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The induced homology map is independent of the chosen comparison lift

Statement

Assume the Axiom of Dependent Choice.

Let $P$ be a supplied projective resolution datum and $F : A \to B$ an additive functor. If $u : A \to B$ is a morphism and ${\tilde{u}}_{∙}, {\hat{u}}_{∙} : P_{∙} (A) \to P_{∙} (B)$ are two comparison lifts of $u$ , then for every $n \in Z$ the induced maps on homology $H_{n} ⁣ (F ({\tilde{u}}_{∙})), H_{n} ⁣ (F ({\hat{u}}_{∙})) : L_{n}^{P} F (A) \to L_{n}^{P} F (B)$ are equal.

Facts & Assumptions

Given: A morphism $u : A \to B$ and two comparison lifts ${\tilde{u}}_{∙}, {\hat{u}}_{∙}$ of $u$ .

[L1]

Two comparison maps lifting the same morphism are chain-homotopic (Projective comparison maps are unique up to chain homotopy).

[L2]

A chain homotopy is given by equations of the form $f_{n} - g_{n} = d_{n + 1} s_{n} + s_{n - 1} d_{n}$ (A chain homotopy).

[L3]

Additive functors preserve sums and zero morphisms (Additive functor, An additive functor preserves zero morphisms).

[L4]

Chain-homotopic maps induce the same map on homology (Chain-homotopic maps induce the same map on homology).

[L5]

The objects $L_{n}^{P} F (A)$ and $L_{n}^{P} F (B)$ are the homology objects of the deleted resolutions after applying $F$ (Left derived objects relative to supplied projective resolution data).

Proof

technique · direct

1.1

By [L1], the two lifts ${\tilde{u}}_{∙}$ and ${\hat{u}}_{∙}$ are chain-homotopic. Let $s$ be such a homotopy.

L1givenconstruct

2.1

The equations in [L2] become $F ({\tilde{u}}_{n}) - F ({\hat{u}}_{n}) = F (d_{n + 1}) F (s_{n}) + F (s_{n - 1}) F (d_{n})$ after applying $F$ , because [L3] lets $F$ preserve sums and zero morphisms. Hence $F (s)$ is a chain homotopy from $F ({\tilde{u}}_{∙})$ to $F ({\hat{u}}_{∙})$ .

L2L3step 1.1algebra

3.1

By [L4], chain-homotopic maps induce the same map on homology. Using [L5] to identify those homology objects with the displayed left derived objects gives $H_{n} ⁣ (F ({\tilde{u}}_{∙})) = H_{n} ⁣ (F ({\hat{u}}_{∙}))$ for every $n$ .

L4L5step 2.1∎

Depends on

Used by

The left derived map relative to supplied resolution data Definition
Independence of two comparison lifts on homology Example
FALSE: the definition of a derived map may depend on the chosen comparison lift False statement
Derived functors are well defined relative to supplied resolution data Remark

Dependency tree · two levels

21 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)