Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




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.


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.


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 (