colimits of spectral sequences

Question

I'm looking for some references about colimits of spectral sequences.

More precisely: let $X : I \longrightarrow \cal{C}$ be a functor from a filtered category $I$ to the category of double cochain complexes of an abelian category $\cal{C}$, in which filtered colimits exist and commute with cohomology.

Let $E_2(X_i)$ be the second page of the first filtration ss associated to $X_i$. Assuming that the $X_i$ are right-half plane double complexes, it weakly converges to $H^*(\mbox{Tot}^\prod X_i)$ for all $i$ (Weibel, "An introduction to homological algebra", page 142):

$$ E_2(X_i) \Longrightarrow H^*(\mbox{Tot}^\prod X_i)\ , $$

where $\mbox{Tot}^\prod$ is the total product complex,

$$ (\mbox{Tot}^\prod X)^n = \prod_{p+q=n} X^{pq} \ . $$

For the same reason:

$$ E_2(\underset{i}{\lim_\longrightarrow} X_i) \Longrightarrow H^*(\mbox{Tot}^\prod \underset{i}{\lim_\longrightarrow} X_i )\ . $$

Then, because of the exactness of $\displaystyle \lim_\longrightarrow$, we have

$$ \underset{i}{\lim_\longrightarrow} E_2 (X_i) = E_2(\underset{i}{\lim_\longrightarrow} X_i) \ . $$

Then my question is: under which conditions can I assure that I have a comparison theorem like

$$ \underset{i}{\lim_\longrightarrow} H^* (\mbox{Tot}^\prod X_i) = H^*(\mbox{Tot}^\prod \underset{i}{\lim_\longrightarrow} X_i) \quad \mbox{?} $$

Any hints or references will be appreciated.

Answer 1

This is a nice question, so I'm not sure why it was never answered- maybe my answer is wrong and this question is harder than I thought? In any event here's my attempt:

Under fairly mild hypotheses on your spectral sequences (i.e. that it converges in the sense of Weibel 5.2.11), we have a comparison theorem which says that if a map of convergent spectral sequences is an isomorphism for any $r$, then it induces an isomorphism on the abuttments. In particular, in this case I think that you definitely have a map of spectral sequences $\text{colim } E(X_i) \rightarrow E(\text{colim } X_i)$, and since it's an isomorphism at the $E_2$ page then it's an isomorphism from then on, so the theorem applies (at least in the case when the double complexes in question are, say, right half-plane or something). The comparison theorem is in Weibel, 5.2.12.