PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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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.
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Negative derived degrees vanish for one-sided resolutions

Statement

Let $P$ and $I$ be supplied projective and injective resolution data, and let $F$ be an additive functor.

If $n < 0$ , then for every object $A$ in the common domain, $L_{n}^{P} F (A) = 0 and R_{I}^{n} F (A) = 0.$

Facts & Assumptions

Given: An object $A$ and an integer $n < 0$ .

[L1]

The object $L_{n}^{P} F (A)$ is the homology of the deleted projective resolution $F (P (A)_{d e l})$ in degree $n$ (Left derived objects relative to supplied projective resolution data).

[L2]

The object $R_{I}^{n} F (A)$ is the cohomology of the deleted injective resolution $F (I (A)_{d e l})$ in degree $n$ (Right derived objects relative to supplied injective resolution data).

Proof

technique · direct

1.1

By [L1], the complex computing $L_{n}^{P} F (A)$ is zero in every negative degree, because a deleted projective resolution is supported only in nonnegative homological degrees. Therefore both its degree- $n$ cycle object and its degree- $n$ boundary object are zero, so $L_{n}^{P} F (A) = 0$ .

L1givenalgebra

1.2

By [L2], the cochain complex computing $R_{I}^{n} F (A)$ is zero in every negative cohomological degree, because a deleted injective resolution begins in degree $0$ . Hence its degree- $n$ cocycle and coboundary objects are zero, so $R_{I}^{n} F (A) = 0$ .

L2givenalgebra

2.1

Steps 1.1 and 1.2 prove the claimed vanishing for both one-sided derived constructions.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)