# Mapping cone (homological algebra)

In homological algebra, the **mapping cone** of a chain map \( f \colon C_{\bullet} \to D_{\bullet} \) is a new chain complex, written \( \Cone(f) \), built from \( C \) and \( D \) so that its homology measures exactly the failure of \( f \) to be a quasi-isomorphism. It is the algebraic analog of the topological mapping cone and the basic ingredient from which distinguished triangles, long exact sequences, and many free resolutions are assembled.

| Key fact | Detail |
|---|---|
| Definition | \( \Cone(f)_n := D_n \oplus C_{n-1} \) with differential \( d^{\Cone(f)}(y,x) := (d^D(y) + f(x),\, -d^C(x)) \)<sup>[1](https://bastiaancnossen.com/notes/stable-homotopy-higher-algebra/section/homological-algebra/derived-categories/index.html)</sup> |
| Canonical sequence | \( 0 \to D_{\bullet} \to \Cone(f) \to C_{\bullet}[1] \to 0 \), natural in \( f \)<sup>[1](https://bastiaancnossen.com/notes/stable-homotopy-higher-algebra/section/homological-algebra/derived-categories/index.html)</sup> |
| Mapping-cone criterion | \( f \) is a quasi-isomorphism if and only if \( \Cone(f) \) is acyclic<sup>[1](https://bastiaancnossen.com/notes/stable-homotopy-higher-algebra/section/homological-algebra/derived-categories/index.html)</sup> |
| Long exact sequence | The connecting homomorphism of the cone's long exact sequence is canonically the map \( f_* \) induced by \( f \) on homology<sup>[2](https://www.math.ucla.edu/~sharifi/notes/homalg-ch02.html)</sup> |
| Triangulated structure | Cones of maps give the distinguished triangles of the homotopy category \( K(\mathcal{A}) \) and derived category \( D(\mathcal{A}) \)<sup>[3](https://stacks.math.columbia.edu/download/derived.pdf)</sup> |
| Functoriality | In a triangulated category the cone is not functorial; in a stable \( \infty \)-category the cone is the cofiber and is functorial<sup>[4](https://ncatlab.org/nlab/show/triangulated%2Bcategory)</sup> |
| Resolutions | Iterated mapping cones build free resolutions of monomial ideals, minimal when the ideal has linear quotients (Herzog–Takayama)<sup>[5](https://www.sciencedirect.com/science/article/pii/S0097316514001149)</sup> |

## Definition

Let \( f \colon C_{\bullet} \to D_{\bullet} \) be a chain map between chain complexes in an abelian category, or of modules over a ring. The mapping cone \( \Cone(f) \) has, in degree \( n \), the direct sum

\[ \Cone(f)_n := D_n \oplus C_{n-1}, \]

so the complex combines \( D \) in its original grading with \( C \) shifted up by one degree. The differential is

\[ d^{\Cone(f)}(y,x) := (d^D(y) + f(x),\, -d^C(x)). \]

The term \( f(x) \) is what distinguishes the cone from a plain direct sum of complexes: it records how \( f \) glues the shifted source to the target. A direct computation shows that two consecutive differentials compose to zero, so \( \Cone(f) \) is a chain complex; the formal verification that \( d \circ d = 0 \) splits into the three summands of the formula, as carried out for instance in the UniMath formalization of mapping cones<sup>[6](https://unimath.github.io/doc/UniMath/d4de26f/UniMath.HomologicalAlgebra.MappingCone.html)</sup>.

For cochain complexes \( f^{\cdot} \colon A^{\cdot} \to B^{\cdot} \), the same construction reads \( \Cone(f)^i = A^{i+1} \oplus B^i \) with differential given in matrix form by

\[ d^{\Cone(f)}_{i} = \begin{pmatrix} -d_A^{i+1} & 0 \\ f^{i+1} & d_B^{i} \end{pmatrix}, \]

which is the cohomological mirror of the homological formula above<sup>[2](https://www.math.ucla.edu/~sharifi/notes/homalg-ch02.html)</sup>. Sign conventions differ between authors; some sources write the differential as \( \partial(v,w) = (\partial_V(v),\, f(v) + \partial_W(w)) \) with the shifted summand listed first<sup>[7](https://escholarship.org/content/qt5t67h1jg/qt5t67h1jg_noSplash_dbde4b3328a6b5b41e4de05a31186148.pdf?t=me7hzl)</sup>, and others as a matrix \( \begin{pmatrix} d_A & 0 \\ f & d_B \end{pmatrix} \) on \( A[1] \oplus B \)<sup>[8](https://notes.dzackgarza.com/Unsorted/mapping%20cone.html)</sup>. These are equivalent up to the choice of shift and sign convention.

