# homological algebra – conditional convergence spectral sequences with outgoing and incoming differentials

I have to deal with unbounded filtrations and I want to use the conditional convergence of the spectral sequences and the results of

(1): J. Michael Boardman, Conditionally Convergent Spectral Sequences, March 1999 (http://hopf.math.purdue.edu/Boardman/ccspseq.pdf)

The article uses cohomological spectral sequences derived from the exact pair resulting from a cochain complex $$C$$ and decreasing filtration $$F$$ of $$C$$. The system of inclusions is $$A ^ s: = H (F_s C) leftarrow A ^ {s + 1}$$ and the pages are noted by $$E ^ s_r$$ for $$s in mathbb {Z}$$ and $$r in mathbb {N}$$ ($$r$$ is the page number and $$s$$ the "degree of filtration"). The symbol $$A ^ infty$$ denotes the limit and the symbol $$A ^ {- infty}$$ the colimit. The symbol $$RA ^ infty$$ denotes the derived module right of the limit. I work mainly on $$mathbb {R}$$.

Here are the two theorems (or their parts) of (1) that interest me:

Theorem 6.1 (p.19): Let $$C$$ to be a filtered cochaine complex. Assume that $$begin {equation} label {Eq: Exit} tag {C1} E ^ s = 0 quad text {for all} s> 0. end {equation}$$ Yes $$A ^ infty = 0$$, then the spectral sequence
converges strongly towards $$A ^ {- infty}$$.

Theorem 7.2 (p.21): Let $$f: C rightarrow bar {C}$$ to be a morphism of filtered cochain complexes and suppose that $$E ^ s$$, resp. $$bar {E} ^ s$$
conditionally converge to $$A ^ {- infty}$$, resp. $$bar {A} ^ {- infty}$$.
Suppose further that $$begin {equation} tag {C2} E ^ s = bar {E} ^ s = 0 quad text {for all} s <0. end {equation}$$ Yes $$f$$ induces the
isomorphisms $$E ^ infty simeq bar {E} ^ infty$$ and $$RE ^ infty simeq R bar {E} ^ infty$$, then he induces isomorphism $$H (C) simeq H ( bar {C})$$.

Let me introduce the standard bigrading (staggered degree) on $$E_r$$ and visualize $$E_r ^ {s, d}$$ as sitting at the coordinate $$(s, d)$$ by plane. The differentials are then
$$d_r: E_r ^ {s, d} rightarrow E_r ^ {s + r, d-r + 1}.$$
My questions are:

1. How does Theorem 6.1 generalize if (C1) is replaced by the following status of existing differentials?
$$E_r text {sit down in a half-plane and correct any coordinates} (s, d), text {then all but a lot}} d_r text {from} (s, d) text {leave the half plane}$$

2. How does Theorem 7.2 generalize if (C2) is replaced by the following differential input condition?
$$E_r text {sit down in a half-plane and correct any coordinates} (s, d), text {then all but a finite number} d_r text {ending in} (s, d) text {start outside the half-plane.}$$

The author of (1) answers the following questions:

1. On page 19, chapter 6 in parentheses just before Theorem 6.1:

…The
the results are generalized appropriately because all the arguments can be treated in degrees; the
The main difficulty is to find a notation that would help rather than hinder the exposure

2. At p.20, chapter 7 brackets a few paragraphs before Theorem 7.2:

… The results remain valid when they are modified appropriately, like all arguments
can be done in degrees; the difficulty is finding the notation
help rather than embarrassment.

How do these theorems become generalized precisely? Has it been done anywhere? Thank you!

P.S. I come from differential geometry and I do not know the proof methods for the spectral sequences at all. I just use it as a black box.

Posted on