# homological

## Articles from Wikipedia

• 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 and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.
• List of homological algebra topics This is a list of homological algebra topics, by Wikipedia page.
• Mapping cone (homological algebra) 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-isomorphism; if we pass to the derived category of complexes, this means that f is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism. If we are working in a t-category, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.
• Homological connectivity In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups. This property is related, but more general, than the properties of graph connectivity and topological connectivity. There are many definitions of homological connectivity of a topological space X.
• Homological dimension Homological dimension may refer to the global dimension of a ring. It may also refer to any other concept of dimension that is defined in terms of homological algebra, which includes:Projective dimension of a module, based on projective resolutions Injective dimension of a module, based on injective resolutions Weak dimension of a module, or flat dimension, based on flat resolutions Weak global dimension of a ring, based on the weak dimension of its modules Cohomological dimension of a group

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