Satellites

Executive summary

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

AlgorithmThe first left satellite S<sub>1</sub>Tin: an additive functor and a class of objects → out: its satellites

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 S_nT for all n, and the family is a connected sequence with long exact sequences.

The point of the construction

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

Enough projectivesSatellites = derived functors

The two constructions agree, naturally. The satellite construction is essentially the dimension-shifting argument performed once at each degree.

No projectivesOnly satellites

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.

Historical role

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

Setting

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.

Setting

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.

Setting

Relative theories

With a projective class rather than genuine projectives, the satellite construction is the natural way to define the relative derived functors directly.

The degree-0 term needs checking

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.

NavigateContinue in this stream

Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Satellites. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Satellites as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—satellites, functors, derived, satellite, section—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.