nLab
satellite

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

If T:XYT \colon X\to Y is an additive functor between abelian categories with sufficiently many projectives and injectives, then one defines its right satellite S 1TS^1 T and left satellite S 1T=S 1T:CDS^{-1}T=S_1 T:C\to D via the formulas

S 1T(A)=ker(T(M)T(P)) S^1 T (A) = ker(T(M)\to T(P))
S 1T(A)=coker(T(Q)T(N)) S_1 T (A) = coker(T(Q)\to T(N))

where 0MPA00\to M\to P\to A\to 0 and 0AQN00\to A\to Q\to N\to 0 are short exact sequences where PP is projective and QQ is injective in XX.

This definition does not depend on the choice of these exact sequences, and moreover S 1S^1 and S 1S_1 extend to functors.

Higher satellites are defined by S nT=(S 1) nTS^n T = (S^1)^n T and S nT=S nT=(S 1) nTS^{-n} T = S_n T = (S_1)^n T for n0n\geq 0. For every exact sequence 0AAA00\to A\to A'\to A''\to 0 there are natural connecting morphisms such that S nTS^n T (with <n<-\infty\lt n\lt\infty) evaluated at A,A,AA,A',A'' compose a long exact sequence.

Properties

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

References

There is also an axiomatic definition of satellites and their relation to derived functors in the case when TT 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)