The Grothendieck Spectral Sequence

Executive summary

The derived functors of a composite, from the composites of derived functors

Given F: A → B and G: B → C left exact, with F carrying injectives to G-acyclic objects, there is a spectral sequence with E₂^p,q = R^pG ∘ R^qF converging to R^p+q(G ∘ F). Nearly every named spectral sequence in the subject is a special case.

Learning objectives

State the theorem and its acyclicity hypothesis.
Explain why the hypothesis is needed.
Identify the edge homomorphisms and the five-term sequence.
Recognise the standard specialisations.

Section 01The statement

E₂^p,q = (R^pG)(R^qF)(A) ⇒ R^p+q(G ∘ F)(A)

The hypothesis is not cosmetic

F must send injective objects of A to G-acyclic objects of B. Without it, applying G to an injective resolution of A gives a complex whose homology is not what is wanted, and the construction collapses. The hypothesis is usually verified via an adjunction: if F has an exact left adjoint, it preserves injectives.

AlgorithmConstructionin: two composable left exact functors → out: the spectral sequence

Take an injective resolution I^• of A in A.
Apply F to get a complex F(I^•) in B, whose terms are G-acyclic by hypothesis.
Take a Cartan–Eilenberg resolution of that complex and apply G, giving a double complex.
Filter the double complex in the two directions.
One filtration collapses because the terms are G-acyclic, computing R^*(G ∘ F); the other gives the E₂ page above.

Section 02Edge maps and the five-term sequence

The edges of the E₂ page give natural maps in low degrees, and truncating gives the five-term exact sequence

0 → R¹G(FA) → R¹(GF)(A) → G(R¹F(A)) → R²G(FA) → R²(GF)(A)

which is usable without any spectral sequence machinery and is the form in which the result is most often applied.

Section 03Specialisations

Named spectral sequences that are instances
Composite	Resulting spectral sequence
Invariants under N, then under G/N	Lyndon–Hochschild–Serre for group cohomology
Direct image, then global sections	Leray spectral sequence of a map of spaces
Restriction, then Hom over the smaller ring	Change-of-rings spectral sequence
Sheafification, then global sections	Čech-to-derived-functor spectral sequence
Invariants under a Lie subalgebra, then the quotient	Hochschild–Serre for Lie algebras
Tensor then Hom	One route to the balance and adjunction spectral sequences

Why so many theorems are one theorem

Recognising a construction as a composite of functors immediately supplies a spectral sequence, its E₂ page and its five-term sequence, with no separate work. This is the main practical reason the Grothendieck formulation is worth knowing rather than the individual cases.

ReferenceFrequently asked questions

How do I verify the acyclicity hypothesis?

Most often by adjointness: a functor with an exact left adjoint preserves injectives, and injectives are acyclic for any left exact functor. For sheaf cohomology, flasque and injective sheaves supply the acyclic class directly.

Is there a homological version?

Yes, for right exact functors and projective resolutions, with the arrows and indices reversed. The hypothesis becomes that the first functor carries projectives to acyclics for the second.

When does the sequence collapse?

When one of the two functors is exact, so that all but one row or column of the E₂ page vanishes. That case reduces to the composition isomorphism for derived functors and is how most change-of-rings isomorphisms are proved.

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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 The Grothendieck Spectral Sequence. 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 The Grothendieck Spectral Sequence as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—sequence, spectral, grothendieck, composite, functors—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.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying The Grothendieck Spectral Sequence?