## The canonical short exact sequence and long exact sequence

The cone fits into a natural short exact sequence of complexes

\[ 0 \longrightarrow D_{\bullet} \longrightarrow \Cone(f) \longrightarrow C_{\bullet}[1] \longrightarrow 0, \]

where \( D \) includes as the summand \( (y,0) \) and the projection sends \( (y,x) \) to \( x \)<sup>[1](https://bastiaancnossen.com/notes/stable-homotopy-higher-algebra/section/homological-algebra/derived-categories/index.html)</sup>. The differential on \( \Cone(f) \) preserves the \( D \)-summand and agrees there with the differential of \( D \), so \( D \) is a subcomplex, and the quotient has \( n \)-th term \( C_{n-1} \) with differential the negative of the one on \( C \), which is canonically the shifted complex \( C[1] \)<sup>[2](https://www.math.ucla.edu/~sharifi/notes/homalg-ch02.html)</sup>.

Any short exact sequence of chain complexes induces a long exact sequence in homology, with connecting maps \( \partial_n \colon H_n(E) \to H_{n-1}(C) \)<sup>[1](https://bastiaancnossen.com/notes/stable-homotopy-higher-algebra/section/homological-algebra/derived-categories/index.html)</sup>. For the cone sequence, the connecting homomorphism is especially simple: it is canonically identified with the map \( f_* \) induced by \( f \) on homology<sup>[2](https://www.math.ucla.edu/~sharifi/notes/homalg-ch02.html)</sup>. The resulting long exact sequence

\[ \cdots \to H_n(D) \to H_n(\Cone(f)) \to H_{n-1}(C) \xrightarrow{f_*} H_{n-1}(D) \to \cdots \]

is the standard tool for extracting homological information from a single chain map, and it is natural in \( f \): a commutative square of chain maps induces a chain map between the cones that respects the short exact sequences<sup>[9](https://web.math.princeton.edu/~petero/GridHomologyBook.pdf)</sup>.

## The mapping-cone criterion

The central computational fact is the mapping-cone criterion: a chain map \( f \colon C_{\bullet} \to D_{\bullet} \) is a quasi-isomorphism, meaning \( H_n(f) \) is an isomorphism for every \( n \), if and only if \( \Cone(f) \) is acyclic, meaning all its homology groups vanish<sup>[1](https://bastiaancnossen.com/notes/stable-homotopy-higher-algebra/section/homological-algebra/derived-categories/index.html)</sup>. The proof reads the long exact sequence of the cone: its connecting maps are \( f_* \), so exactness forces \( f_* \) to be an isomorphism in every degree precisely when \( H(\Cone(f)) = 0 \)<sup>[1](https://bastiaancnossen.com/notes/stable-homotopy-higher-algebra/section/homological-algebra/derived-categories/index.html)</sup>. The same statement appears across the literature in both homological and cohomological indexing<sup>[10](https://math.uchicago.edu/~amathew/chhomological.pdf)</sup><sup> • </sup><sup>[11](https://ssather.people.clemson.edu/DOCS/han.pdf)</sup>.

This criterion gives the cone its role as a detector: to check whether two complexes have the same homology via \( f \), one computes the homology of a single new complex. It also explains why the cone is the right object to invert in the derived category, where quasi-isomorphisms become isomorphisms.

## Triangles and the derived category

In the homotopy category \( K(\mathcal{A}) \) of complexes over an abelian category, the cone supplies the distinguished triangles. For a map \( a \colon A_{\bullet} \to B_{\bullet} \), the triple \( (A, B, \Cone(a), a, i, -p) \), with \( i \) the inclusion of \( B \) and \( p \) the projection to \( A[1] \), is a distinguished triangle, and the localization functor \( K(\mathcal{A}) \to D(\mathcal{A}) \) being exact carries these to distinguished triangles in the derived category<sup>[3](https://stacks.math.columbia.edu/download/derived.pdf)</sup>. Short exact sequences of complexes also give distinguished triangles in \( D(\mathcal{A}) \): given \( 0 \to A \to B \to C \to 0 \), one maps the cone \( \Cone(a) \) to \( C \) by a quasi-isomorphism and composes with its inverse to obtain the third map of the triangle<sup>[3](https://stacks.math.columbia.edu/download/derived.pdf)</sup>.

