Now Let be a filtered module and be -linear s.t. . We call a defferential filtered module if
. The theory of spectral sequences essentially is meant to
construct successive approximation to using the filtration. Use the
convension for .
for . Define
Note that if is a cycle (ie. ) then defines an element
and if is a boundary then its image in
zero for large .
For every there exists a derivation of of degre such that the
module is canonically isomorphic to .
Proof:The map restricted to
. Hence we have a map
. Put ,
On the other hand,
we are done.
Proof:We have a commutative diagram:
Hence a commutative diagram with exact rows: