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If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig ...
Every equivalence or duality of abelian categories is exact.. The most basic examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then F A (X) = Hom A (A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. [1]
An exact sequence (or exact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A short exact sequence is a bounded exact sequence in which only the groups A k, A k+1, A k+2 may be nonzero. For example, the following chain complex is a short exact sequence.
And although the visual part of the illustration does a fine job of illustrating the general concept of an exact sequence, the textual part is more or less illegible at normal image viewing sizes. — David Eppstein ( talk ) 20:27, 2 February 2021 (UTC) [ reply ]
In mathematics, specifically in category theory, an exact category is a category equipped with short exact sequences.The concept is due to Daniel Quillen and is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence.
For example it is common to take A to be Z/2Z, so that coefficients are modulo 2. This becomes straightforward in the absence of 2- torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers b i of X and the Betti numbers b i , F with coefficients in a field F .
The above short exact sequences specifying the relative chain groups give rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence
The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology .