ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-05

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Two universal delta functors and their unique isomorphism

Example

Assume the Axiom of Dependent Choice.

Let $A$ and $B$ be abelian categories, suppose $A$ has enough projectives, and let $F : A \to B$ be additive and right exact. If $P$ and $Q$ are two supplied projective resolution data on all objects of $A$ , then the two universal homological delta functors $(L_{n}^{P} F)_{n \geq 0} and (L_{n}^{Q} F)_{n \geq 0}$ are uniquely isomorphic. After choosing the standard natural identifications $L_{0}^{P} F ≅ F ≅ L_{0}^{Q} F$ , the isomorphism is the unique one whose degree-zero component corresponds to ${id}_{F}$ .

Facts & Assumptions

Given: Two supplied projective resolution data $P$ and $Q$ on all objects of $A$ for the same right exact functor $F$ .

[L1]

Each of the two derived constructions is a universal homological delta functor (Derived functors are universal delta functors).

[L2]

Two universal delta functors with the same degree-zero term are uniquely isomorphic (Universal delta functors extending the same degree-zero functor are uniquely isomorphic).

[L3]

For every supplied projective resolution datum, the zeroth left derived functor is naturally isomorphic to the original right exact functor (Left derived functors form a homological delta functor).

Verification

technique · direct

1.1L1L3given

By [L1], both $(L_{n}^{P} F)$ and $(L_{n}^{Q} F)$ are universal homological delta functors, and [L3] supplies their natural degree-zero identifications with $F$ .

2.1L2step 1.1∎

Choose the natural degree-zero identifications with $F$ from step 1.1. Applying [L2] yields the unique isomorphism of delta functors whose degree-zero part corresponds under those identifications to ${id}_{F}$ ; its higher components are then forced.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)