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Universal delta functors extending the same degree-zero functor are uniquely isomorphic

Statement

Let $F$ be a fixed degree-zero functor.

If $S$ and $T$ are universal homological delta functors equipped with chosen natural isomorphisms $σ : S_{0} \tilde{\to} F, τ : T_{0} \tilde{\to} F,$ then there is a unique isomorphism of homological delta functors $S \tilde{\to} T$ whose degree-zero part is $τ^{- 1} σ : S_{0} \tilde{\to} T_{0}$ .

If $S$ and $T$ are universal cohomological delta functors equipped with chosen natural isomorphisms $σ : S^{0} \tilde{\to} F, τ : T^{0} \tilde{\to} F,$ then there is a unique isomorphism of cohomological delta functors $S \tilde{\to} T$ whose degree-zero part is $τ^{- 1} σ : S^{0} \tilde{\to} T^{0}$ .

Facts & Assumptions

Given: Two universal delta functors together with chosen degree-zero identifications to $F$ .

[L1]

Universality for homological and cohomological delta functors is the unique extension property from degree zero (Universal delta functor).

[L2]

Morphisms of delta functors are the degreewise maps compatible with the connecting morphisms (Morphism of homological delta functors, Morphism of cohomological delta functors).

Proof

technique · direct

1.1

In the homological case, set $t_{0} = τ^{- 1} σ : S_{0} \to T_{0}$ . Applying [L1] to $t_{0}$ gives a morphism $ϕ : S \to T$ , and applying [L1] to $t_{0}^{- 1}$ gives a morphism $ψ : T \to S$ . Their composites extend ${id}_{S_{0}}$ and ${id}_{T_{0}}$ respectively, so uniqueness in [L1] forces $ψ ϕ = {id}_{S}$ and $ϕ ψ = {id}_{T}$ . Hence $ϕ$ is the unique isomorphism extending $τ^{- 1} σ$ .

L1L2givenalgebra

2.1

The cohomological case is identical, with the maps oriented out of the universal functors as required by [L1].

L1L2step 1.1∎

Depends on

Used by

Two universal delta functors and their unique isomorphism Example
Satellites give the first derived functor Proposition

Dependency tree · two levels

6 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)