The cohomological reindexing of a homological spectral sequence

The homological numerators at x on pages 1 and 2 are ℤx since dx lies in $F_1{C}_1=F_0{C}_1$; their denominators are zero. At y the numerators are ℤy and the boundary sources $F_0{C}_2,F_1{C}_2$ vanish. Thus $E^1=E^2$ has the two groups at (2,0),(0,1), $d^1=0$ by its absent target, and $d^2$ sends x to y. At r=3, the x numerator vanishes and the y denominator is $d(F_2{C}_2)=\mathrm{Z}y$.

Under [F1], the generators lie in $K^{-2}$ and $K^{-1}$; x is present in $F^p$ precisely for p≤-2 and y precisely for p≤0, so the cochain differential preserves the decreasing filtration. Negating (2,0) and (0,1) gives (-2,0) and (0,-1). The index change is (2,-1), total degree +1, and the map is still the identity on coefficients. Both compositions with the inverse coefficient map send the indicated generator to itself.

The identity complex has zero homology by [F2], hence zero cohomology after reindexing. Its decreasing filtrations are degreewise finite. The $E_3$ and stable terms are zero by step 1.1 and [F1], agreeing with the associated graded of the zero cohomology abutment.