Abutment to a filtered object

An abutment to a filtered family $H_n$ consists of specified isomorphisms $E_{p,q}^{\mathrm{\infty }}\stackrel{\sim }{\to }{\mathrm{gr}}_p^F{H}_{p+q}=F_p{H}_{p+q}\mathrm{/}{F}_{p-1}{H}_{p+q}\mathrm{.}$ Use Associated graded object of a filtered object for the graded pieces, and Limiting cycles boundaries and e infinity or Spectral sequence stabilization at a bidegree for the stable terms. On this page every asserted convergence has a finite filtration in each total degree. In particular it is exhaustive and separated. The associated graded family is not identified with the unfiltered target. A morphism of abutments is a filtered map on the $H_n$ whose maps on these quotients agree with the stable-page maps under the specified identifications. For decreasing cohomological filtrations replace the quotient by $F^p{H}^{p+q}\mathrm{/}{F}^{p+1}{H}^{p+q}$.

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.