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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.


Comments (2)

Comment #2036 by Yu-Liang Huang on

Missing a subscript in (2).

There are also:

  • 1 comment(s) on Section 12.24: Spectral sequences: filtered complexes

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