TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Localisation of modules is exact

Statement

0 ⟶ M^{'} \overset{f}{\to} M \overset{g}{\to} M^{''} ⟶ 0

is a short exact sequence of $R$ -modules, then

0 ⟶ S^{- 1} M^{'} \overset{S^{- 1} f}{\to} S^{- 1} M \overset{S^{- 1} g}{\to} S^{- 1} M^{''} ⟶ 0

is a short exact sequence of $S^{- 1} R$ -modules.

Facts & Assumptions

Given: A commutative ring $R$ , a multiplicative subset $S \subseteq R$ , and a short exact sequence $0 \to M^{'} \overset{f}{\to} M \overset{g}{\to} M^{''} \to 0$ .

[L1]

Localisation is naturally $(S^{- 1} R) \otimes_{R} -$ (Localisation of modules is extension of scalars).

[L2]

Tensoring a right-exact sequence with a fixed module preserves right exactness (Tensoring is right exact).

[L3]

Injective module homomorphisms remain injective after localisation (Injective module maps remain injective after localisation).

[L4]

A short exact sequence is exact, with left map injective and right map surjective (Exact sequences and short exact sequences of modules).

Proof

technique · direct

1.1

By [L4], the tail $M^{'} \overset{f}{\to} M \overset{g}{\to} M^{''} \to 0$ is exact. Using [L1] to identify localisation with tensor product, [L2] gives an exact sequence $S^{- 1} M^{'} \overset{S^{- 1} f}{\to} S^{- 1} M \overset{S^{- 1} g}{\to} S^{- 1} M^{''} \to 0$ .

L1L2L4

1.2

The map $f$ is injective by [L4], so [L3] makes $S^{- 1} f$ injective.

L3L4

2.1

Steps 1.1 and 1.2 show that the localised sequence is short exact.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Theorem 12.20 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, Proposition 5.11 (standard reference, not scraped)