mathlib documentation

topology.algebra.valuation

The topology on a valued ring #

In this file, we define the non archimedean topology induced by a valuation on a ring. The main definition is a valued type class which equips a ring with a valuation taking values in a group with zero (living in the same universe). Other instances are then deduced from this.

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@[class]
structure valued (R : Type u) [ring R] :
Type (u+1)

A valued ring is a ring that comes equipped with a distinguished valuation.

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theorem valued.subgroups_basis {R : Type u_1} [ring R] [valued R] :
ring_subgroups_basis (λ (γ : (valued.Γ₀ R)ˣ), valued.v.lt_add_subgroup γ)

The basis of open subgroups for the topology on a valued ring.

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@[protected, instance]
def valued.topological_space {R : Type u_1} [ring R] [valued R] :
topological_space R
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theorem valued.mem_nhds {R : Type u_1} [ring R] [valued R] {s : set R} {x : R} :
s ∈ 𝓝 x ↔ ∃ (γ : (valued.Γ₀ R)ˣ), {y : R | ⇑valued.v (y - x) < ↑γ} ⊆ s
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theorem valued.mem_nhds_zero {R : Type u_1} [ring R] [valued R] {s : set R} :
s ∈ 𝓝 0 ↔ ∃ (γ : (valued.Γ₀ R)ˣ), {x : R | ⇑valued.v x < ↑γ} ⊆ s
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theorem valued.loc_const {R : Type u_1} [ring R] [valued R] {x : R} (h : ⇑valued.v x ≠ 0) :
{y : R | ⇑valued.v y = ⇑valued.v x} ∈ 𝓝 x
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@[protected, instance]
def valued.uniform_space {R : Type u_1} [ring R] [valued R] :
uniform_space R

The uniform structure on a valued ring.

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@[protected, instance]
def valued.uniform_add_group {R : Type u_1} [ring R] [valued R] :
uniform_add_group R

A valued ring is a uniform additive group.

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theorem valued.cauchy_iff {R : Type u_1} [ring R] [valued R] {F : filter R} :
cauchy F ↔ F.ne_bot ∧ ∀ (γ : (valued.Γ₀ R)ˣ), ∃ (M : set R) (H : M ∈ F), ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ⇑valued.v (y - x) < ↑γ