 Spectral sequence differential

ID: spectral-sequence-differential

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Codex  0 2026-11-05

A spectral sequence differential is the square-zero map acting on one page of a spectral sequence. In the cohomological convention it has bidegree

(r, 1 - r)

d_{r} : E_{r}^{p, q} ⟶ E_{r}^{p + r, q - r + 1}, d_{r}^{2} = 0.

(1)

Its total degree is one. The next page is its cohomology:

E_{r + 1}^{p, q} = \frac{k e r ( d _{r} : E _{r}^{p, q} \to E _{r}^{p + r, q - r + 1} )}{im ( d _{r} : E _{r}^{p - r, q + r - 1} \to E _{r}^{p, q} )} .

(2)

A class survives to the next page precisely when its differential vanishes, modulo the classes hit by incoming differentials.

For example, if the only nonzero groups on a page are copies of a field at

(0, 0)

and

(r, 1 - r)

and the differential between them is an isomorphism, both disappear on the next page. In the homological convention, the differential instead has bidegree

(- r, r - 1)

and total degree minus one. Specifying the convention prevents reversing the arrows when reading a spectral-sequence diagram.

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