Regular spectral sequence

For a homological spectral sequence as in Homological spectral sequence, two-sided regularity means that for every $(p,q)$ there is an integer $R\ge r_0$ such that for every $r\ge R$ both $d_{p,q}^r:E_{p,q}^r⟶E_{p-r,q+r-1}^r,d_{p+r,q-r+1}^r:E_{p+r,q-r+1}^r⟶E_{p,q}^r$ are zero. On this page the design term regular means this two-sided condition. The bound may depend on $(p,q)$; there need not be a single collapse page.

This differs from the convention on the prerequisite spectral-sequences page and in Stacks, Definition 12.24.7: there regular means eventual outgoing vanishing alone and coregular means eventual incoming vanishing. We call these outgoing regularity and incoming regularity when only one is intended. Neither may silently replace the two-sided hypothesis.

When both maps vanish the specified next-page isomorphism identifies $E_{p,q}^{r+1}$ with $E_{p,q}^r$ itself, since its kernel is the whole term and its incoming image is zero. Thus the condition gives canonical pointwise stationarity, as in Degree reasons force stabilization in a bounded region. For first-quadrant support, the outgoing target is zero for $r>p$, and the incoming source is zero for $r>q+1$. Outside that support every page term is zero. These bounds include the axes and the entirely zero sequence, without a choice of representatives or an assumption of AC.