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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:

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

Then, because of the exactness of $\varinjlim$, \displaystyle \lim_\longrightarrow$, we have $$\varinjlim_i underset{i}{\lim_\longrightarrow} E_2 (X_i) = E_2(\varinjlim_i E_2(\underset{i}{\lim_\longrightarrow} X_i) \ .$$ Then my question is: under which conditions can I assure that I have a comparison theorem like $$\varinjlim underset{i}{\lim_\longrightarrow} H^* (\mbox{Tot}^\prod X_i) = H^*(\mbox{Tot}^\prod \varinjlim_i underset{i}{\lim_\longrightarrow} X_i) \quad \mbox{?}$$ Any hints or references will be appreciated. 2 edited tags 1 # colimits of spectral sequences 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: $$E_2(\varinjlim_i X_i) \Longrightarrow H^*(\mbox{Tot}^\prod \varinjlim_i X_i )\ .$$ Then, because of the exactness of$\varinjlim\$, we have

$$\varinjlim_i E_2 (X_i) = E_2(\varinjlim_i X_i) \ .$$

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

$$\varinjlim H^* (\mbox{Tot}^\prod X_i) = H^*(\mbox{Tot}^\prod \varinjlim_i X_i) \quad \mbox{?}$$

Any hints or references will be appreciated.