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The change-of-projective-resolution isomorphisms are natural

Statement

Assume the Axiom of Dependent Choice.

With the notation of Objectwise comparison of two projective resolution data induces an isomorphism on derived objects, the isomorphisms $θ_{P, Q} (A)$ are natural in $A$ .

Facts & Assumptions

Given: A morphism $u : A \to B$ and an integer $n$ .

[L1]

The supplied projective data admit comparison lifts of $u$ on both sides (A morphism has a comparison lift between the supplied projective resolutions).

[L2]

The objectwise comparison maps induce isomorphisms on derived objects (Objectwise comparison of two projective resolution data induces an isomorphism on derived objects).

[L3]

Two projective comparison maps lifting the same morphism are chain-homotopic, and chain-homotopic maps induce the same homology map (Projective comparison maps are unique up to chain homotopy, Chain-homotopic maps induce the same map on homology).

Proof

technique · direct

1.1L1L2givenconstruct

Choose objectwise comparison maps $c_{A} : P_{∙} (A) \to Q_{∙} (A)$ and $c_{B} : P_{∙} (B) \to Q_{∙} (B)$ that define the isomorphisms in [L2], and choose comparison lifts ${\tilde{u}}^{P} : P_{∙} (A) \to P_{∙} (B)$ and ${\tilde{u}}^{Q} : Q_{∙} (A) \to Q_{∙} (B)$ from [L1].

2.1L2L3step 1.1∎

Both composites $c_{B} {\tilde{u}}^{P}$ and ${\tilde{u}}^{Q} c_{A}$ are comparison maps from $P_{∙} (A)$ to $Q_{∙} (B)$ lifting the same morphism $u$ , so [L3] makes them chain-homotopic. Passing to homology gives $θ_{P, Q} (B) \circ L_{n}^{P} F (u) = L_{n}^{Q} F (u) \circ θ_{P, Q} (A) .$ Therefore the family $θ_{P, Q} (A)$ is natural.

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors`