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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 ...
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 ...
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
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
Download as PDF; Printable version; In other projects ... (2003), Methods of Homological Algebra, Berlin, New York ... An introduction to homological algebra ...
Download as PDF; Printable version; In other projects ... is a method of mathematical proof used especially in homological algebra, ... , sometimes called the free ...
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 ...
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 ...