Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-28
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Localization of modules gives an exact functor between module categories

Example

If S is a multiplicative subset of a commutative ring R, the localization functor

S−1(−):R-Mod⟶S−1R-Mod

is exact.

Facts & Assumptions

Given: A ring R and a multiplicative subset S.

[L1]

Module categories are abelian (Modules over a ring form an abelian category).

[L2]

Localization of modules preserves short exact sequences (Localisation of modules is exact).

[L3]

Exact functors between abelian categories are defined by additivity plus left and right exactness (Exact functor between abelian categories).

Verification

technique · direct
1.1L2

By [L2], localization carries every short exact sequence of R-modules to a short exact sequence of S−1R-modules.

2.1L1L2L3step 1.1∎

Since both source and target are abelian by [L1], the short-exact-sequence criterion makes localization exact, which is exactly the notion in [L3].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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