CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-01‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Every Grothendieck category has enough injectives, and every object admits an injective resolution

Statement

Assume the Axiom of Choice.

Every locally small Grothendieck category has enough injectives, and every object in it admits an injective resolution.

Facts & Assumptions

Given: A locally small Grothendieck category $A$ and an object $A$ of $A$ .

[L1]

Grothendieck categories admit functorial injective embeddings (Grothendieck abelian categories have functorial injective embeddings).

[L2]

A chosen injective embedding of the current cokernel extends a partial coaugmented resolution by one exact step (One-step extension of a partial injective resolution).

[L3]

Enough injectives means that every object embeds in an injective object (A category with enough projectives and with enough injectives).

[L4]

An injective resolution is an exact coaugmented complex of injectives (Injective resolutions in an abelian category).

Proof

technique · direct

1.1

By [L1], every object $A$ admits a monomorphism into an injective object. Therefore $A$ has enough injectives in the sense of [L3].

L1L3

1.2

Starting from the functorial embedding $η_{A} : A ↣ E (A)$ from [L1], let $C^{0}$ be its cokernel and iterate the same functorial construction on successive cokernels. Applying [L2] at each stage yields an exact coaugmented complex $0 \to A \to E (A) \to E (C^{0}) \to E (C^{1}) \to \dots$ of injectives.

L1L2construct

2.1

By [L4], the complex from step 1.2 is an injective resolution of $A$ , including when $A = 0$ . Because the embedding functor in [L1] is functorial, no additional arbitrary sequence of choices is introduced.

L1L4step 1.2∎

Depends on

Used by

FALSE: every abelian category has enough projectives and enough injectives False statement

Dependency tree · two levels

14 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)