Properties
Suppose now that we are working over an abelian category, so that the cohomology 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 triangle
where the maps are the projections onto the direct summands (see Homotopy category of chain complexes). Since this is a triangle, it gives rise to a long exact sequence on cohomology 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 cohomology groups, and hence (again by definition) is a quasi-isomorphism.
This fact recalls the usual alternative characterization of isomorphisms in an abelian category as those maps whose kernel and cokernel both vanish. This appearance of a cone as a combined kernel and cokernel is not accidental; in fact, under certain circumstances the cone literally embodies both. Say for example that we are working over an abelian category and have only one nonzero term in degree 0:
and therefore is just (as a map of objects of the underlying abelian category). Then the cone is just
(Underset text indicates the degree of each term.) The cohomology of this complex is then
This is not an accident and in fact occurs in every t-category.
Read more about this topic: Mapping Cone (homological Algebra)
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)