Anticommuting differentials, one total differential
A double complex is a bigraded object with a horizontal and a vertical differential. The two are required to anticommute — or to commute, with a sign inserted when forming the total differential — so that the total differential squares to zero. Totalising by sum or by product gives a single complex, and the two obvious filtrations give two spectral sequences converging to the same thing. Almost every comparison theorem in the subject is this observation applied to a well-chosen double complex.
Learning objectives
- Define a double complex and the sign convention.
- Form the total complex by sum and by product.
- Explain when the two totalisations differ.
- Identify the two filtrations and what they compute.
Section 01The sign convention
Let Cp,q carry d′ (horizontal) and d″ (vertical). The total differential is
and DD = 0 requires the two differentials to anticommute. If the given differentials commute, the sign in the total differential produces the anticommutation; both conventions appear in the literature and describe the same structure.
Omitting the sign gives DD ≠ 0 and every subsequent computation is meaningless. The check DD = 0 is one line and should be performed whenever a double complex is constructed by hand.
Section 02Totalisation
(Tot C)n = ⊕p+q=n Cp,q. The right choice for first-quadrant complexes, where the sum is finite in each degree.
(Tot C)n = ∏p+q=n Cp,q. Needed for unbounded complexes; can differ from the sum, and convergence questions become delicate.
For a double complex concentrated in the first quadrant each total degree involves only finitely many terms, so the two agree and no issue arises. This is the case for the double complexes built from resolutions, which is why the distinction is often not mentioned.
Given complexes C and D, the double complex Cp ⊗ Dq totalises to the tensor product complex C ⊗ D, with differential ∂⊗1 + (−1)p1⊗∂. The Künneth formula computes its homology.
Section 03Two filtrations, two spectral sequences
- Stage 01Filter by columnsTake the first differential first. The E1 page is the homology in the vertical direction.
- Stage 02Filter by rowsTake the second differential first. The E1 page is the homology in the horizontal direction.
- Stage 03Both convergeTo the homology of the total complex, when convergence conditions hold.
- Stage 04CompareIf one spectral sequence collapses, it computes the total homology outright, and the other then yields information about the object of interest.
Balance of Ext, the Künneth formula, the Grothendieck spectral sequence and the Lyndon–Hochschild–Serre sequence are all instances: build a double complex whose two filtrations compute two different things, then compare. Recognising the pattern makes each of these look like one theorem rather than four.
ReferenceFrequently asked questions
Do I need spectral sequences to use double complexes?
Not always. When one filtration collapses immediately — because a row or column is exact — the comparison is a direct isomorphism. Balance of Ext is proved this way with no spectral sequence machinery.
When do sum and product totalisation differ?
Only when infinitely many terms contribute to some total degree, which requires the double complex to be unbounded in an anti-diagonal direction. First-quadrant and bounded complexes are unaffected.
Is the sign convention standardised?
No. Some texts put the sign on the vertical differential, some on the horizontal, and some define double complexes with anticommuting differentials from the start. Comparisons between sources should check which is in use before concluding formulas disagree.
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