The triangulated-category framework was introduced by [Jean-Louis Verdier](https://www.edgepedia.org/jean-louis-verdier), a student of Alexandre Grothendieck, in his 1963 thesis, which axiomatized the properties of derived categories and of the cone construction; the thesis was published only in 1996 as Astérisque 239<sup>[4](https://ncatlab.org/nlab/show/triangulated%2Bcategory)</sup><sup> • </sup><sup>[12](https://smf.emath.fr/publications/des-categories-derivees-des-categories-abeliennes)</sup>.

**Functoriality.** In a triangulated category the cone is not functorial: there is no functor \( T^{[1]} \to T \) sending a map to its cone, only cones unique up to (non-canonical) isomorphism<sup>[4](https://ncatlab.org/nlab/show/triangulated%2Bcategory)</sup>. This is among the most cited inconveniences of triangulated categories, alongside the absence of homotopy limits and colimits<sup>[13](https://escholarship.org/content/qt8bh298gg/qt8bh298gg_noSplash_e56eabb5e48d979316a9482c21b8c7d1.pdf?t=rfzhoj)</sup>. In a stable \( \infty \)-category, which remembers the homotopical information that the homotopy category discards, the cone of a morphism is simply its cofiber, the pushout of the morphism with the unique map to the zero object, and cofibers form an actual functor \( \mathrm{cofib} \colon \mathcal{C}^{\Delta^1} \to \mathcal{C} \)<sup>[13](https://escholarship.org/content/qt8bh298gg/qt8bh298gg_noSplash_e56eabb5e48d979316a9482c21b8c7d1.pdf?t=rfzhoj)</sup><sup> • </sup><sup>[14](https://math.stackexchange.com/questions/4471234/why-use-infty-categories-over-model-categories?noredirect=1)</sup>. The homotopy category of a stable \( \infty \)-category is triangulated, and its distinguished triangles are exactly those arising from cofiber sequences<sup>[13](https://escholarship.org/content/qt8bh298gg/qt8bh298gg_noSplash_e56eabb5e48d979316a9482c21b8c7d1.pdf?t=rfzhoj)</sup>.

## Iterated mapping cones and resolutions

The cone is also a construction tool. If \( I \subset R \) is an ideal generated by \( f_1, \ldots, f_n \) and \( I_j = (f_1, \ldots, f_j) \), there are exact sequences

\[ 0 \to R/(I_{j-1} : f_j) \to R/I_{j-1} \to R/I_j \to 0. \]

Given free resolutions \( F \) of \( R/I_{j-1} \) and \( G \) of \( R/(I_{j-1} : f_j) \), the mapping cone of a complex homomorphism \( \psi \colon G \to F \) lifting \( R/(I_{j-1} : f_j) \to R/I_{j-1} \) is a free resolution of \( R/I_j \)<sup>[15](https://www.maths.tcd.ie/EMIS/journals/HHA/volumes/2002/n2a12/v4n2a12.pdf)</sup>. Iterating over the generators resolves \( R/I \). The result need not be minimal in general, but Jürgen Herzog and Takayama showed that when \( I \) has linear quotients, meaning the colon ideals \( I_{j-1} : f_j \) are generated by regular sequences (in the monomial case, by subsets of the variables), each step cones a [Koszul complex](https://www.edgepedia.org/koszul-complex) onto the previously built resolution and the procedure yields a minimal resolution<sup>[15](https://www.maths.tcd.ie/EMIS/journals/HHA/volumes/2002/n2a12/v4n2a12.pdf)</sup><sup> • </sup><sup>[5](https://www.sciencedirect.com/science/article/pii/S0097316514001149)</sup>. Stable, squarefree stable, and matroidal monomial ideals all belong to this class<sup>[15](https://www.maths.tcd.ie/EMIS/journals/HHA/volumes/2002/n2a12/v4n2a12.pdf)</sup>.

Well-known resolutions arise this way, including the Eliahou–Kervaire resolution of stable monomial ideals and the Taylor resolution<sup>[15](https://www.maths.tcd.ie/EMIS/journals/HHA/volumes/2002/n2a12/v4n2a12.pdf)</sup>. Subsequent work realized these resolutions cellularly: a regular CW-complex supports the Herzog–Takayama resolution when the ideal has a regular decomposition function, and varying the chain map recovers other known cellular resolutions, such as the box of complexes resolutions of Corso, Nagel, and Reiner<sup>[5](https://www.sciencedirect.com/science/article/pii/S0097316514001149)</sup>. Mapping cones of two-sided DG-modules over DG-algebras carry DG-algebra structures under suitable hypotheses, which is what allows the iterated construction to produce DG-algebra resolutions<sup>[15](https://www.maths.tcd.ie/EMIS/journals/HHA/volumes/2002/n2a12/v4n2a12.pdf)</sup>.

