# Spectral Sequences

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 .

Theorem 3   For every there exists a derivation of of degre such that the module is canonically isomorphic to .

Proof:The map restricted to gives

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:

Theorem 4   Let be the image of in . Then is a filtration of , and

Proof:We have a commutative diagram:

Hence a commutative diagram with exact rows:

.