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The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product). This article takes the latter ...
For a left split sequence, the map t × r: B → A × C gives an isomorphism, so B is a direct sum (3.), and thus inverting the isomorphism and composing with the natural injection C → A × C gives an injection C → B splitting r (2.). However, if a short exact sequence of groups is right split (2.), then it need not be left split or a ...
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 ...
Alternatively, one can argue that every short exact sequence of k-vector spaces splits, and any additive functor turns split sequences into split sequences.) If X is a topological space , we can consider the abelian category of all sheaves of abelian groups on X .
Assume that A is an abelian group such that every short exact sequence. must split if B is also abelian. The Whitehead problem then asks: must A be free? This splitting requirement is equivalent to the condition Ext 1 (A, Z) = 0.
A module P is projective if and only if every short exact sequence of modules of the form . is a split exact sequence.That is, for every surjective module homomorphism f : B ↠ P there exists a section map, that is, a module homomorphism h : P → B such that f h = id P .
The question of what groups are extensions of by is called the extension problem, and has been studied heavily since the late nineteenth century.As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups {}, where each {+} is an extension of {} by some simple group.
The preceding interpretation of outer automorphisms as symmetries of a Dynkin diagram follows from the general fact, that for a complex or real simple Lie algebra, 𝔤, the automorphism group Aut(𝔤) is a semidirect product of Inn(𝔤) and Out(𝔤); i.e., the short exact sequence. 1 Inn(𝔤) Aut(𝔤) Out(𝔤) 1. splits.