= Spectral sequence differential
{title2=$d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$}
A spectral sequence differential is the square-zero map acting on one page of a <spectral sequence>. In the cohomological convention it has bidegree $(r,1-r)$:
$$
d_r:E_r^{p,q}\longrightarrow E_r^{p+r,q-r+1},\qquad d_r^2=0.
$$
Its total degree is one. The next page is its cohomology:
$$
E_{r+1}^{p,q}=\frac{\ker(d_r:E_r^{p,q}\to E_r^{p+r,q-r+1})}{\operatorname{im}(d_r:E_r^{p-r,q+r-1}\to E_r^{p,q})}.
$$
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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