A pathological filtration on a bounded chain complex

Let A be any non-zero abelian group, let C0 be the group of (infinite) sequences of elements of A, and let C1 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 H0(C) is an example of a filterpower (of A), a generalisation of ultrapowers.

Now, consider the (ascending, integer-indexed) filtration FC where FmC0 is the subgroup of sequences whose first m entries are zero and FmC1 = FmC0C1. This is an exhaustive, separated filtration, albeit unbounded below. It is not hard to see that the inclusions F-(m + 1)CFmC are all quasi-isomorphisms, so the associated graded chain complex must be acyclic.

In particular, the E2 page of the homology spectral sequence associated with FC 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 FC is separable.


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