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Illustration of an exact sequence of groups using Euler diagrams. In mathematics, an exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
These groups are abelian for but for = form the top group of a crossed module with bottom group (). There is also a long exact sequence of relative homotopy groups that can be obtained via the Puppe sequence :
The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext 1 classifies extensions of one module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934).
In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram: where the rows are exact sequences and 0 is the zero object. Then there is an exact sequence relating the kernels and cokernels of a, b, and c:
Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G. The zeroth cohomology of G with coefficients in A is defined to be the subgroup
The five lemma is often applied to long exact sequences: when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object. This alone is often not sufficient to determine ...
A sheaf E of abelian groups on a topological space X is called acyclic if H j (X,E) = 0 for all j > 0. By the long exact sequence of sheaf cohomology, the cohomology of any sheaf can be computed from any acyclic resolution of E (rather than an injective resolution). Injective sheaves are acyclic, but for computations it is useful to have other ...
In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category.