Derived functors without resolutions
Satellites construct a connected sequence of functors from a single additive functor without using resolutions at all: embed an object in a short exact sequence, apply the functor, and take the kernel or cokernel that appears. Iterating gives the satellites in each degree. When the category has enough projectives the satellites agree with the classical derived functors; when it does not, they are what remains, and they are what makes relative homological algebra work in settings with no projectives at all.
Learning objectives
- Construct the first left and right satellites.
- Explain why the construction is independent of the chosen embedding.
- State when satellites agree with derived functors.
- Identify the settings where satellites are the only option.
Section 01The construction
- Choose a short exact sequence 0 → K → P → A → 0 with P relatively projective, or more generally with P in a chosen class.
- Apply the additive functor T, giving T(K) → T(P) → T(A) → 0 when T is right exact.
- Define S1T(A) as the kernel of T(K) → T(P). This is exactly what right exactness fails to control.
- Independence of the chosen sequence follows from a comparison argument using additivity of T.
- Iterating on K gives SnT for all n, and the family is a connected sequence with long exact sequences.
No resolution is built — only one short exact sequence at a time. That makes the construction available whenever a suitable class of objects exists, even if no resolution of finite or infinite length can be assembled.
Section 02Relation to derived functors
The two constructions agree, naturally. The satellite construction is essentially the dimension-shifting argument performed once at each degree.
The classical construction is unavailable, but satellites still produce a connected sequence with long exact sequences and the correct degree-0 term.
In the axiomatic language of the derived-functors stream, the satellites of T form the universal δ-functor extending T, when one exists. Grothendieck's effaceability criterion is the modern statement of what the satellite construction achieves by hand.
Satellites predate the general theory of derived functors and were how Cartan and Eilenberg organised the subject before the universal δ-functor formulation. They remain the natural tool in relative settings and in categories built from adjunctions rather than from modules.
Section 03Where satellites are the only option
Categories without projectives
Sheaves of modules, and many functor categories, have enough injectives but no projectives. Left derived functors are unavailable; left satellites relative to a chosen class are not.
Half-exact functors
A functor that is neither left nor right exact cannot be derived classically. Satellites still produce a connected sequence, though degree 0 need not recover the functor.
Relative theories
With a projective class rather than genuine projectives, the satellite construction is the natural way to define the relative derived functors directly.
For a functor that is neither left nor right exact, S0T need not be T. The satellites still form a connected sequence, but the identification with the original functor — automatic in the classical case — must be verified rather than assumed.
ReferenceFrequently asked questions
Are satellites unique?
Up to natural isomorphism, given the class of objects used in the construction. Changing the class changes the answer, which is exactly the point in relative homological algebra — the class is part of the data.
Why is the construction independent of the chosen short exact sequence?
Because any two such sequences can be compared by a map constructed from the lifting property, and additivity of the functor makes the induced comparison an isomorphism. The argument is the same one that underlies the comparison theorem for resolutions.
Do satellites give the long exact sequences?
Yes, for the allowable short exact sequences. That is what makes them a connected sequence rather than an unrelated family, and it is why they deserve to be called derived functors at all.
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