Let *A* be any non-zero abelian group, let *C*_{0} be the group of (infinite) sequences of elements of *A*, and let *C*_{1} be the subgroup consisting of those sequences that are almost-everywhere zero. Then *C*_{•} is a bounded chain complex with non-zero homology, concentrated in degree 0; the homology group *H*_{0}(*C*) is an example of a filterpower (of *A*), a generalisation of ultrapowers.

Now, consider the (ascending, integer-indexed) filtration *F*_{•}*C*_{•} where *F*_{–m}*C*_{0} is the subgroup of sequences whose first *m* entries are zero and *F*_{–m}*C*_{1} = *F*_{–m}*C*_{0} ∩ *C*_{1}. This is an exhaustive, separated filtration, albeit unbounded below. It is not hard to see that the inclusions *F*_{-(m + 1)}*C*_{•} → *F*_{–m}*C*_{•} are all quasi-isomorphisms, so the associated graded chain complex must be acyclic.

In particular, the *E*^{2} page of the homology spectral sequence associated with *F*_{•}*C*_{•} is zero! But this should not be surprising; after all, the induced filtration on *H*_{∗}(*C*) is trivial. One can make various modifications to this example, but the essential point is that the filtration on *H*_{∗}(*C*) can fail to be separable even if *F*_{•}*C*_{•} is separable.