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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]
In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal -bundle over a manifold, where is a Lie group, is the Lie algebroid of the gauge groupoid of . Explicitly, it is given by the following short exact sequence of vector bundles over M {\displaystyle M} :
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
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 most basic example of a semisimple module is a module over a field, i.e., a vector space. On the other hand, the ring Z of integers is not a semisimple module over itself, since the submodule 2Z is not a direct summand. Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules.