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Concept
Mapping cone (homological algebra)
Résumé
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined and cokernel: if the chain complexes take their terms in an , so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi-isomorphism; if we pass to the of complexes, this means that f is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism. If we are working in a , then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.
The cone may be defined in the category of cochain complexes over any (i.e., a category whose morphisms form abelian groups and in which we may construct a direct sum of any two objects). Let be two complexes, with differentials i.e.,
and likewise for
For a map of complexes we define the cone, often denoted by or to be the following complex:
on terms,
with differential
(acting as though on column vectors).
Here is the complex with and .
Note that the differential on is different from the natural differential on , and that some authors use a different sign convention.
Thus, if for example our complexes are of abelian groups, the differential would act as
Suppose now that we are working over an , so that the homology of a complex is defined. The main use of the cone is to identify quasi-isomorphisms: if the cone is acyclic, then the map is a quasi-isomorphism. To see this, we use the existence of a
where the maps are given by the direct summands (see ). Since this is a triangle, it gives rise to a long exact sequence on homology groups:
and if is acyclic then by definition, the outer terms above are zero. Since the sequence is exact, this means that induces an isomorphism on all homology groups, and hence (again by definition) is a quasi-isomorphism.
Source officielle
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