Jump to: navigation , search The branch of algebra whose main study is derived functors on various categories of algebraic objects modules over a given ring, sheaves, etc. 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.

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Jump to: navigation , search The branch of algebra whose main study is derived functors on various categories of algebraic objects modules over a given ring, sheaves, etc. 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. 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. 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. 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.

The same result will be obtained in all cases. The establishment of the new relations considerably extended and advanced the theory of extensions of modules. The generalization of this observation resulted in the development of the general theory of torsion. The homology theory of algebraic systems forms part of the general scheme of derived functors. Appropriate cohomology and homology groups of monoids, abelian groups, algebras, graded algebras, rings, etc.

The guideline in each case is the fact that the second cohomology group is the group of extensions for the type of algebraic systems under consideration.

In turn, the homology groups of algebraic systems form the subject of study of relative homological algebra. In concrete cases, derived functors of functors are usually computed by means of an explicit resolution. The resolution may be finite e. There has long been interest in the length of the shortest resolution this length is called the homological dimension.

The first significant result in this direction is the Hilbert syzygy theorem appearing at the end of the 19th century. Homological-dimension theory is one of the actively-developing branches of homological algebra. Thus, any group is the inductive limit of its finitely-generated subgroups. Every compact totally-disconnected group is representable as the projective limit of its finite quotient groups. Interest in these groups stems from their connection with Galois theory.

The derived functors of these functors are used in homological dimension theory. Derived functors for non-additive functors have been studied e. The principal means of computations in homological algebra, other than the resolutions already mentioned, are spectral sequences and the homology product.

The former, which are a most powerful tool in the study of derived functors, approximate the homology groups of a group by the homology groups of a subgroup and a quotient group of it.

Methods of homological algebra are now extensively employed in very different branches of mathematics, like functional analysis, the theory of functions of a complex variable, differential equations, etc. References H. Cartan, S. Press MR Zbl Zbl MR [4] A. MR [5] S. Eilenberg, N. Steenrood ed. References P. Hilton, U. Encyclopedia of Mathematics.

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## Homological algebra

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## Cartan–Eilenberg resolution

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