Is the filtration of double complex Hausdorff?

Question

Let $K^{\bullet,\bullet}$ be a double complex in a general abelian category. I'm wondering if the filtration $$F_I^p(\text{Tot}^n(K^{\bullet , \bullet })) = \bigoplus_{i + j = n, i \geq p} K^{i, j}$$ on the direct sum total complex is Hausdorff ($\bigcap_pF_I^p(\text{Tot}^n(K^{\bullet , \bullet })=0$). The filtration is obviously Hausdorff in $R$-Mod; but in a general abelian category, I cannot prove it nor find a counterexample.

Answer 1

I don't know about general truth. Obviously the question is only interesting for $K$ which is unbounded to the right i.e. not (up to a translation) a left half plane cohomological double complex.

If you use product totalisation, the filtration is always Hausdorff! That's an easy limit calculation; the key is that maps into $\prod$ are determined by their factors. Of course this is not true for $\oplus$ for general categories; the universal property for $\oplus$ faces the wrong direction.

However, there is a discussion here about when this is true for $\oplus$ in an Abelian category; one common situation is a complete Abelian category satisfying Grothendieck's axiom AB$5$, such as the category $\mathsf{RMod}$.

Thus my partial answer is: to seek a counterexample, you have to search among Abelian categories satisfying AB$3$ but not AB$5$. Much of the machinery of spectral sequences for unbounded complexes does require stricter axioms on your Abelian category anyway, so it is not so unreasonable to demand AB$5$ holds as well.

Answer 2

Taking your abelian category $\mathcal{A}$ to be the opposite category of the category of abelian groups (or of the module category of any nonzero ring) gives a counterexample.

Take $K^{i,j}=\mathbb{Z}$ if $i+j=n$ and $K^{i,j}=0$ otherwise (with all differentials necessarily zero).

The direct sum in $\mathcal{A}$ is the direct product in the category of abelian groups, and inclusion of direct sums in $\mathcal{A}$ is projection of direct products of abelian groups. So the chain of inclusions in $\mathcal{A}$ that arises in the question becomes the chain of projections $$\cdots\leftarrow\prod_{i\geq p+1}\mathbb{Z} \leftarrow\prod_{i\geq p}\mathbb{Z} \leftarrow\prod_{i\geq p-1}\mathbb{Z}\leftarrow\cdots$$ and the intersection in $\mathcal{A}$ becomes the direct limit of abelian groups.

But this direct limit is nonzero, as any element of $\prod_{i\geq p}\mathbb{Z}$ represents a nonzero element of the direct limit if it has infinitely many nonzero components.