Neighborhood bases for non-archimedean rings and modules

This files contains special families of filter bases on rings and modules that give rise to non-archimedean topologies.

The main definition is RingSubgroupsBasis which is a predicate on a family of additive subgroups of a ring. The predicate ensures there is a topology RingSubgroupsBasis.topology which is compatible with a ring structure and admits the given family as a basis of neighborhoods of zero. In particular the given subgroups become open subgroups (bundled in RingSubgroupsBasis.openAddSubgroup) and we get a non-archimedean topological ring (RingSubgroupsBasis.nonarchimedean).

A special case of this construction is given by SubmodulesBasis where the subgroups are sub-modules in a commutative algebra. This important example gives rise to the adic topology (studied in its own file).

structure RingSubgroupsBasis {A : Type u_1} {ι : Type u_2} [Ring A] (B : ι → AddSubgroup A) : Prop

A family of additive subgroups on a ring A is a subgroups basis if it satisfies some axioms ensuring there is a topology on A which is compatible with the ring structure and admits this family as a basis of neighborhoods of zero.

• inter : ∀ (i j : ι), ∃ (k : ι), B k ≤ B i ⊓ B j

  Condition for B to be a filter basis on A.

• mul : ∀ (i : ι), ∃ (j : ι), ↑(B j) * ↑(B j) ⊆ ↑(B i)

  For each set B in the submodule basis on A, there is another basis element B' such that the set-theoretic product B' * B' is in B.

• leftMul : ∀ (x : A) (i : ι), ∃ (j : ι), ↑(B j) ⊆ (fun (x_1 : A) => x * x_1) ⁻¹' ↑(B i)

  For any element x : A and any set B in the submodule basis on A, there is another basis element B' such that B' * x is in B.

• rightMul : ∀ (x : A) (i : ι), ∃ (j : ι), ↑(B j) ⊆ (fun (x_1 : A) => x_1 * x) ⁻¹' ↑(B i)

  For any element x : A and any set B in the submodule basis on A, there is another basis element B' such that x * B' is in B.

theorem RingSubgroupsBasis.inter {A : Type u_1} {ι : Type u_2} [Ring A] {B : ι → AddSubgroup A} (self : RingSubgroupsBasis B) (i : ι) (j : ι) : ∃ (k : ι), B k ≤ B i ⊓ B j

Condition for B to be a filter basis on A.

theorem RingSubgroupsBasis.mul {A : Type u_1} {ι : Type u_2} [Ring A] {B : ι → AddSubgroup A} (self : RingSubgroupsBasis B) (i : ι) : ∃ (j : ι), ↑(B j) * ↑(B j) ⊆ ↑(B i)

For each set B in the submodule basis on A, there is another basis element B' such that the set-theoretic product B' * B' is in B.

theorem RingSubgroupsBasis.leftMul {A : Type u_1} {ι : Type u_2} [Ring A] {B : ι → AddSubgroup A} (self : RingSubgroupsBasis B) (x : A) (i : ι) : ∃ (j : ι), ↑(B j) ⊆ (fun (x_1 : A) => x * x_1) ⁻¹' ↑(B i)

For any element x : A and any set B in the submodule basis on A, there is another basis element B' such that B' * x is in B.

theorem RingSubgroupsBasis.rightMul {A : Type u_1} {ι : Type u_2} [Ring A] {B : ι → AddSubgroup A} (self : RingSubgroupsBasis B) (x : A) (i : ι) : ∃ (j : ι), ↑(B j) ⊆ (fun (x_1 : A) => x_1 * x) ⁻¹' ↑(B i)

For any element x : A and any set B in the submodule basis on A, there is another basis element B' such that x * B' is in B.

theorem RingSubgroupsBasis.of_comm {A : Type u_3} {ι : Type u_4} [CommRing A] (B : ι → AddSubgroup A) (inter : ∀ (i j : ι), ∃ (k : ι), B k ≤ B i ⊓ B j) (mul : ∀ (i : ι), ∃ (j : ι), ↑(B j) * ↑(B j) ⊆ ↑(B i)) (leftMul : ∀ (x : A) (i : ι), ∃ (j : ι), ↑(B j) ⊆ (fun (y : A) => x * y) ⁻¹' ↑(B i)) : RingSubgroupsBasis B

def RingSubgroupsBasis.toRingFilterBasis {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) : RingFilterBasis A

Every subgroups basis on a ring leads to a ring filter basis.

