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.