Filtered chain complex

A filtered chain complex $(C,d,F)$ in an abelian category is a chain complex with increasing filtrations $F_p{C}_n\subseteq C_n$ for all $n$, whose pieces form subcomplexes. Explicitly $d_n$ factors as $F_p{C}_n\to F_p{C}_{n-1}$ on each piece. These restrictions square to zero since their composites with the monic inclusion into $C_{n-2}$ are $d_{n-1}{d}_n=0$. The conventions are Increasing and decreasing filtration of an object and Subcomplex. Thus, in modules, the condition is exactly $d_n(F_p{C}_n)\subseteq F_p{C}_{n-1}$; a factorization implies the inclusion, and the inclusion defines the restricted function. No finite-bound hypothesis is implicit.

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.