PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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An exact functor has vanishing positive derived functors

Statement

Let $P$ be a supplied projective resolution datum on a class $D_{P}$ , let $I$ be a supplied injective resolution datum on a class $D_{I}$ , and let $F : A \to B$ be an exact functor between abelian categories. Then for every $A \in D_{P}$ and every $n > 0$ , $L_{n}^{P} F (A) = 0,$ and for every $B \in D_{I}$ and every $n > 0$ , $R_{I}^{n} F (B) = 0.$

Facts & Assumptions

Given: An integer $n > 0$ , an object $A \in D_{P}$ , and an object $B \in D_{I}$ .

[L1]

The left and right derived objects are the homology or cohomology of the deleted chosen resolutions after applying $F$ (Left derived objects relative to supplied projective resolution data, Right derived objects relative to supplied injective resolution data).

[L2]

Exact functors commute with homology of chain complexes (An exact functor commutes with homology).

[L3]

Exactness means that $F$ is exact on both the projective and injective resolution complexes (Exact functor between abelian categories).

Proof

technique · direct

1.1L1L3givenalgebra

The deleted projective resolution of $A$ is exact in every positive degree. By [L3], applying $F$ preserves that exactness, so the resulting chain complex has zero homology in every positive degree. Using [L1], this says $L_{n}^{P} F (A) = 0$ for $n > 0$ .

1.2L1L2L3algebra

Read the deleted injective resolution of $B$ as a reindexed chain complex. It is exact in every positive cohomological degree, and [L3] preserves that exactness after applying $F$ . By [L2], the resulting homology, hence cohomology, is zero in every positive degree. Therefore $R_{I}^{n} F (B) = 0$ for $n > 0$ .

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove the claimed vanishing on both sides.

Depends on

Used by

Grothendieck collapse when one functor is exact Corollary
Acyclic open cover for a sheaf Definition
Cohomological dimension relative to a sheaf class Definition
Compact support and softness on a finite discrete space Example
The left derived functors of an exact functor Example
FALSE: an acyclic resolution is the same thing as an injective resolution False statement
An exact base functor has the trivial universal delta functor Proposition
A point has no higher sheaf cohomology Theorem
Grothendieck vanishing on a Noetherian space Theorem

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

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