Equations

• hB.toRingFilterBasis = RingFilterBasis.mk ⋯ ⋯ ⋯

theorem RingSubgroupsBasis.mem_addGroupFilterBasis_iff {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) {V : Set A} : V ∈ RingFilterBasis.toAddGroupFilterBasis ↔ ∃ (i : ι), V = ↑(B i)

theorem RingSubgroupsBasis.mem_addGroupFilterBasis {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) (i : ι) : ↑(B i) ∈ RingFilterBasis.toAddGroupFilterBasis

def RingSubgroupsBasis.topology {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) : TopologicalSpace A

The topology defined from a subgroups basis, admitting the given subgroups as a basis of neighborhoods of zero.

Equations

• hB.topology = RingFilterBasis.toAddGroupFilterBasis.topology

theorem RingSubgroupsBasis.hasBasis_nhds_zero {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) : (nhds 0).HasBasis (fun (x : ι) => True) fun (i : ι) => ↑(B i)

theorem RingSubgroupsBasis.hasBasis_nhds {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) (a : A) : (nhds a).HasBasis (fun (x : ι) => True) fun (i : ι) => {b : A | b - a ∈ B i}

def RingSubgroupsBasis.openAddSubgroup {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) (i : ι) : OpenAddSubgroup A

Given a subgroups basis, the basis elements as open additive subgroups in the associated topology.

Equations

• hB.openAddSubgroup i = let x := hB.topology; let __src := B i; { toAddSubgroup := __src, isOpen' := ⋯ }

theorem RingSubgroupsBasis.nonarchimedean {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) : NonarchimedeanRing A

structure SubmodulesRingBasis {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (B : ι → Submodule R A) : Prop

A family of submodules in a commutative R-algebra A is a submodules basis if it satisfies some axioms ensuring there is a topology on A which is compatible with the ring structure and admits this family as a basis of neighborhoods of zero.

• inter : ∀ (i j : ι), ∃ (k : ι), B k ≤ B i ⊓ B j

  Condition for B to be a filter basis on A.

• leftMul : ∀ (a : A) (i : ι), ∃ (j : ι), a • B j ≤ B i

  For any element a : A and any set B in the submodule basis on A, there is another basis element B' such that a • B' is in B.

• mul : ∀ (i : ι), ∃ (j : ι), ↑(B j) * ↑(B j) ⊆ ↑(B i)

  For each set B in the submodule basis on A, there is another basis element B' such that the set-theoretic product B' * B' is in B.

theorem SubmodulesRingBasis.inter {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} (self : SubmodulesRingBasis B) (i : ι) (j : ι) : ∃ (k : ι), B k ≤ B i ⊓ B j

Condition for B to be a filter basis on A.

theorem SubmodulesRingBasis.leftMul {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} (self : SubmodulesRingBasis B) (a : A) (i : ι) : ∃ (j : ι), a • B j ≤ B i

For any element a : A and any set B in the submodule basis on A, there is another basis element B' such that a • B' is in B.

theorem SubmodulesRingBasis.mul {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} (self : SubmodulesRingBasis B) (i : ι) : ∃ (j : ι), ↑(B j) * ↑(B j) ⊆ ↑(B i)

For each set B in the submodule basis on A, there is another basis element B' such that the set-theoretic product B' * B' is in B.

theorem SubmodulesRingBasis.toRing_subgroups_basis {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} (hB : SubmodulesRingBasis B) : RingSubgroupsBasis fun (i : ι) => (B i).toAddSubgroup

def SubmodulesRingBasis.topology {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} [Nonempty ι] (hB : SubmodulesRingBasis B) : TopologicalSpace A

The topology associated to a basis of submodules in an algebra.

Equations

• hB.topology = ⋯.topology

structure SubmodulesBasis {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] (B : ι → Submodule R M) : Prop

A family of submodules in an R-module M is a submodules basis if it satisfies some axioms ensuring there is a topology on M which is compatible with the module structure and admits this family as a basis of neighborhoods of zero.

• inter : ∀ (i j : ι), ∃ (k : ι), B k ≤ B i ⊓ B j

  Condition for B to be a filter basis on M.

