TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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A chain map induces a well-defined map on homology

Statement

Let $f : C_{∙} \to D_{∙}$ be a chain map. For every $n \in Z$ there is a unique morphism $H_{n} (f) : H_{n} (C) \to H_{n} (D)$ such that the quotient maps from cycles to homology commute with $Z_{n} (f)$ .

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ and an integer $n$ .

[L1]

The maps $Z_{n} (f) : Z_{n} (C) \to Z_{n} (D)$ and $B_{n} (f) : B_{n} (C) \to B_{n} (D)$ exist and are compatible with the boundary and cycle inclusions (A chain map carries cycles to cycles and boundaries to boundaries).

[L2]

$H_{n} (C)$ and $H_{n} (D)$ are the cokernels of the canonical maps $B_{n} (C) \to Z_{n} (C)$ and $B_{n} (D) \to Z_{n} (D)$ (Homology object of a chain complex).

[L3]

A cokernel is universal among arrows that kill the map being quotiented (Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers).

Proof

technique · direct

1.1L1L2givenalgebra

Let $β_{C} : B_{n} (C) \to Z_{n} (C)$ and $β_{D} : B_{n} (D) \to Z_{n} (D)$ be the boundary-to-cycle maps. Compatibility in [L1] means $Z_{n} (f) β_{C} = β_{D} B_{n} (f) .$ Therefore the composite $q_{D} Z_{n} (f)$ kills $β_{C}$ , where $q_{D} : Z_{n} (D) \to H_{n} (D)$ is the homology quotient.

2.1L2L3step 1.1constructdischarge-construct∎

Since $q_{D} Z_{n} (f)$ annihilates $β_{C}$ , the cokernel property [L3] for $q_{C} : Z_{n} (C) \to H_{n} (C)$ gives a unique morphism $H_{n} (f) : H_{n} (C) \to H_{n} (D)$ with $H_{n} (f) q_{C} = q_{D} Z_{n} (f) .$ By [L2], this is exactly the induced map on homology.

Depends on

Used by

Gaussian cancellation preserves homotopy type and abelian-category homology Corollary
The long exact homology sequence is natural Corollary
Induced filtration on homology Definition
Morphism of spectral sequences Definition
Quasi-isomorphism Definition
Termwise Hochschild homology and iterated homology Definition
A chain map computed on cycles, boundaries, and homology Example
FALSE: a chain map is determined by its maps on homology False statement
Derived tensor product of abelian sheaves Lemma
Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
A natural transformation induces natural transformations of left derived functors Proposition
A natural transformation induces natural transformations of right derived functors Proposition
Homology respects identities and composition Proposition
Relative homology is invariant under homotopy equivalence of arrows Proposition
A chain map is a quasi-isomorphism exactly when its cone is acyclic Theorem
Chain-homotopic maps induce the same map on homology Theorem
Functoriality and coefficient long exact sequences for Hochschild homology Theorem
Homology is an additive functor Theorem
Pullback induces a well defined map on de rham cohomology Theorem
Termwise Hochschild cyclicity for bounded projective bimodule complexes Theorem
Termwise Hochschild homology respects bimodule chain homotopies Theorem
The cone long exact sequence Theorem
The de Rham map on cohomology is well defined Theorem
The long exact sequence in homology Theorem
The long exact sequence of relative homology for a composable pair Theorem
The PID Kunneth sequence admits a section after choices Theorem

Dependency tree · two levels

10 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)