Spectral sequence stabilization at a bidegree

A Homological spectral sequence stabilizes at $(p,q)$ if there is R such that for every r≥R both the incoming and outgoing differentials at that position are zero. Then their homology quotient is canonically $E_{p,q}^r$, and α supplies identifications with every subsequent term. Isomorphism of the underlying objects alone does not assert this vanishing.

For an abstract sequence, its terms can be represented as nested subquotients of the initial term at each position. Start with $Z_{r_0}=E_{p,q}^{r_0}$ and $B_{r_0}=0$. Given the quotient $Z_r\twoheadrightarrow Z_r\mathrm{/}{B}_r\cong E_{p,q}^r$, define $Z_{r+1}$ as the inverse image of the outgoing kernel and $B_{r+1}$ as the inverse image of the incoming image. Square zero puts $B_r\subseteq B_{r+1}\subseteq Z_{r+1}\subseteq Z_r$. The transition α identifies $Z_{r+1}\mathrm{/}{B}_{r+1}$ with $E_{p,q}^{r+1}$. When both differentials vanish, these inverse images are respectively $Z_r$ and $B_r$, so stabilization makes both families stationary. Their eventual quotient is denoted $E_{p,q}^{\mathrm{\infty }}$.

Weibel, Chapter 5, §5.2 pp.122–125 and Definitions 5.4.2–4 pp.132–133; Sharifi, §4.1 pp.87–89 and Definition 4.2.1 p.90.