [H : TopologicalSpace R] (J : Ideal R) :

          Given a topology on a ring R and an ideal J, IsAdic J means the topology is the J-adic one.

          Equations
          Instances For
            theorem isAdic_iff {R : Type u_1} [CommRing R] [top : TopologicalSpace R] [TopologicalRing R] {J : Ideal R} :
            IsAdic J (∀ (n : ), IsOpen (J ^ n)) snhds 0, ∃ (n : ), (J ^ n) s

            A topological ring is J-adic if and only if it admits the powers of J as a basis of open neighborhoods of zero.

            theorem is_ideal_adic_pow {R : Type u_1} [CommRing R] [TopologicalSpace R] [TopologicalRing R] {J : Ideal R} (h : IsAdic J) {n : } (hn : 0 < n) :
            IsAdic (J ^ n)
            class WithIdeal (R : Type u_2) [CommRing R] :
            Type u_2

            The ring R is equipped with a preferred ideal.

            Instances
              @[instance 100]
              Equations
              @[instance 100]
              Equations
              • =
              @[instance 100]
              Equations
              • =

              The adic topology on an R module coming from the ideal WithIdeal.I. This cannot be an instance because R cannot be inferred from M.

              Equations
              Instances For