Exhaustive separated bounded and finite filtration

For an increasing filtration as in Increasing and decreasing filtration of an object, exhaustive means ${\bigvee }_{p\in \mathbb{Z}}{F}_{p}A=A$ and separated means ${\bigwedge }_{p\in \mathbb{Z}}{F}_{p}A=0$, when those joins and meets exist in the subobject order. In modules these are the usual union and intersection. A filtration is bounded below if $F_{a}A=0$ for some $a$, bounded above if $F_{b}A=A$ for some $b$, and finite if both hold; the witnesses may be enlarged to give $a<b$. Thus finite does not require distinct consecutive subobjects.

For a complex, degreewise bounded means $F_{a_n}{C}_n=0$ and $F_{b_n}{C}_n=C_n$ for each $n$, with bounds permitted to depend on $n$; uniformly bounded requires the same bounds for all degrees. For decreasing filtrations finite means $F^{a}A=A$ and $F^{b}A=0$ for $a<b$. For orientation only, completeness means that the canonical map is an isomorphism, when that inverse limit exists. No assertion about convergence of an unbounded spectral sequence is part of these definitions.

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. Sharifi 4.1.14 reverses the decreasing endpoints in its prose; the endpoints above agree with its Definition 4.2.1(b).