 Mapping cone (homological algebra)

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 kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasiisomorphism; if we pass to the derived category 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 tcategory, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.
Contents
Definition
The cone may be defined in the category of chain complexes over any additive category (i.e., a category whose morphisms form abelian groups and in which we may construct a direct sum of any two objects). Let A,B be two complexes, with differentials d_{A},d_{B}; i.e.,
and likewise for B.
For a map of complexes we define the cone, often denoted by or C(f), to be the following complex:
 on terms,
with differential
 (acting as though on column vectors).
Here A[1] is the complex with A[1]^{n} = A^{n + 1} and . Note that the differential on C(f) 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
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 quasiisomorphisms: if the cone is acyclic, then the map is a quasiisomorphism. 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 C(f) is acyclic then by definition, the outer terms above are zero. Since the sequence is exact, this means that f ^{*} induces an isomorphism on all cohomology groups, and hence (again by definition) is a quasiisomorphism.
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 A,B 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 tcategory.
Mapping cylinder
A related notion is the mapping cylinder: let f: A → B be a morphism of complexes, let further g : Cone(f)[1] → A be the natural map. The mapping cylinder of f is by definition the mapping cone of g.
Topological inspiration
This complex is called the cone in analogy to the mapping cone of a continuous map of topological spaces : the complex of singular chains of the topological cone cone(ϕ) is homotopy equivalent to the cone (in the chaincomplexsense) of the induced map of singular chains of X to Y. The mapping cylinder of a map of complexes is similarly related to the mapping cylinder of continuous maps.
References
 Manin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: SpringerVerlag, ISBN 9783540435839
 Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, ISBN 9780521559874, OCLC 36131259, MR1269324
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
Mapping cone — In mathematics, especially homotopy theory, the mapping cone is a construction Cf of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated Cf. Contents 1 Definition 1.1 Example of circle … Wikipedia
Cone — This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article … Wikipedia
List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… … Wikipedia
Triangulated category — A triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t category is a triangulated category with a t… … Wikipedia
Homotopy category of chain complexes — In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain… … Wikipedia
Derived category — In mathematics, the derived category D(C) of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C. The construction proceeds on the… … Wikipedia
Vector space — This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is… … Wikipedia
List of algebraic topology topics — This is a list of algebraic topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Contents 1… … Wikipedia
Homogeneous coordinate ring — In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring R = K[X0, X1, X2, ..., XN]/I where I is the homogeneous ideal… … Wikipedia
Mayer–Vietoris sequence — In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to… … Wikipedia