ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-28

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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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A directed union of subgroups distributes over intersection with a fixed subgroup

Example

Let $A = ⨁_{n \geq 1} Z e_{n}$ , let $B_{n} = ⟨ e_{1}, \dots, e_{n} ⟩$ , and let $C = ⟨ e_{1} + e_{2}, e_{3}, e_{4}, \dots ⟩$ . Then $(B_{n})$ is directed, $⋁_{n} B_{n} = A$ , and

$(⋁_{n} B_{n}) \land C = C = ⋁_{n} (B_{n} \land C) .$

This is the AB5 lattice identity in a concrete module calculation.

Facts & Assumptions

Given: The subgroup chain $(B_{n})$ and the fixed subgroup $C$ displayed in the statement.

[L1]

AB5 is the directed-join distributivity law for subobjects (The axioms AB5 and AB5*).

[L2]

Module categories are Grothendieck, hence satisfy AB5 (Module categories are Grothendieck categories).

Verification

technique · direct

1.1L1algebra

The subgroups $B_{n}$ are directed by inclusion and their union is all of $A$ , since every element of $A$ has finite support. Also $B_{n} \cap C = ⟨ e_{1} + e_{2}, e_{3}, \dots, e_{n} ⟩$ for $n \geq 2$ , because those are exactly the generators of $C$ lying in the first $n$ coordinates.

2.1L1L2step 1.1∎

Taking the union of the intersections from step 1.1 recovers all of $C$ , so $⋁_{n} (B_{n} \land C) = C = (⋁_{n} B_{n}) \land C .$ This is precisely the AB5 identity [L1], exactly as the abstract theorem [L2] predicts for module categories.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Alexandre Grothendieck, Sur quelques points d'algèbre homologique, Barr translation, Section 1.5 (standard reference, not scraped)