LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-29

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A sufficiently long generator-extension iteration is injective

Statement

Assume the Axiom of Choice. Let $A$ be a locally small Grothendieck category with generator $U$ , and let $(M_{α})$ be the transfinite iteration of the one-step generator extension functor starting at an object $M$ . Let $κ$ bound the cardinalities of the sets of subobjects of all subobjects $N \subseteq U$ . If $λ$ is a limit ordinal with $cf (λ) > κ$ , then $M_{λ}$ is injective.

Facts & Assumptions

Given: The Axiom of Choice, the transfinite tower $(M_{α})$ in a locally small Grothendieck category with generator $U$ , the bound $κ$ from [L2], and a limit ordinal $λ$ with $cf (λ) > κ$ .

[A1]

AC supplies the choice hypothesis in both the transfinite factorization and the generator-subobject injectivity test (The Axiom of Choice).

[L1]

Extension from subobjects of the fixed generator detects injectivity (Extension from subobjects of a generator detects injectivity).

[L2]

If $cf (λ) > κ$ , every map from a subobject of the generator to $M_{λ}$ factors through an earlier stage, and all transition maps are monic (Transfinite iteration of the generator extension preserves monomorphisms and factorizes small-source maps).

[L3]

Every map from a subobject of the generator into one stage extends across the generator at the next stage (The one-step generator map is a functorial monomorphism).

Proof

technique · direct

1.1L2L3givenconstruct

Let $f : N \to M_{λ}$ with $N \subseteq U$ . By [L2], write $f = j_{α, λ} g$ for some $α < λ$ and $g : N \to M_{α}$ . Since $λ$ is a limit ordinal, $α + 1 < λ$ . By [L3], there is $h : U \to M_{α + 1}$ whose restriction to $N$ is $η_{M_{α}} g$ . Compatibility of the transition maps gives $j_{α + 1, λ} h ∣_{N} = j_{α + 1, λ} η_{M_{α}} g = j_{α, λ} g = f,$ so $j_{α + 1, λ} h$ extends $f$ across $N \subseteq U$ .

2.1A1L1step 1.1∎

Thus every map from every subobject of $U$ extends to $U$ . Under [A1], [L1] makes $M_{λ}$ injective.

Depends on

Used by

Grothendieck abelian categories have functorial injective embeddings Theorem

Dependency tree · two levels

15 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

The Stacks Project, Section 19.11: Injectives in Grothendieck categories (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)