When reading various literature on spectral sequences one always comes across two setups:


* A chain complex with an increasing filtration
* A cochain complex with a decreasing filtration


My question is why the other two options are never mentioned:


* A chain complex with an decreasing filtration
* A cochain complex with a increasing filtration


I worked through the construction of a spectral sequence associated to an increasing filtration on a cochain complex, and it seems to work just fine.


Is the reason for this convention in the literature just that the treated cases show up more frequently in practice or is there some deeper reason?



Why only consider decreasing filtrations on cochain complexes?
There is not a conceptual reason why increasing filtrations cannot happen on cochain complexes, or vice versa. A prominent example of this type of spectral sequence is the Eilenberg-Moore spectral sequence
$$
Tor^{H^\*(Z)}\_{\*\*}(H^\* Y, H^\* X) \Rightarrow H^\*(Y \times\_Z X)
$$
for the cohomology groups of a (homotopy) pullback. Another one of this type, though it doesn't really arise from filtering a *complex*, is the Adams spectral sequence
$$
Ext\_{A^\*}^{\*\*}(H^\* X, \Bbb Z/p) \Rightarrow \pi\_\*^s(X)^\wedge\_p
$$
computing the stable homotopy groups of $X$ using $Ext$ over the Steenrod algebra.


The most common spectral sequences can be derived from an expression of a space $X$ as the colimit of a sequence of subspaces, which naturally lends itself to an increasing chain filtration or a decreasing cochain filtration. On the other hand, many spectral sequences like the ones I just mentioned are second-quadrant or fourth-quadrant spectral sequences. They have attendant convergence issues that make them a little less easy to use (discussed at length in Boardman's paper that Mike Miller mentioned), and hence they are not as well known to nonspecialists.


