False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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FALSE: Baer sum only takes the direct sum of middle objects

Statement

Facts & Assumptions

Given: Two extensions $0 \to N \to E_{i} \to M \to 0$ for $i = 1, 2$ .

Refutation

technique · direct

1.1givenalgebra

Taking their direct sum gives $0 \to N \oplus N \to E_{1} \oplus E_{2} \to M \oplus M \to 0$ , whose endpoints are $N \oplus N$ and $M \oplus M$ , not the required $N$ and $M$ .

2.1step 1.1algebra∎

Pulling this sequence back along $Δ : M \to M \oplus M$ restores the quotient $M$ , and pushing out along $\nabla : N \oplus N \to N$ restores the subobject $N$ . These two operations, not merely the direct sum of middle terms, define the Baer sum.

Depends on

Used by

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Dependency tree · two levels

7 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapters 3–4 (standard reference, not scraped)