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  2. Homological algebra - Wikipedia

    en.wikipedia.org/wiki/Homological_algebra

    Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ) and abstract algebra (theory of modules and syzygies ) at the end of the 19th century, chiefly by ...

  3. Coherent duality - Wikipedia

    en.wikipedia.org/wiki/Coherent_duality

    Assuming is a localization of a finite type -algebra, existence of a rigid dualizing complex over relative to was first proved by Yekutieli and Zhang [5] assuming is a regular noetherian ring of finite Krull dimension, and by Avramov, Iyengar and Lipman [6] assuming is a Gorenstein ring of finite Krull dimension and is of finite flat dimension ...

  4. Homotopical algebra - Wikipedia

    en.wikipedia.org/wiki/Homotopical_algebra

    In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects as special cases. . The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categor

  5. Quasi-isomorphism - Wikipedia

    en.wikipedia.org/wiki/Quasi-isomorphism

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

  6. Homotopy category of chain complexes - Wikipedia

    en.wikipedia.org/wiki/Homotopy_category_of_chain...

    Download as PDF; Printable version; In other projects ... (2003), Methods of Homological Algebra, Berlin, New York ... An introduction to homological algebra ...

  7. Commutative diagram - Wikipedia

    en.wikipedia.org/wiki/Commutative_diagram

    Download as PDF; Printable version; In other projects ... is a method of mathematical proof used especially in homological algebra, ... , sometimes called the free ...

  8. Mapping cone (homological algebra) - Wikipedia

    en.wikipedia.org/wiki/Mapping_cone_(homological...

    In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology.In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi ...

  9. Homology (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Homology_(mathematics)

    In other cases, such as for group homology, there are multiple common methods to compute the same homology groups. In the language of category theory , a homology theory is a type of functor from the category of the mathematical object being studied to the category of abelian groups and group homomorphisms, or more generally to the category ...