Mapping cone (homological algebra)

Tool in 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 quasi-isomorphism; 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 t-category, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.

## Definition

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The cone may be defined in the category of cochain 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 {\displaystyle A,B} be two complexes, with differentials  d A , d B ; {\displaystyle d_{A},d_{B};} i.e.,

A = ⋯ -> A n − 1 -> d A n − 1 A n -> d A n A n + 1 -> ⋯ {\displaystyle A=\dots \to A^{n-1}{\xrightarrow {d_{A}^{n-1}}}A^{n}{\xrightarrow {d_{A}^{n}}}A^{n+1}\to \cdots }

and likewise for  B . {\displaystyle B.}

For a map of complexes  f : A -> B , {\displaystyle f:A\to B,} we define the cone, often denoted by  Cone ⁡ ( f ) {\displaystyle \operatorname {Cone} (f)} or  C ( f ) , {\displaystyle C(f),} to be the following complex:

C ( f ) = A [ 1 ] ⊕ B = ⋯ -> A n ⊕ B n − 1 -> A n + 1 ⊕ B n -> A n + 2 ⊕ B n + 1 -> ⋯ {\displaystyle C(f)=A[1]\oplus B=\dots \to A^{n}\oplus B^{n-1}\to A^{n+1}\oplus B^{n}\to A^{n+2}\oplus B^{n+1}\to \cdots } on terms,

with differential

d C ( f ) = ( d A [ 1 ] 0 f [ 1 ] d B ) {\displaystyle d_{C(f)}={\begin{pmatrix}d_{A[1]}&0\\\f[1]&d_{B}\end{pmatrix}}} (acting as though on column vectors).

Here  A [ 1 ] {\displaystyle A[1]} is the complex with  A [ 1 ] n = A n + 1 {\displaystyle A[1]^{n}=A^{n+1}} and  d A [ 1 ] n = − d A n + 1 {\displaystyle d_{A[1]}^{n}=-d_{A}^{n+1}} . Note that the differential on  C ( f ) {\displaystyle C(f)} is different from the natural differential on  A [ 1 ] ⊕ B {\displaystyle A[1]\oplus B} , and that some authors use a different sign convention.

Thus, if for example our complexes are of abelian groups, the differential would act as

d C ( f ) n ( a n + 1 , b n ) = ( d A [ 1 ] n 0 f [ 1 ] n d B n ) ( a n + 1 b n ) = ( − d A n + 1 0 f n + 1 d B n ) ( a n + 1 b n ) = ( − d A n + 1 ( a n + 1 ) f n + 1 ( a n + 1 ) + d B n ( b n ) ) = ( − d A n + 1 ( a n + 1 ) , f n + 1 ( a n + 1 ) + d B n ( b n ) ) . {\displaystyle {\begin{array}{ccl}d_{C(f)}^{n}(a^{n+1},b^{n})&=&{\begin{pmatrix}d_{A[1]}^{n}&0\\\f[1]^{n}&d_{B}^{n}\end{pmatrix}}{\begin{pmatrix}a^{n+1}\\\b^{n}\end{pmatrix}}\\\&=&{\begin{pmatrix}-d_{A}^{n+1}&0\\\f^{n+1}&d_{B}^{n}\end{pmatrix}}{\begin{pmatrix}a^{n+1}\\\b^{n}\end{pmatrix}}\\\&=&{\begin{pmatrix}-d_{A}^{n+1}(a^{n+1})\\\f^{n+1}(a^{n+1})+d_{B}^{n}(b^{n})\end{pmatrix}}\\\&=&\left(-d_{A}^{n+1}(a^{n+1}),f^{n+1}(a^{n+1})+d_{B}^{n}(b^{n})\right).\end{array}}}

## Properties

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Suppose now that we are working over an abelian category, 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 triangle

A -> f B -> C ( f ) -> A [ 1 ] {\displaystyle A{\xrightarrow {f}}B\to C(f)\to A[1]}

where the maps  B -> C ( f ) , C ( f ) -> A [ 1 ] {\displaystyle B\to C(f),C(f)\to A[1]} are given by the direct summands (see Homotopy category of chain complexes). Since this is a triangle, it gives rise to a long exact sequence on homology groups:

⋯ -> H i − 1 ( C ( f ) ) -> H i ( A ) -> f ∗ H i ( B ) -> H i ( C ( f ) ) -> ⋯ {\displaystyle \dots \to H_{i-1}(C(f))\to H_{i}(A){\xrightarrow {f^{*}}}H_{i}(B)\to H_{i}(C(f))\to \cdots }

and if  C ( f ) {\displaystyle C(f)} is acyclic then by definition, the outer terms above are zero. Since the sequence is exact, this means that  f ∗ {\displaystyle f^{*}} induces an isomorphism on all homology 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  A , B {\displaystyle A,B} have only one nonzero term in degree 0:

A = ⋯ -> 0 -> A 0 -> 0 -> ⋯ , {\displaystyle A=\dots \to 0\to A_{0}\to 0\to \cdots ,}
B = ⋯ -> 0 -> B 0 -> 0 -> ⋯ , {\displaystyle B=\dots \to 0\to B_{0}\to 0\to \cdots ,}

and therefore  f : A -> B {\displaystyle f\colon A\to B} is just  f 0 : A 0 -> B 0 {\displaystyle f_{0}\colon A_{0}\to B_{0}} (as a map of objects of the underlying abelian category). Then the cone is just

C ( f ) = ⋯ -> 0 -> A 0 [ − 1 ] -> f 0 B 0 [ 0 ] -> 0 -> ⋯ . {\displaystyle C(f)=\dots \to 0\to {\underset {[-1]}{A_{0}}}{\xrightarrow {f_{0}}}{\underset {[0]}{B_{0}}}\to 0\to \cdots .}

(Underset text indicates the degree of each term.) The homology of this complex is then

H − 1 ( C ( f ) ) = ker ⁡ ( f 0 ) , {\displaystyle H_{-1}(C(f))=\operatorname {ker} (f_{0}),}
H 0 ( C ( f ) ) = coker ⁡ ( f 0 ) , {\displaystyle H_{0}(C(f))=\operatorname {coker} (f_{0}),}
H i ( C ( f ) ) = 0 for i ≠ − 1 , 0. {\displaystyle H_{i}(C(f))=0{\text{ for }}i\neq -1,0.\ }

This is not an accident and in fact occurs in every t-category.

## Mapping cylinder

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A related notion is the mapping cylinder: let  f : A -> B {\displaystyle f\colon A\to B} be a morphism of chain complexes, let further  g : Cone ⁡ ( f ) [ − 1 ] -> A {\displaystyle g\colon \operatorname {Cone} (f)[-1]\to A} be the natural map. The mapping cylinder of f is by definition the mapping cone of g.

## Topological inspiration

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This complex is called the cone in analogy to the mapping cone (topology) of a continuous map of topological spaces ϕ : X -> Y {\displaystyle \phi :X\rightarrow Y} : the complex of singular chains of the topological cone  c o n e ( ϕ ) {\displaystyle cone(\phi )} is homotopy equivalent to the cone (in the chain-complex-sense) 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

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* Manin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43583-9
* Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38\. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
* Joeseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See chapter 9)