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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, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold, and write O M for the sheaf of holomorphic functions on M. Let O M * be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups.
Noncommutative algebraic geometry. ... and form direct limits, you obtain a short exact sequence ... (consider for example the category of finite sets, ...
The above short exact sequences specifying the relative chain groups give rise to a chain complex of short exact sequences. ... Another insightful geometric example ...
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.
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]
If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to split. In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that: C ≅ B/ker r ≅ B/q(A) (i.e., C isomorphic to the coimage of r or cokernel of q) to: