TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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A direct summand of a projective is projective

Statement

Every direct summand of a projective object in an abelian category is projective.

Facts & Assumptions

Given: A projective object $P$ with a decomposition $P ≅ Q \oplus Q^{'}$ .

[L1]

Projective objects are exactly those with the lifting property against epimorphisms (Projective object characterisations).

Proof

technique · direct

1.1L1construct

Let $q : E ↠ M$ be epic and let $f : Q \to M$ be any morphism. Write $i : Q \to P$ and $r : P \to Q$ for the split inclusion and retraction, so $r i = 1_{Q}$ . The composite $f r : P \to M$ lifts across $q$ by [L1] to a map $\tilde{g} : P \to E$ .

2.1L1step 1.1∎

Put $\tilde{f} : = \tilde{g} i : Q \to E$ . Then $q \tilde{f} = q \tilde{g} i = f r i = f .$ So $Q$ has the lifting property against every epimorphism, hence is projective by [L1].

Depends on

Projective object characterisations

Used by

The complex of a braid word Definition
The two-sided projective bimodules Uᵢ and their tensor functors Definition
Matrix-unit rotation for a k–Matₙ(k) Morita pair Example
FALSE: a generator is automatically projective False statement
Bounded two-sided projective bimodule complexes act on Cₘ Lemma
Double bar comparison for cyclic bimodule tensor products Lemma
Finite graded projectives are sums of shifted vertex projectives Lemma
Corner computations: the Uᵢ satisfy the Temperley-Lieb relations Theorem
Derived cyclicity of Hochschild hyperhomology Theorem
Finite graded projectives are finite shifted-free summands Theorem

Dependency tree · two levels

3 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

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.6 (standard reference, not scraped)