One of the origins of homological algebra is the singular homology theory of topological spaces. This makes it possible, in a number of cases, to reduce the study of topological objects to the study of certain algebraic objects, as is done in analytic geometry with the difference that the transition from geometry to algebra in homology theory is irreversible. In algebra, in turn, in the theory of groups cf. Extensive preparatory material was developed in the theory of associative algebras, the theory of Lie algebras, the theory of finite-dimensional algebras, the theory of rings and the theory of quadratic forms.

The language of homological algebra arose mainly from the process of studying homology groups.

There appeared arrows as symbols for mappings and commutative diagrams if, in a diagram, any two paths with a common beginning and end give rise to the same composite mapping, then the diagram is said to be commutative. Sequences of homomorphisms in which the kernel of each outgoing homomorphism coincides with the image of the incoming one were encountered; such sequences are called exact.

It became customary to specify mathematical objects together with their mappings; the correspondences most preferred were those between objects that preserve the mappings.

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These correspondences became known as functors. The principal advantages of this language — the amount of information conveyed, naturalness and clarity — were soon recognized. For example, the language of homological algebra was employed [5] in the axiomatic exposure of the fundamentals of algebraic topology. Nowadays, this language is used in numerous studies, including those not employing homological methods.

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The principal domain of application of homological algebra is the category of modules over a ring. Most of the results known for modules may be applied to abelian categories with certain restrictions this is because such categories are embeddable into categories of modules. In the most fruitful extension of the domain of application of homological algebra [4] , the latter was extended so as to apply to arbitrary abelian categories with enough injective objects, and became applicable to arithmetical algebraic geometry and to the theory of functions in several complex variables cf.

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Grothendieck category. The base of the theory is the study of derived functors , which may be constructed as follows. In a certain sense, the derived functors are a measure of the deviation of the functor from exactness. They are not affected by the arbitrariness involved in the construction of a resolution. In such a case, one can fix one argument and construct a resolution for the other, or, having constructed resolutions of both arguments, one can construct a binary complex.

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Manufacturers, suppliers and others provide what you see here, and we have not verified it. See our disclaimer. Homological algebra first arose as a language for describing topological prospects of geometrical objects.

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As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory sheaf cohomology, spectral sequences, etc. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented.

The book addresses people who want to learn a modern approach to homological algebra and to use it in their work.

## Homological Algebra

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