Quasi-isomorphism

A quasi-isomorphism is a chain map that induces isomorphisms on all cohomology groups. Unlike chain homotopy equivalences, quasi-isomorphisms need not have inverses as chain maps. The derived category is constructed precisely by formally inverting quasi-isomorphisms.

Definition

Definition4.9Quasi-isomorphism

A chain map $f : A^\bullet \to B^\bullet$ is a quasi-isomorphism (abbreviated qis) if $H^n(f) : H^n(A^\bullet) \to H^n(B^\bullet)$ is an isomorphism for all $n \in \mathbb{Z}$ .

RemarkQuasi-isomorphism vs homotopy equivalence

Every chain homotopy equivalence is a quasi-isomorphism, but the converse fails. In the derived category $D(\mathcal{A})$ , quasi-isomorphisms become isomorphisms, but in the homotopy category $K(\mathcal{A})$ , only homotopy equivalences are isomorphisms. The passage from $K(\mathcal{A})$ to $D(\mathcal{A})$ is precisely the localization at quasi-isomorphisms.

Examples

ExampleProjective resolution as a qis

A projective resolution $P_\bullet \to M$ gives a quasi-isomorphism $P_\bullet \to M[0]$ : the cohomology of $P_\bullet$ is $M$ in degree $0$ and $0$ elsewhere.

ExampleInjective resolution as a qis

An injective resolution $M \to I^\bullet$ gives a quasi-isomorphism $M[0] \to I^\bullet$ : the cohomology of $I^\bullet$ is $M$ in degree $0$ .

ExampleQis that is not a homotopy equivalence

The augmentation $\varepsilon : \cdots \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{0} \mathbb{Z} \to 0$ (a free resolution of $\mathbb{Z}/2\mathbb{Z}$ ) gives a quasi-isomorphism to $\mathbb{Z}/2\mathbb{Z}[0]$ , but this is not a homotopy equivalence since $\mathbb{Z}/2\mathbb{Z}$ has no projective resolution of length zero.

ExampleAcyclic complex = qis to zero

$A^\bullet$ is acyclic iff the zero map $A^\bullet \to 0$ is a quasi-isomorphism, equivalently iff $0 \to A^\bullet$ is a quasi-isomorphism.

ExampleMapping cone and qis

A chain map $f : A^\bullet \to B^\bullet$ is a quasi-isomorphism iff $\mathrm{Cone}(f)$ is acyclic. This follows from the long exact sequence $\cdots \to H^n(A) \xrightarrow{H^n(f)} H^n(B) \to H^n(\mathrm{Cone}(f)) \to \cdots$ .

ExampleQuasi-isomorphism of de Rham and singular complexes

The de Rham theorem states that integration of differential forms over singular chains gives a quasi-isomorphism $\Omega^\bullet(M) \to C^\bullet_{\mathrm{sing}}(M; \mathbb{R})^*$ , establishing $H^n_{\mathrm{dR}}(M) \cong H^n_{\mathrm{sing}}(M; \mathbb{R})$ .

ExampleCech resolution

For a good cover $\mathcal{U}$ of $X$ and a sheaf $\mathcal{F}$ , the natural map from the Cech complex $\check{C}^\bullet(\mathcal{U}, \mathcal{F})$ to the Godement resolution computes the same cohomology, connected by quasi-isomorphisms.

ExampleFlat resolution

For a module $M$ , a flat resolution $F_\bullet \to M$ gives a quasi-isomorphism $F_\bullet \to M[0]$ . While not projective, flat resolutions suffice for computing Tor.

ExampleTruncation functors

The stupid truncation $\sigma_{\geq n} A^\bullet$ (set $A^k = 0$ for $k < n$ ) is not a quasi-isomorphism in general. The canonical truncation $\tau_{\leq n} A^\bullet$ (replace $A^n$ by $\ker d^n$ and set higher terms to zero) satisfies $H^k(\tau_{\leq n} A) = H^k(A)$ for $k \leq n$ and $= 0$ for $k > n$ .

ExampleQuasi-isomorphism between resolutions

If $P_\bullet \to M$ and $Q_\bullet \to M$ are projective resolutions, the comparison theorem gives a chain map $f : P_\bullet \to Q_\bullet$ that is a quasi-isomorphism (and in fact a homotopy equivalence). However, this $f$ need not be a degreewise isomorphism.

ExampleDerived equivalence

An exact equivalence $D^b(\mathcal{A}) \simeq D^b(\mathcal{B})$ of bounded derived categories sends quasi-isomorphisms to quasi-isomorphisms (since they become isomorphisms in both derived categories).

ExampleQuasi-isomorphism and spectral sequences

A morphism of filtered complexes that is a quasi-isomorphism on each graded piece induces an isomorphism on the associated spectral sequences (from the $E_1$ page onward).

Properties

Theorem4.4Properties of quasi-isomorphisms

The composition of quasi-isomorphisms is a quasi-isomorphism.
A chain homotopy equivalence is a quasi-isomorphism.
A quasi-isomorphism between bounded below complexes of projectives is a homotopy equivalence.
A quasi-isomorphism between bounded above complexes of injectives is a homotopy equivalence.

RemarkWhitehead's theorem for complexes

Property (3) is the algebraic analogue of Whitehead's theorem in topology: a weak homotopy equivalence between CW-complexes (= cofibrant objects) is a homotopy equivalence. In the model category structure on $\mathbf{Ch}(\mathcal{A})$ , projective complexes are cofibrant and injective complexes are fibrant.

RemarkLooking ahead

The derived category $D(\mathcal{A}) = K(\mathcal{A})[\mathrm{qis}^{-1}]$ is obtained by formally inverting all quasi-isomorphisms in the homotopy category. This is the Verdier localization.