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?