Lemma of the snake
The lemma of the snake , in Mathematical, and in particular in Homology, a valid statement in all abelian Category, is a tool of most important for the construction of exact continuations, omnipresent objects in homology and its applications, for example in algebraic Topologie. The Homomorphisme S thus built are generally called connecting homomorphisms .
Statement
In an abelian category (for example the category of a abelian Group or that of a vector Space on a body), let us consider the commutative Diagramme according to:
where the line are exact continuations and 0 are the Zero of the structure concerned. Then there exists an exact continuation binding the cores and the Conoyau X of has , B , and C :
Moreover, if the morphism F is a Monomorphisme, then the morphism ker has → ker B is also, and if g' is a epimorphism, then coker B → coker C is too.
Sources
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