Quasi-isomorphism

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In homological algebra, a branch of mathematics, a quasi-isomorphism is a morphism A → B of chain complexes (respectively, cochain complexes) such that the induced morphisms

H_n(A_\bullet) \to H_n(B_\bullet)\ (\text{respectively, } H^n(A^\bullet) \to H^n(B^\bullet))\

of homology groups (respectively, of cohomology groups) are isomorphisms for all n ≥ 0.

[edit] Applications

In the theory of model categories, quasi-isomorphisms are sometimes used as the class of weak equivalences when the objects of the category are chain or cochain complexes. This results in a homology-local theory, in the sense of Bousfield localization in homotopy theory.

Quasi-isomorphisms play the fundamental role in defining the derived category of an abelian category.

[edit] References

  • Gelfand, Manin. Methods of Homological Algebra, 2nd ed. Springer, 2000.