• smul : ∀ (m : M) (i : ι), ∀ᶠ (a : R) in nhds 0, a • m ∈ B i

  For any element m : M and any set B in the basis, a • m lies in B for all a sufficiently close to 0.

theorem SubmodulesBasis.inter {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] {B : ι → Submodule R M} (self : SubmodulesBasis B) (i : ι) (j : ι) : ∃ (k : ι), B k ≤ B i ⊓ B j

Condition for B to be a filter basis on M.

theorem SubmodulesBasis.smul {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] {B : ι → Submodule R M} (self : SubmodulesBasis B) (m : M) (i : ι) : ∀ᶠ (a : R) in nhds 0, a • m ∈ B i

For any element m : M and any set B in the basis, a • m lies in B for all a sufficiently close to 0.

def SubmodulesBasis.toModuleFilterBasis {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] [Nonempty ι] {B : ι → Submodule R M} (hB : SubmodulesBasis B) : ModuleFilterBasis R M

The image of a submodules basis is a module filter basis.

Equations

• One or more equations did not get rendered due to their size.

def SubmodulesBasis.topology {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] [Nonempty ι] {B : ι → Submodule R M} (hB : SubmodulesBasis B) : TopologicalSpace M

The topology associated to a basis of submodules in a module.

Equations

• hB.topology = hB.toModuleFilterBasis.topology

def SubmodulesBasis.openAddSubgroup {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] [Nonempty ι] {B : ι → Submodule R M} (hB : SubmodulesBasis B) (i : ι) : OpenAddSubgroup M

Given a submodules basis, the basis elements as open additive subgroups in the associated topology.

Equations

• hB.openAddSubgroup i = let x := hB.topology; let __src := (B i).toAddSubgroup; { toAddSubgroup := __src, isOpen' := ⋯ }

theorem SubmodulesBasis.nonarchimedean {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] [Nonempty ι] {B : ι → Submodule R M} (hB : SubmodulesBasis B) : NonarchimedeanAddGroup M

theorem SubmodulesRingBasis.toSubmodulesBasis {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] [TopologicalSpace R] {B : ι → Submodule R A} (hB : SubmodulesRingBasis B) (hsmul : ∀ (m : A) (i : ι), ∀ᶠ (a : R) in nhds 0, a • m ∈ B i) : SubmodulesBasis B

structure RingFilterBasis.SubmodulesBasis {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] (BR : RingFilterBasis R) (B : ι → Submodule R M) : Prop

Given a ring filter basis on a commutative ring R, define a compatibility condition on a family of submodules of an R-module M. This compatibility condition allows to get a topological module structure.

• inter : ∀ (i j : ι), ∃ (k : ι), B k ≤ B i ⊓ B j

  Condition for B to be a filter basis on M.

• smul : ∀ (m : M) (i : ι), ∃ U ∈ BR, U ⊆ (fun (x : R) => x • m) ⁻¹' ↑(B i)

  For any element m : M and any set B i in the submodule basis on M, there is a U in the ring filter basis on R such that U * m is in B i.

theorem RingFilterBasis.SubmodulesBasis.inter {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] {BR : RingFilterBasis R} {B : ι → Submodule R M} (self : BR.SubmodulesBasis B) (i : ι) (j : ι) : ∃ (k : ι), B k ≤ B i ⊓ B j

Condition for B to be a filter basis on M.

theorem RingFilterBasis.SubmodulesBasis.smul {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] {BR : RingFilterBasis R} {B : ι → Submodule R M} (self : BR.SubmodulesBasis B) (m : M) (i : ι) : ∃ U ∈ BR, U ⊆ (fun (x : R) => x • m) ⁻¹' ↑(B i)

For any element m : M and any set B i in the submodule basis on M, there is a U in the ring filter basis on R such that U * m is in B i.

theorem RingFilterBasis.submodulesBasisIsBasis {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] (BR : RingFilterBasis R) {B : ι → Submodule R M} (hB : BR.SubmodulesBasis B) : SubmodulesBasis B

def RingFilterBasis.moduleFilterBasis {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [Nonempty ι] (BR : RingFilterBasis R) {B : ι → Submodule R M} (hB : BR.SubmodulesBasis B) : ModuleFilterBasis R M

The module filter basis associated to a ring filter basis and a compatible submodule basis. This allows to build a topological module structure compatible with the given module structure and the topology associated to the given ring filter basis.

Equations

• BR.moduleFilterBasis hB = ⋯.toModuleFilterBasis