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 .

Let
. Put

Evidently for . Define

Note that if is a cycle (ie. ) then defines an element of and if is a boundary then its image in is zero for large .

and and . Hence we have a map

if then , ie. . Put , then and , ie. . Hence ).

On the other hand,
.

Now, . Hence,

we are done.

**Notations:**

Hence a commutative diagram with exact rows:

.

Suman Bandyopadhyay 2005-03-10