Van Hove singularity: Difference between revisions

Revision as of 20:12, 2 May 2013

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_{∙}) \to H_{n} (B_{∙}) (respectively, H^{n} (A^{∙}) \to H^{n} (B^{∙}))

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

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.

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+In [[homological algebra]], a branch of [[mathematics]], a '''quasi-isomorphism''' is a morphism ''A'' → ''B'' of [[chain complex]]es (respectively, cochain complexes) such that the induced morphisms
+:<math>H_n(A_\bullet) \to H_n(B_\bullet)\ (\text{respectively, } H^n(A^\bullet) \to H^n(B^\bullet))\ </math>
+of [[homology (mathematics)|homology]] groups (respectively, of cohomology groups) are isomorphisms for all ''n''.
+In the theory of [[model category|model categories]], quasi-isomorphisms are sometimes used as the class of [[weak equivalence (homotopy theory)|weak equivalence]]s 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]].
+==References==
+*Gelfand, Manin. ''Methods of Homological Algebra'', 2nd ed. Springer, 2000.
+[[Category:Algebraic topology]]
+[[Category:Homological algebra]]