CounterexampleConstruction: AI-generatedVerification: AI-adaptedprecheck passaudited 2026-09-04‡ 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.

L_0 of a non-right-exact functor need not recover the functor

Statement refuted

For an additive functor, the zeroth left derived object always agrees with the original functor value.

Facts & Assumptions

Given: The functor $F (M) = {Hom}_{Z} (Z / 2 Z, M)$ , and supplied projective resolution data $P$ on a class containing $Z / 2 Z$ that assigns it the standard projective resolution below.

[L1]

The companion false statement is false (FALSE: every additive functor has L_0 naturally isomorphic to itself).

[L2]

Hom is left exact in each variable (Hom is left exact in each variable).

[L3]

Right exactness is sufficient for the natural recovery of $F$ from $L_{0}$ (The zero-th left derived functor of a right exact functor recovers the functor).

[L4]

Left derived objects are computed from the homology of an applied deleted projective resolution (Left derived objects relative to supplied projective resolution data).

Counterexample

technique · direct

1.1

The functor $F$ is additive and, by [L2], left exact. On the standard resolution $0 \to Z \overset{\times 2}{\to} Z \to Z / 2 \to 0$ assigned by $P$ , both groups ${Hom}_{Z} (Z / 2, Z)$ vanish. Therefore [L4] gives $L_{0}^{P} F (Z / 2) = H_{0} (0 \to 0) = 0.$

L2L4givenalgebra

2.1

On the other hand, $F (Z / 2) = {Hom}_{Z} (Z / 2, Z / 2) \neq 0$ . So $L_{0}^{P} F (Z / 2) \neq F (Z / 2)$ . This concrete computation realises the failure announced by [L1] and shows that the right-exactness hypothesis in the recovery theorem [L3] cannot simply be omitted.

L1L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)