Chain Homotopy and the Comparison Theorem

Executive summary

Homotopic chain maps induce the same map on homology

Two chain maps are homotopic when their difference is a boundary in the algebraic sense: f − g = ∂s + s∂ for some degree-raising family s. Homotopic maps induce the same map on homology, which is what makes derived functors well defined: any two projective resolutions of a module are homotopy equivalent, so the homology of the resolved complex does not depend on which resolution was chosen.

Learning objectives

Define chain homotopy and homotopy equivalence.
Prove that homotopic maps agree on homology.
State and apply the comparison theorem.
Distinguish homotopy equivalence from quasi-isomorphism.

Section 01Homotopy

f − g = ∂s + s∂, s_n: C_n → D_n+1

The family s need not commute with anything; it is simply a degree-raising map of graded modules. On a cycle z, (f − g)(z) = ∂s(z) is a boundary, so f and g agree on homology classes.

This is the only reason derived functors exist

Additive functors preserve the identity ∂s + s∂, so they carry homotopies to homotopies. Hence the homology of the image complex is independent of the chosen resolution. Additivity of the functor is exactly what this argument needs, which is why it is a standing hypothesis.

Section 02The comparison theorem

AlgorithmComparison theorem for projective resolutionsin: two resolutions and a map → out: a lift, unique up to homotopy

Let P_• ↠ M be a projective resolution and Q_• ↠ N any resolution, with f: M → N given.
Construct a chain map lifting f by induction: at each stage the projectivity of P_n lifts the required map through the surjection onto the cycles of Q_•. Projectivity supplies the lift; exactness of Q supplies the surjection.
Any two such lifts are chain homotopic, by the same induction applied to their difference.
Taking M = N and f = 1 gives: any two projective resolutions of M are chain homotopy equivalent.
Therefore the derived functors computed from either agree, canonically.

The horseshoe lemma

Given a short exact sequence of modules and projective resolutions of the outer two, the horseshoe lemma builds a compatible resolution of the middle term, producing a short exact sequence of complexes. That is the step converting a short exact sequence of modules into the long exact sequence of derived functors.

Section 03Homotopy equivalence versus quasi-isomorphism

Homotopy equivalenceStronger

Chain maps both ways whose composites are homotopic to identities. Preserved by every additive functor.

Quasi-isomorphismWeaker

A chain map inducing isomorphisms on all homology. NOT generally preserved by additive functors.

The distinction is not academic

A quasi-isomorphism can be destroyed by applying a functor, whereas a homotopy equivalence cannot. This is precisely why derived functors are computed with projective or injective resolutions rather than arbitrary quasi-isomorphic complexes, and why the derived category has to be constructed by formally inverting quasi-isomorphisms rather than by taking homotopy classes.

Between bounded-below complexes of projectives the two notions coincide, which is why the comparison theorem is available in exactly the setting where resolutions live.

ReferenceFrequently asked questions

Is chain homotopy the algebraic shadow of topological homotopy?

Yes, and historically that is where it came from. Homotopic maps of spaces induce chain homotopic maps on singular chains, so they agree on homology. The algebraic notion was abstracted from exactly this.

Does a contracting homotopy prove exactness?

Yes — if the identity of a complex is homotopic to zero, all homology vanishes. The converse fails in general, but holds for bounded-below complexes of projectives, which is again the resolution setting.

Why must the functor be additive?

Because the homotopy identity involves a sum of two composites. A non-additive functor need not preserve that sum, so homotopic maps could have non-homotopic images and the whole independence argument would collapse.

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