# nLab satellite

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

If $T:X\to Y$ is an additive functor between abelian categories with sufficiently many projectives and injectives, then one defines its right satellite ${S}^{1}T$ and left satellite ${S}^{-1}T={S}_{1}T:C\to D$ via the formulas

${S}^{1}T\left(A\right)=\mathrm{ker}\left(T\left(M\right)\to T\left(P\right)\right)$S^1 T (A) = ker(T(M)\to T(P))
${S}_{1}T\left(A\right)=\mathrm{coker}\left(T\left(Q\right)\to T\left(N\right)\right)$S_1 T (A) = coker(T(Q)\to T(N))

where $0\to M\to P\to A\to 0$ and $0\to A\to Q\to N\to 0$ are short exact sequences where $P$ is projective and $Q$ is injective in $X$.

This definition does not depend on the choice of these exact sequences, and moreover ${S}^{1}$ and ${S}_{1}$ extend to functors.

Higher satellites are defined by ${S}^{n}T=\left({S}^{1}{\right)}^{n}T$ and ${S}^{-n}T={S}_{n}T=\left({S}_{1}{\right)}^{n}T$ for $n\ge 0$. For every exact sequence $0\to A\to A\prime \to A″\to 0$ there are natural connecting morphisms such that ${S}^{n}T$ (with $-\infty ) evaluated at $A,A\prime ,A″$ compose a long exact sequence.

## Properties

If $T$ is right exact then ${S}^{n}T=0$ for all $n>0$ and if $T$ is left exact then ${S}^{n}T=0$ for all $n<0$. If $T$ is covariant and $A$ projective then $\left({S}^{n}T\right)\left(A\right)=0$ for $n<0$, for contravariant $T$ or injective $A$ replace $n<0$ with $n>0$ in the conclusion.

## References

There is also an axiomatic definition of satellites and their relation to derived functors in the case when $T$ is half exact. See

There are generalizations to non-additive categories. See

• G. Z. Janelidze, On satellites in arbitrary categories, Bull. Georgian Acad. Sci. 82 (1976), no. 3, 529-532, in Russian, with a reprint translated in English at arXiv:0809.1504.

for early work; and also in a bit different setup recent

or for a shorter presentation, Ch. 3 or so of the survey

• A. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57 pdf

Revised on April 3, 2013 01:11:16 by Urs Schreiber (82.169.65.155)