## You are here

Homedelta functor

## Primary tabs

# delta functor

The concept of a $\delta$-functor is used to formalize the procedure of constructing long exact sequences from short exact sequences. Examples include derived functors and cohomology.

Definition. Let $\mathcal{A}$ and $\mathcal{B}$ be Abelian categories. A
*delta functor* ($\delta$-functor) from $\mathcal{A}$ to
$\mathcal{B}$ consists of a family of covariant additive
functors

$F^{n}\colon\mathcal{A}\to\mathcal{B}\quad(n=0,1,2,\ldots)$ |

and for each exact sequence

$0\to A\to B\to C\to 0$ |

of objects in $\mathcal{A}$ a family of homomorphisms

$\delta^{n}\colon F^{n}(C)\to F^{{n+1}}(A)\quad(n=0,1,2,\ldots)$ |

such that the following two conditions hold:

1. For any exact sequence $0\to A\to B\to C\to 0$ as above, there is a corresponding long exact sequence

2. For any morphism between exact sequences

$\xymatrix{0\ar[r]&A\ar[r]\ar[d]&B\ar[r]\ar[d]&C\ar[r]\ar[d]&0\\ 0\ar[r]&A^{{\prime}}\ar[r]&B^{{\prime}}\ar[r]&C^{{\prime}}\ar[r]&0\\ }$ and all integers $n\geq 0$ the diagram

$\xymatrix{F^{n}(C)\ar[r]^{{\delta^{n}}}\ar[d]&F^{{n+1}}(A)\ar[d]\\ F^{n}(C^{{\prime}})\ar[r]^{{\delta^{n}}}&F^{{n+1}}(A^{{\prime}})\\ }$ is commutative.

Keywords:

exact sequence, homological algebra

Related:

DerivedFunctor

Synonym:

$\delta$-functor

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

18G10*no label found*18G99

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections