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

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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

1.1L2L4givenalgebra

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 $L_{0}^{P} F (Z / 2) = H_{0} (0 \to 0) = 0.$

2.1L1L3step 1.1∎

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.

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)