CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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A localisation is unique up to a unique isomorphism compatible with the map from $R$

Statement

Let $λ : R \to L$ and $λ^{'} : R \to L^{'}$ be unital homomorphisms of commutative rings. Suppose each map sends every $s \in S$ to a unit and has the localisation universal property: every homomorphism from $R$ that inverts $S$ factors uniquely through it. Then there is a unique ring isomorphism $Φ : L \to L^{'}$ such that $Φ λ = λ^{'}$ .

Facts & Assumptions

Given: Two objects $(L, λ)$ and $(L^{'}, λ^{'})$ satisfying the stated universal property for the same pair $(R, S)$ .

[F1]

A homomorphism from $R$ that takes $S$ to units factors uniquely through a localisation map (Universal property of localisation: maps that invert $S$ factor uniquely through $S^{- 1} R$ ).

Proof

technique · direct universal-property argument

1.1

Apply the universal property of $L$ to $λ^{'}$ and that of $L^{'}$ to $λ$ . This gives unique homomorphisms $Φ : L \to L^{'}$ and $Ψ : L^{'} \to L$ with $Φ λ = λ^{'}$ and $Ψ λ^{'} = λ$ .

2.1

Both $Ψ Φ$ and ${id}_{L}$ compose with $λ$ to give $λ$ . Uniqueness for $L$ gives $Ψ Φ = {id}_{L}$ ; similarly $Φ Ψ = {id}_{L^{'}}$ . Thus $Φ$ is an isomorphism.

F1step 1.1

3.1

Any isomorphism compatible with the maps from $R$ is, in particular, a factorisation of $λ^{'}$ through $λ$ , so it equals $Φ$ by uniqueness.

F1step 1.1∎

Depends on

Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$

Used by

Localising twice is localising once at the multiplicative set generated by both denominator sets Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

The Stacks Project, Proposition 10.9.3 (standard reference, not scraped)

A localisation is unique up to a unique isomorphism compatible with the map from RR