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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 ...
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.
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.
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]
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
In a regular category, a diagram of the form is said to be an exact sequence if it is both a coequalizer and a kernel pair. The terminology is a generalization of exact sequences in homological algebra : in an abelian category , a diagram
In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma.It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well.
An infinite sequence of real numbers (in blue). This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded. In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms).