DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Split short exact sequences, sections, and retractions

Definition

In a short exact sequence $0 \to A \overset{i}{\to} B \overset{p}{\to} C \to 0,$ a section of $p$ is a homomorphism $s : C \to B$ with $p \circ s = {id}_{C}$ , and a retraction of $i$ is a homomorphism $r : B \to A$ with $r \circ i = {id}_{A}$ .

The sequence splits if it has a section, equivalently, as proved in The splitting lemma for short exact sequences of modules ↗, if it has a retraction or if its middle term is isomorphic to $A \oplus C$ compatibly with $i$ and $p$ .

Depends on

Exact sequences and short exact sequences of modules

Used by

0→ℤxrightarrow×2ℤ→ℤ/2ℤ→0 does not split Counterexample
0→ A→ A⊕ C→ C→0 is canonically split Example
Every short exact sequence of modules splits False statement
The splitting lemma for short exact sequences of modules Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 5 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

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)