R cycles and r boundaries of an increasingly filtered complex

For a filtered chain complex as in Filtered chain complex, put $n=p+q$ and, for $r\ge 0$, $A_{p,n}^r=F_p{C}_n\cap d_n^{-1}(F_{p-r}{C}_{n-1})\mathrm{.}$ Inverse images and intersections are the pullbacks of Spectral sequence subquotient and local lifting calculus. Since $d$ preserves $F$, $A_{p,n}^0=F_p{C}_n$. For $r\ge 1$ define the subobjects $Z_{p,q}^r=A_{p,n}^r,B_{p,q}^r=A_{p-1,n}^{r-1}+d_{n+1}(A_{p+r-1,n+1}^{r-1})\mathrm{.}$ The image in the second summand lies in $F_p{C}_n$ since $(p+r-1)-(r-1)=p$. Separately set $Z_{p,q}^0=F_p{C}_n$, $B_{p,q}^0=F_{p-1}{C}_n$. The symbols $B^r$ include lower-filtration approximate cycles, not just actual boundaries.

For comparison with projected conventions, let ${\bar{Z}}_{p,q}^r$ and ${\bar{B}}_{p,q}^r$ be the images of $A_{p,n}^r$ and $d(A_{p+r-1,n+1}^{r-1})$ in ${\mathrm{gr}}_p{C}_n$ for $r\ge 1$; set ${\bar{Z}}^0={\mathrm{gr}}_p{C}_n$ and ${\bar{B}}^0=0$.

Weibel, Chapter 5, Construction 5.4.6 and Lemma 5.4.7, pp.133–134; Sharifi, Theorem 4.2.3, pp.91–92. Increasing homological indices are used here.