A nonzero d two differential in a small filtered complex

The filtration is by subcomplexes because d sends filtration 2 into filtration 0. $E^0$ has ℤx at (2,0) and ℤy at (0,1). $d^0=0$ since the only nonzero chain differential drops filtration. For r=1 and r=2 the x numerator is ℤx (dx lies in $F_1{C}_1=F_0{C}_1=\mathrm{Z}y$) and its denominator is zero. The y denominator is zero since its potential sources $F_0{C}_2$ and $F_1{C}_2$ are zero. Thus $E^1=E^2$ have the same two terms.

The possible $d^1$ from x targets (1,0), a zero group; all $d^1$ vanish. At r=2, [F2] instead gives [x]↦[y], an isomorphism between the two nonzero terms, whose inverse sends [y] to [x]. In particular the generator has nonzero image. At r=3 the x numerator is zero because $F_{-1}{C}_1=0$, and the y denominator contains $d(F_2{C}_2)=\mathrm{Z}y$. All terms on $E^3$ and later are zero.

The unfiltered complex has kernel and cokernel of the identity equal to zero, so all $H_n(C)=0$. Its image filtration and every associated graded homology piece are zero, exactly matching $E^{\mathrm{\infty }}$ under [F3].