# Spectral sequences and short exact sequences

Suppose I take a short exact sequence of filtered chain complexes: $$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$ We assume that $$p$$ and $$q$$ are filtration-preserving, so that $$p(F_rA)\subseteq F_rB$$ and $$q(F_rB)\subseteq F_rC$$ for all $$r$$. We also assume that $$p$$ and $$q$$ are strict, so that $$p^{-1}(F_rB) = F_rA$$ and $$q(F_rB) = F_rC$$ for all $$r$$.

Question: Let $$r\geq 0$$. In what ways, if any, is the above situation reflected on the $$r$$-th pages of the spectral sequences associated to $$A$$, $$B$$ and $$C$$?

Our assumption that $$p$$ and $$q$$ are strict means that we have a short exact sequence of filtration quotients: $$0 \to F_iA/F_{i-1}A\to F_iB/F_{i-1}B\to F_iC/F_{i-1}C\to 0$$ But this precisely means that we have a short exact sequence of $$E^0$$-pages: $$0 \to E^0_{i,j}A\xrightarrow{p^0} E^0_{i,j}B\xrightarrow{q^0} E^0_{i,j}C\to 0$$ (I'm writing $$E^r_{i,j}A$$ for the terms of the $$r$$-th page of the spectral sequence associated to $$A$$, and similarly for $$B$$ and $$C$$. I'm writing $$p^r$$ and $$q^r$$ for the maps of $$r$$-th pages induced by $$p$$ and $$q$$. And I'm using the convention that $$E^0_{i,j} = (F_i/ F_{i-1})_{i+j}$$ so that $$d^0\colon E^0_{i,j}\to E^0_{i,j-1}$$ and more generally $$d^r\colon E^r_{i,j}\to E^r_{i-r,j+r-1}$$.)

Since $$p^0$$ and $$q^0$$ respect the differentials of the $$E^0$$ pages, the short exact sequences above turn into long exact sequences of homology groups, or in other words, long exact sequences of $$E^1$$ pages: $$\cdots \to E^1_{i,j+1}C\to E^1_{i,j}A\xrightarrow{p^1} E^1_{i,j}B\xrightarrow{q^1} E^1_{i,j}C\to E^1_{i,j-1}A\xrightarrow{p^1}\cdots$$ But what happens next?

Refined sub-question: For $$r\geq 2$$, is there any structure at all that relates the image of $$p^r$$ with the kernel of $$q^r$$?

I wonder if there is some kind of obstruction-theory flavoured result involving with primary, secondary, tertiary, ... connecting homomorphisms that all have a different degree shift.

If you instead work with a cofiber sequence of filtered spectra, then I gave a sufficient condition in Proposition 5.4 of https://www.mn.uio.no/math/personer/vit/rognes/papers/highfix.pdf (JPAA, 1999) to ensure that a short exact sequence of $$E_1$$-terms persists to all $$E_r$$-terms. I believe Section 5 can be read independently of the rest of that paper.
This is a pretty natural question that doesn't have an immediately straightforward answer. The long exact sequence you wrote for the $$E_1$$-pages is special, because there we only have one differential to cope with. "It's complicated" is not a great answer in general; here is an example to show some of what can happen.
Let $$B$$ be a filtered complex, fix an $$n$$, and define $$A = B_{\leq n} = F_n A.$$ and $$C = B_{> n} = A / F_n A$$ These have induced filtrations: $$F_k A = \begin{cases} F_k B &\text{if }k \leq n\\F_n B &\text{if }k > n\end{cases}$$ $$F_k C = \begin{cases} 0 &\text{if }k \leq n\\F_k B / F_n B &\text{if }k > n\end{cases}$$ On the level of $$E_0$$-pages, we get the $$E_0$$-page of $$A$$ by taking the $$E_0$$-page of $$B$$ and throwing away the parts above filtration $$n$$; we similarly get the $$E_0$$-page of $$C$$ by throwing away the parts in filtration $$n$$ and below. On the level of spectral sequences, these spectral sequences have "the same" differentials as the spectral sequence for $$B$$, except that differentials which cross the line between filtrations $$n$$ and $$n+1$$ no longer appear.
What about $$E_r$$-pages? The $$E_r$$-page for $$A$$ maps to the $$E_r$$-page for $$B$$; the image of this map is the entirety of filtrations $$n$$ and below, but the kernel consists of classes in the $$E_r$$-page for $$A$$ which were supposed to be hit by a differential coming from a filtration above $$n$$. Similarly, the $$E_r$$-page for $$B$$ maps to the $$E_r$$-page for $$C$$; this map is injective above filtration $$n$$, but the cokernel detects classes in the $$E_r$$-page of $$C$$ which were supposed to support a differential to filtration $$n$$ or below.
So in this example, we do have exactness here of $$E_r A \to E_r B \to E_r C$$, and the kernel of the first map is certainly related to the cokernel of the last map. However, information about the relationship between this kernel and cokernel is spread across various filtrations and can be difficult to state how it should be assembled.