False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-04

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FALSE: every additive functor has L_0 naturally isomorphic to itself

Statement

Every additive functor $F$ has $L_{0} F$ naturally isomorphic to $F$ .

Facts & Assumptions

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

[L1]

If $F$ is right exact, the zero-th left derived functor recovers $F$ naturally (The zero-th left derived functor of a right exact functor recovers the functor).

[L2]

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

[L3]

Additivity means preservation of sums on hom-groups (Additive functor).

Refutation

technique · direct

1.1L2L3givenalgebra

The functor $F$ is additive by [L3], but it is enough to compute its value on the standard projective resolution $0 \to Z \overset{\times 2}{\to} Z \to Z / 2 Z \to 0.$ Applying $F$ to the deleted resolution gives $0 \to {Hom}_{Z} (Z / 2, Z) \overset{\times 2}{\to} {Hom}_{Z} (Z / 2, Z) \to 0,$ and both displayed Hom groups are $0$ .

2.1L1step 1.1∎

Therefore $L_{0}^{P} F (Z / 2) = 0$ , while $F (Z / 2) = {Hom}_{Z} (Z / 2, Z / 2) \neq 0$ . So $L_{0}^{P} F$ is not naturally isomorphic to $F$ for this supplied datum and additive functor. Thus additivity alone does not guarantee recovery; [L1] records right exactness as a sufficient hypothesis, and the displayed claim is false.

Depends on

Used by

L₀ of a non-right-exact functor need not recover the functor Counterexample

Dependency tree · two levels

12 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)