Difference between revisions of "Homology sequence"
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| − | + | An [[Exact sequence|exact sequence]], infinite on both sides, of homology groups of three complexes, connected by a short exact sequence. Let $ 0 \rightarrow K _ {\mathbf . } \rightarrow L _ {\mathbf . } \rightarrow M _ {\mathbf . } \rightarrow 0 $ | |
| + | be an exact sequence of chain complexes in an Abelian category. Then there are morphisms | ||
| − | + | $$ | |
| + | \partial _ {n} : H _ {n} ( M _ {\mathbf . } ) \rightarrow H _ {n - 1 } ( K _ {\mathbf . } ) | ||
| + | $$ | ||
| + | |||
| + | defined for all $ n $. | ||
| + | They are called connecting (or boundary) morphisms. Their definition in the category of modules is especially simple: For $ h \in H _ {n} ( M _ {\mathbf . } ) $ | ||
| + | a pre-image $ x \in L _ {n} $ | ||
| + | is chosen; $ d x $ | ||
| + | will then be the image of an element $ z \in Z _ {n-} 1 ( K _ {\mathbf . } ) $ | ||
| + | whose homology class is $ \partial _ {n} ( h) $. | ||
| + | The sequence of homology groups | ||
| + | |||
| + | $$ | ||
| + | \dots \rightarrow ^ { {\partial _ { n} + 1 } } \ | ||
| + | H _ {n} ( K _ {\mathbf . } ) \rightarrow H _ {n} ( L _ {\mathbf . } ) \rightarrow \ | ||
| + | H _ {n} ( M _ {\mathbf . } ) \rightarrow ^ { {\partial _ n } } \ | ||
| + | H _ {n - 1 } ( K _ {\mathbf . } ) \rightarrow \dots , | ||
| + | $$ | ||
constructed with the aid of the connecting morphisms, is exact; it is called the homology sequence. Thus, the homology groups form a [[Homology functor|homology functor]] on the category of complexes. | constructed with the aid of the connecting morphisms, is exact; it is called the homology sequence. Thus, the homology groups form a [[Homology functor|homology functor]] on the category of complexes. | ||
Latest revision as of 22:11, 5 June 2020
An exact sequence, infinite on both sides, of homology groups of three complexes, connected by a short exact sequence. Let $ 0 \rightarrow K _ {\mathbf . } \rightarrow L _ {\mathbf . } \rightarrow M _ {\mathbf . } \rightarrow 0 $
be an exact sequence of chain complexes in an Abelian category. Then there are morphisms
$$ \partial _ {n} : H _ {n} ( M _ {\mathbf . } ) \rightarrow H _ {n - 1 } ( K _ {\mathbf . } ) $$
defined for all $ n $. They are called connecting (or boundary) morphisms. Their definition in the category of modules is especially simple: For $ h \in H _ {n} ( M _ {\mathbf . } ) $ a pre-image $ x \in L _ {n} $ is chosen; $ d x $ will then be the image of an element $ z \in Z _ {n-} 1 ( K _ {\mathbf . } ) $ whose homology class is $ \partial _ {n} ( h) $. The sequence of homology groups
$$ \dots \rightarrow ^ { {\partial _ { n} + 1 } } \ H _ {n} ( K _ {\mathbf . } ) \rightarrow H _ {n} ( L _ {\mathbf . } ) \rightarrow \ H _ {n} ( M _ {\mathbf . } ) \rightarrow ^ { {\partial _ n } } \ H _ {n - 1 } ( K _ {\mathbf . } ) \rightarrow \dots , $$
constructed with the aid of the connecting morphisms, is exact; it is called the homology sequence. Thus, the homology groups form a homology functor on the category of complexes.
Cohomology sequences are defined in a dual manner.
References
| [1] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) |
Homology sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_sequence&oldid=15902