## Software implementations

The construction is implemented in computer algebra systems. Macaulay2 provides a `cone` command that produces the mapping cone of a map of chain complexes; in a worked example over \( \mathbb{Z}/101[x,y,z] \), coning the comparison map between two resolutions produces a complex whose \( H_0 \) vanishes and whose \( H_1 \) is isomorphic to \ \mathfrak{m}/\mathfrak{m}^2 \), as expected from the criterion above<sup>[16](https://web.stanford.edu/~mluciano/M2-help/1325.html)</sup>. Sage offers chain complexes, their morphisms, chain homotopies, and homology computation as part of its algebraic topology toolset<sup>[17](https://doc.sagemath.org/html/en/reference/homology/index.html)</sup>, and the UniMath library formalizes the cone and the vanishing of \( d \circ d \) in a proof assistant<sup>[6](https://unimath.github.io/doc/UniMath/d4de26f/UniMath.HomologicalAlgebra.MappingCone.html)</sup>.

## References

1. [Section 6.1: Derived ∞-categories, Stable homotopy theory and higher algebra (B. Cnossen)](https://bastiaancnossen.com/notes/stable-homotopy-higher-algebra/section/homological-algebra/derived-categories/index.html)
2. [Romyar Sharifi, Homological Algebra, Chapter 2: Abelian Categories](https://www.math.ucla.edu/~sharifi/notes/homalg-ch02.html)
3. [The Stacks Project, Chapter on Derived Categories](https://stacks.math.columbia.edu/download/derived.pdf)
4. [Triangulated category, nLab](https://ncatlab.org/nlab/show/triangulated%2Bcategory)
5. [Cellular resolutions from mapping cones, Journal of Combinatorial Theory, Series A 128 (2014), 180–206](https://www.sciencedirect.com/science/article/pii/S0097316514001149)
6. [UniMath.HomologicalAlgebra.MappingCone](https://unimath.github.io/doc/UniMath/d4de26f/UniMath.HomologicalAlgebra.MappingCone.html)
7. [Requisite Homological Algebra, University of California eScholarship](https://escholarship.org/content/qt5t67h1jg/qt5t67h1jg_noSplash_dbde4b3328a6b5b41e4de05a31186148.pdf?t=me7hzl)
8. [Mapping cone, notes of D. Zack Garza](https://notes.dzackgarza.com/Unsorted/mapping%20cone.html)
9. [Grid Homology Book, Appendix A: Homological Algebra, Princeton](https://web.math.princeton.edu/~petero/GridHomologyBook.pdf)
10. [CRing Project, Chapter 14: Homological Algebra](https://math.uchicago.edu/~amathew/chhomological.pdf)
11. [Homological Algebra Notes (S. Sather-Wagstaff)](https://ssather.people.clemson.edu/DOCS/han.pdf)
12. [Des catégories dérivées des catégories abéliennes, Jean-Louis Verdier, Astérisque 239, SMF](https://smf.emath.fr/publications/des-categories-derivees-des-categories-abeliennes)
13. [Understanding stability and stabilization of model categories, University of California eScholarship](https://escholarship.org/content/qt8bh298gg/qt8bh298gg_noSplash_e56eabb5e48d979316a9482c21b8c7d1.pdf?t=rfzhoj)
14. [Why use ∞-categories over model categories?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/4471234/why-use-infty-categories-over-model-categories?noredirect=1)
15. [Resolutions by mapping cones, Homology, Homotopy and Applications 4(2) (2002), 277–294](https://www.maths.tcd.ie/EMIS/journals/HHA/volumes/2002/n2a12/v4n2a12.pdf)
16. [cone — mapping cone of a chain map, Macaulay2 documentation](https://web.stanford.edu/~mluciano/M2-help/1325.html)
17. [Chain complexes and homology, Sage 10.8 Reference Manual](https://doc.sagemath.org/html/en/reference/homology/index.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Homological algebra and K-theory › Mapping cones, cylinders, and homotopy constructions*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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