The Stacks project

Definition 12.24.7. Let $\mathcal{A}$ be an abelian category. Let $(E_ r, d_ r)_{r \geq r_0}$ be a spectral sequence of bigraded objects of $\mathcal{A}$ with $d_ r$ of bidegree $(r, -r + 1)$. We say such a spectral sequence is

1. regular if for all $p, q \in \mathbf{Z}$ there is a $b = b(p, q)$ such that the maps $d_ r^{p, q} : E_ r^{p, q} \to E_ r^{p + r, q - r + 1}$ are zero for $r \geq b$,

2. coregular if for all $p, q \in \mathbf{Z}$ there is a $b = b(p, q)$ such that the maps $d_ r^{p - r, q + r - 1} : E_ r^{p - r, q + r - 1} \to E_ r^{p, q}$ are zero for $r \geq b$,

3. bounded if for all $n$ there are only a finite number of nonzero $E_{r_0}^{p, n - p}$,

4. bounded below if for all $n$ there is a $b = b(n)$ such that $E_{r_0}^{p, n - p} = 0$ for $p \geq b$.

5. bounded above if for all $n$ there is a $b = b(n)$ such that $E_{r_0}^{p, n - p} = 0$ for $p \leq b$.